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A New Robust Optimization Approach Induced by Flexible Uncertainty Set: Optimization under Continuous Uncertainty Yi Zhang, Yiping Feng, and Gang Rong Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b02989 • Publication Date (Web): 16 Dec 2016 Downloaded from http://pubs.acs.org on December 20, 2016

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Industrial & Engineering Chemistry Research

1

A New Robust Optimization Approach

2

Induced by Flexible Uncertainty Set:

3

Optimization under Continuous Uncertainty

4

Yi ZHANG, Yiping FENG, Gang RONG*

5

State Key Laboratory of Industrial Control Technology, Department of Control

6

Science and Engineering, Zhejiang University, Hangzhou, 310027, China

7 8

ABSTRACT

9

In the real-world optimization problems, continuous uncertainties (such as field

10

uncertainty, demand uncertainty) are usually unbounded and with unknown probability

11

density functions, which have significant influences on the fluctuation in production.

12

As the classical robust counterpart optimization approach is mainly for dealing with

13

bounded uncertainties, the concept of “flexible uncertainty set” is proposed based on

14

the definitions of the classical box, ellipsoidal and polyhedral uncertainty sets. Then the

15

continuous uncertainties can be transformed into bounded ones, which are connected

16

with a predefined “confidence level”. Moreover, their corresponding robust

17

formulations and a priori probability bounds of constraints’ violation are derived and

18

proved in detail. Several numerical examples (including a real-world process industry 1

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1

example) are introduced to illustrate the new formulations induced by flexible

2

uncertainty sets, which have been proved less conservative and with tight probability

3

bounds of constraints’ violation.

4

KEYWORDS: Robust optimization; Uncertainty set; Flexible; Continuous uncertainty

5

1. INTRODUCTION

6

In general, uncertainties in industrial optimization problems can be classified into

7

discrete and continuous uncertainties. Discrete uncertainty commonly refers to

8

emergency incidents, such as the breakdown of equipment and safety accidents. Once

9

the uncertainty appears in the production process, reactive scheduling techniques are

10

applied the most, which means the optimization model needs to be modified and

11

resolved. In process industry, uncertainties typically have continuous sources, i.e., the

12

data generating process has a continuous support (e.g., product demand, processing

13

yield). Different from the discrete random events, continuous uncertainties can be

14

considered in the optimization model before making decisions. According to the

15

sources, continuous uncertainties are divided into model-inherent, process-inherent and

16

external uncertainty (Li and Ierapetritou 1). However, continuous uncertain parameters

17

are always difficult to be described, and many of them cannot be measured directly. As

18

a result, fuzzy programming, stochastic programming (SP) and robust optimization are

19

proposed for dealing with problems subject to uncertainty, which can be called

20

preventive scheduling approaches. By the above approaches, uncertainties can be

21

described by expert experience, scenario-trees or bounded uncertainty sets.

22

In fact, if uncertainties can be measured from the real world, their probability density 2

23

functions (PDF) could be obtained through historical data. Sahin and Diwekar

24

proposed a new optimization approach, containing a kernel-based reweighting strategy, 2


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which used Gaussian kernel density estimation (KDE) to estimate the PDF for

2

uncertainties. Thus, when the PDF of uncertainty is considered, the constraints are

3

permitted to be violated at a certain level, where the chance-constrained approach is

4

applied (Charnes and Cooper 3) and improved (Cooper, et al. 4). Driven by historical

5

data of uncertainties, Jiang and Guan 5 proposed the concept of “data-driven confidence

6

set” and its corresponding chance-constrained formulations were derived. After

7

introducing KDE into data-driven chance-constrained optimization approaches, Calfa,

8

et al. 6 solved individual and joint chance constrained problems under right-hand side

9

(RHS) uncertainties. Then the data-driven method was further applied to problems

10

under matrix uncertainties by Zhang, et al. 7, where the chance constraints were

11

reformulated into robust constraints.

12

In many cases, uncertain parameters can be regarded as bounded uncertainties, and

13

robust optimization provides another view for the discussion of infeasibility of

14

constraints (Ben-Tal and Nemirovski 8; Beyer and Sendhoff 9; Bertsimas, et al. 10). In

15

1995, Mulvey, et al. 11 proposed the concept of robust optimization, where the model

16

robustness with conflicting objectives was discussed. In the original robust model, the

17

objective was divided into two parts: feasible part and another part for measuring

18

infeasibility, and this means the traditional approach allowed a certain degree of

19

constraints’ violation. Then Lin, et al. 12 regarded the uncertain parameter as a nominal

20

value with random fluctuation, which means uncertainties can be considered in two

21

parts: nominal part and uncertain part. Concentrating on the uncertain part, Bertsimas,

22

et al. 13 proposed the concept of uncertainty set, trying to interpret the possible ranges

23

of uncertain parameters. In this view, the traditional scenario-based approach only

24

focuses on a subset of the whole uncertainty set. 3



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1

To find solutions which are immune against infeasibility at a certain level, the size of

2

uncertainty set has a significant influence on the conservatism and robustness of

3

transformed robust models. For continuous uncertainties, uncertainty sets can be

4

defined in many ways. Some sets are defined according to the distance between

5

estimated result and exact probability density function, such as the data-driven

6

confidence set (Jiang and Guan 5); some are defined based on the norm of uncertainty,

7

such as the classical box, ellipsoidal and polyhedral uncertainty sets (14). Uncertainty

8

sets with tighter bounds can help us get less conservative solutions. The traditional

9

approach in robust optimization is to choose a proper robust parameter to define the

10

size of uncertainty set. Thus uncertainties are controlled within a bounded space,

11

covering part of the uncertain scenarios. Uncertain parameters in the same constraint

12

can be regarded as a random vector, and normally the parameters are limited to the same

13

uncertainty set with the same adjustable parameter, which makes the uncertainties

14

coupled inside the uncertainty set.

15

Assuming that continuous uncertainties’ probability density functions can be

16

obtained from historical data, we define flexible uncertainty sets based on the classical

17

norm-based sets for both bounded and unbounded uncertainties. Here, ‘flexible’ means

18

that each uncertainty can be considered at different weight and adjustable parameters

19

can be chosen independently. With tighter and changeable bounds on every dimension,

20

the flexible sets are no longer limited to the natural shape of classical uncertainty sets

21

from the geometric view, which will be illustrated in section 3 in detail. The flexible

22

sets induced formulations are derived for linear problems (LP) and mixed-integer linear

23

problems (MILP) in section 4. Based on the researches of probability bound on

24

constraints’ violation (Bertsimas and Sim 14, Janak, et al. 15, Guzman, et al. 16), a priori 4


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1

probability bound of constraints’ satisfaction for the new robust formulation has also

2

been analyzed and discussed in section 5. Finally, in section 6, the proposed robust

3

optimization approach is proved less conservative and with tight probability bounds on

4

the constraints’ violation through the classic State-Task-Network (STN) example and a

5

real-world process industry example.

6

2. PROBLEM DESCRIPTION

7

In this paper, we mainly deal with optimization problems under continuous

8

uncertainty, which can be classified into bounded and unbounded ones. A new robust

9

optimization approach induced by flexible uncertainty sets is proposed to extend the

10

classical sets induced robust formulations under bounded uncertainties. The boundary

11

of uncertainty sets is flexible and related to the uncertainties’ probability density

12

functions. At first, the robust counterpart optimization and uncertainty sets induced

13

robust formulations are briefly reviewed.

14

2.1 Robust Counterpart Optimization

15 16

For optimization problem under bounded uncertainties, Bertsimas, et al. 13 proposed the concept of robust counterpart for robust linear programming problem as

max 17

s.t.

z  cx  Ax  b

 A, b  U

(2.1)

x X 18

where x  R n1 , c  R n1 and X represents a computable bounded convex set in  n ;

19

 A  R m n is the matrix of uncertain coefficients and b  R m1 is the vector of uncertain

20

parameters on the right side, and they all belong to a known uncertainty set U.

21

If the robust model is feasible, the constraints need to be satisfied for any realization

22

of the uncertain parameter, which means all the scenarios in the uncertainty set U should 5



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1

be acceptable for the constraints in Eq.(2.1). In this paper, we consider the uncertain

2

parameters a ij in A and b i in b as bounded symmetric uncertainties in the

3

following forms:

a ij  aij  ij a ij

4

j  J i

b i  bi  i 0 b i

(2.2)

5

where aij and bi represent the nominal value of the uncertain parameter; a ij and

6

b i are positive, representing constant perturbation; ij and  i 0 are newly-defined

7

variables, controlling the range of uncertainties; J i represents the set of coefficients

8

in row i that are subject to uncertainty. Thus the equivalent formulation of the ith

9

constraint in Eq.(2.1) is

a x

10

ij

j

j

     max   ij a ij x j  i 0 b i   bi  U  jJi  

(2.3)

11

According to the position of uncertain parameters in the constraints, the robust

12

formulation can be discussed in several parts: only LHS uncertainty; only RHS

13

uncertainty; simultaneous LHS and RHS uncertainty and uncertainty in the objective.

14

For interpreting the concept of flexible uncertainty sets and discussing the solution of

15

the new robust formulations, we mainly consider LHS uncertainty in the numerical case,

16

whose formulation is:

17

a x ij

j

18

j

     max   ij a ij x j   bi  U  jJi  

2.2 Uncertainty Set Induced Robust Formulation

6


(2.4)

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1

As presented in the literature (Li, et al. 17), there are several classical definitions of

2

uncertainty sets for bounded uncertainties, especially for the uncertainties in robust

3

counterpart optimization, which are defined as shown in Table 1.

4

Table 1 Norm-based Uncertainty Sets. Type

Mathematical Form

Adjustable parameter

Box Ellipsoidal

Polyhedral



U   





U2   

 

     j   , j  J i

2

     



 U1    1     









  j2    jJ i

 



  j    jJ i 



5

When the above uncertainty sets are combined, several new sets are obtained with

6

better performance in robust optimization (Li, et al. 17). Here the robust formulations

7

induced by classical uncertainty sets are briefly reviewed as following:

8 9

1) The equivalent formulation of robust counterpart in Eq.(2.4) induced by box uncertainty set is:

 aij x j    a ij u j  bi  jJ i  j u j  x j  u j 

10

11 12 13 14

(2.5)

where  is the adjustable parameter and the suggested range is   1 ; u j is a positive intermediate variable. 2) The equivalent formulation of robust counterpart in Eq.(2.4) induced by ellipsoidal uncertainty set is:

7



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a x

1

ij

j



j

2

 a jJ i

2 ij

x 2j  bi

where  is the adjustable parameter and the suggested range is  

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(2.6)

Ji ; Ji is

3

the cardinality of the uncertainty set, which also represents the number of uncertain

4

parameters in the ith constraint.

5 6

3) The equivalent formulation of robust counterpart in Eq.(2.4) induced by polyhedral uncertainty set is:

 aij x j  pi  bi  j  pi  a ij u j   u  x  u j j j  

7

j  J i j  J i

(2.7)

8

where  is the adjustable parameter and the suggested range is   J i ; pi and u j

9

are positive intermediate variables.

10

The constraints in the classical sets induced robust formulations are all dependent on

11

adjustable parameters (  ,  ,  ), which means uncertainties in the same constraint are

12

controlled by the unique parameter. To some extent, these uncertainties are jointly

13

constrained by the uncertainty set. The robust counterpart optimization is mainly for

14

solving problems under bounded uncertainties, which can also be extended for

15

unbounded ones. Our research is mainly for improving the applicability of traditional

16

approaches and helping decision makers to select adjustable parameters for real-world

17

optimization problems better.

18

3. FLEXIBLE UNCERTAINTY SETS

8


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1

According to the uncertainty sets defined by the Euclidean norm, uncertain

2

parameters can be considered within a multi-dimensional space, where the dimension

3

is equal to the cardinality of uncertainty sets. The size of uncertainty set is decided by

4

the selected norm and its adjustable parameter.

5 6

For two uncertainties a 1  a1  1 a 1 , a 2  a2  2 a 2 , their norm-based uncertainty sets can be presented from the geometric view, which are shown in Figure 1. 2

2

0

-

7

(a)







-

2



1

-

0



1

-

0

- (b)

 1

- (c)

8

Figure 1. Geometric view of classical uncertainty sets: (a) Box uncertainty set; (b)

9

Ellipsoidal uncertainty set; (c) Polyhedral uncertainty set.

10

By choosing the adjustable parameters (  ,  ,  ), the above uncertainty sets define

11

the boundary of all the uncertain parameters at the same time. However, in real-world

12

problems, uncertainties prefer to be considered at different weights, which means some

13

uncertainties have greater influences on the objective. The constraints under these

14

uncertainties should be satisfied at first. In fact, the adjustable parameters cannot reflect

15

the exact percentage of uncertain scenarios considered in the reformulated robust model.

16

For example, when using the traditional robust formulation, we can easily come to a

17

conclusion that the more uncertainties are considered, the more conservative the

18

solution is. Sometimes the adjustable parameter is chosen bigger than one, more than

19

100% of uncertain scenarios will be considered, and this is meaningless. If the decision

20

maker decides to consider 80% of uncertain values of a 1 and a 2 , the adjustable 9



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1

parameter for the classical sets would not satisfy this requirement. Thus uncertainty sets

2

with flexible bounds for each uncertainty is needed in this case.

3

Normally, continuous uncertainties have their probability density functions (PDF),

4

but most of them are difficult to be measured or estimated. Here the kernel-based

5

estimation approach of continuous uncertainty is briefly reviewed at first. If these

6

uncertainties are regarded as bounded ones, then all the random values lying within the

7

bounds will be taken as uncertain scenarios for the robust model. Thus a method to

8

transform unbounded continuous uncertainties into bounded ones is also worth being

9

discussed.

10

3.1 Kernel-based Estimation of Continuous Uncertainty

11

Assuming that uncertain values of continuous uncertainty can be measured from the

12

real world, the dynamic changes of uncertainties can be stored by the real-time database.

13

Normally, the fluctuation in continuous parameters (such as field, demand) is always

14

regarded as ruleless, which seems to be meaningless for the production. However,

15

through analyzing a period of historical data, these uncertainties can be found with an

16

irregular probability density function.

17 18

19

In our research, the kernel density estimation (KDE) is introduced to estimate the unknown density function, which can be defined as Eq.(3.1). n  x  Xi  f  x   1 K  h  nh i 1  h 

(3.1)

20

where f h  x  represents the estimated density function of uncertain parameter x ;

21

h is a positive smoothing parameter, representing the bandwidth; n refers to the size

22

of sampled points X i ; K    is the kernel, which is a non-negative function that 10


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1

integrates to one and has mean zero. There are several classical functions can be used

2

as the kernel, such as Gaussian, Logistic. If Gaussian is chosen as the kernel, the

3

bandwidth h can be calculated by the formulas Eq.(3.2), proposed by Silverman 18. 1

1  4 5  5    1.06 n 5 h  3n   

4

5 6 7 8 9 10 11 12 13 14

(3.2)

Thus, the density function of continuous uncertainty can be estimated through the following approach, containing seven steps: STEP 1: Selection of the kernel function (Gaussian, Logistic, etc.) and determining the period T and frequency f of the sampling process; STEP 2: Initialization for parameters: t=0, t represents the number of sampled values; k=0, k represents the number of valid values. Then start sampling. STEP 3: Sampling and counting for the valid values. If the sampled point is outlier, t=t+1, otherwise t=t+1 and k=k+1. STEP 4: If t>Tf is satisfied, the sampling process will be terminated, otherwise return to STEP 3.

15

STEP 5: Calculation of the bandwidth h and the size of valid values n=k;

16

STEP 6: Kernel-based estimation of the sampled data. Here, the valid historical data

17

should be preprocessed and imported into R platform, then the ks package is called for

18

the estimation. Finally, the estimated density function for uncertainty x can be

19

obtained.

20

When applying the above approach in the real world production process, the

21

parameters are chosen according to the time scales of optimization. For example, the

22

sampling period T is usually consistent with the scheduling period; the frequency f

23

is related to the period of batch cycle in the production process. Thus the estimated 11



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1

density function can be better introduced to the optimization approach, and this will be

2

illustrated through a real-world example in section 6.

3

3.2 Continuous Uncertainty with Confidence Level

4

Based on the assumption that the probability density functions of continuous

5

uncertainties can be obtained, the percentage of uncertain values lying within a certain

6

range is easy to be calculated. But for unbounded uncertainties, there are no clear

7

bounds on the perturbation of uncertainty, which means there is still the possibility for

8

the existence of extreme values. However, these extreme scenarios for uncertainties are

9

trivial for some optimization problems. In this section, unbounded continuous

10

uncertainties can be transformed to a bounded uncertainty with a “confidence level”,

11

and some extreme uncertain values outside the boundaries can be ignored. In fact, a

12

bounded uncertainty can be regarded as an interval with uncertain bounds (Zhang, et al.

13

7

14

for the ith constraint is predefined. In fact, the confidence level can be chosen

15

independently for every uncertainty as 1   ij , and in our research, confidence levels

16

are uniformly described as 1   i . Thus the uncertainty a ij can be described as

). Here, the bounds can be chosen as quantile values when the “confidence level” 1   i

1 i a ij  aij1 i  ij a ij

17

18

1 i ij

where a



Faij1   i 2  Faij1 1   i 2 2

1 i ij

, a



(3.3)

Faij1 1   i 2   Faij1   i 2  2

, and the

19

uncertain range is decided by ij . When ij equals to the unit interval [-1, 1], the

20

1  i 1 i bounded form of continuous uncertainty a ij is  aij1 i  a ij , aij1 i  a ij  . Here, a  

21

ratio parameter cij  a ij

1 i

max a ij for any j  J i is defined to represent the ratio of

12


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1

max considered width and the exact width of the uncertain interval. a ij represents the

2

1 maximum range of uncertainty, which equals to a ij when the confidence level

3

max 1   =1 for bounded uncertainties. Here, for unbounded uncertainties, the a ij

4

1 equals to a ij when the confidence level is bigger than 99% (which means 1    1 ).

5

When the confidence level of a ij equals to 1, all the uncertain values are contained in

6

the bounded form and cij  1 . Similarly, as the confidence level falls to zero, more

7

uncertain values outside the uncertain interval are ignored, and cij  0 means a ij is

8

regarded as a constant. Thus the newly-defined adjustable parameter can be used for

9

determining the changeable range of every individual uncertainty, such as max a ij  aij1 i   ij a ij ， ij   cij , cij 

10

11 12

(3.4)

Thus each uncertainty can be controlled by the individual adjustable parameter cij , which can reflect the percentage of uncertain values lying in the uncertainty set.

13

For an unbounded continuous uncertainty   N  5, 0.6  , its transformed bounded

14

uncertainties can be found in Figure 2, where the bounds are decided by the confidence

15

level.

13



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1 2

Figure 2. Bounded uncertainties transformed from continuous uncertainty.

3

If we take the maximum range of the uncertainty  max  1.5 , then the corresponding

4

parameter c1 0.95  1.18 1.5 0  0.78 . Similarly, for the other bounded uncertainties,

5

when the confidence level is 0.90, 0.85, 0.80, their corresponding ratio parameter

6

c1 0.90  0.66 , c1 0.85  0.57 , c1 0.80  0.51 .

7

As the concept of “confidence level” is introduced to the norm-based uncertainty sets,

8

we can control the size of the sets and the percentage of random values lying in the sets.

9

If all the uncertainties can be transformed into uncertain intervals with independent

10

confidence levels, the bounds on each dimension of the uncertainty set become flexible

11

and can be determined when the confidence level is defined.

12

3.3 Flexible Uncertainty Set

13

Based on the definition of bounded uncertainty in Eq.(3.3), the focus on all the

14

continuous uncertainties in the optimization model can be turned to the vector of

15

uncertain variable ij . As a result, the flexible uncertainty sets are still defined based

16

on the Euclidean norms of vectors related to  . The boundaries of sets are related to

17

the chosen confidence level and estimated results of uncertainties. When we introduce 14


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1

cij to the classical uncertainty sets, the traditional adjustable parameter can be

2

influenced by every ratio parameter of uncertainty within the uncertainty set. However,

3

uncertainties are still coupled in these transformed sets (defined as flexible uncertainty

4

set I), which will be regarded as a transition form from classical sets to the real flexible

5

uncertainty set (defined as flexible uncertainty set II).

6

3.2.1 Flexible Box Uncertainty Set

7

Definition 3.1 (Flexible Box Uncertainty Set I). The first kind of flexible box

8

uncertainty set is described based on the original box uncertainty set as follows:



U I   

9



  ,   max cij   , j   J i j



(3.5)

10

where the adjustable parameter  is decided by the maximum of ratio parameter cij  ,

11

which means the size of the uncertainty set is controlled by independent confidence

12

levels of uncertainty. If any ratio parameter cij  is equal to one or the confidence level

13

of any uncertainty equals to one, the uncertainty set U I will become the interval

14

uncertainty set (classical box uncertainty set when =1 ).

15

Definition 3.2 (Flexible Box Uncertainty Set II). The second kind of flexible box

16

uncertainty set is described using the  -norm of newly-defined uncertain vector as

17

follows:

18

  U    c II 



1 i   ij a ij     1    1, j  J i    ij  cij , cij  max , j  J i    cij a ij   

(3.6)

19

Different from the flexible box uncertainty set I, ij in set II is controlled by the ratio

20

parameter cij independently, which is like that every random variable has its own

15



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Page 16 of 67

1

adjustable parameter. When all the uncertainties’ confidence level are equal to one,

2

cij  1 for any j  J i and the uncertainty set U II is the same as the interval

3

uncertainty set.

4

3.2.2 Flexible Ellipsoidal Uncertainty Set

5

Definition 3.3 (Flexible Ellipsoidal Uncertainty Set I). The first kind of flexible

6

ellipsoidal uncertainty set is described based on the original ellipsoidal uncertainty set

7

as follows:

 U 2I    

8

9

2

 ,  

 2 c  ij  jJi 

(3.7)

where the adjustable parameter  is decided by the 2-norm of the vector of ratio

10

parameter cij . Because 0  cij  1 ,  

11

J i . The confidence level of any uncertainty would influence the size of the uncertainty

12

set U 2I . When all the ratio parameters are equal to one, the flexible set covers all the

13

possible values of uncertainty.

J i , where Ji is the cardinality of the set

14

Definition 3.4 (Flexible Ellipsoidal Uncertainty Set II). The second kind of

15

flexible ellipsoidal uncertainty set is described using the 2-norm of the newly-defined

16

uncertain vector as follows:

17

  U 2II    c

 2

   J i      

 ij   jJ i  cij

1 i  a ij  J i , cij  max , j  J i  a ij  

2

   

(3.8)

18

For uncertainty set U 2II , the bounds of uncertainty on different dimensions are not the

19

same anymore. Every dimension has independent bounds, which are also decided by 16


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1

the confidence level of each uncertainty. If all the ratio parameters are equal to one, the

2

set U 2II becomes the same as the set U 2I (when cij  1, j  J i ) and U 2 (when the

3

adjustable parameter  

J i ).

4

3.2.3 Flexible Polyhedral Uncertainty Set

5

Definition 3.5 (Flexible Polyhedral Uncertainty Set I). The first kind of flexible

6

polyhedral uncertainty set is described based on the original polyhedral uncertainty set

7

as follows:

  U1I    1  ,    cij  jJ i  

8

(3.9)

9

where the adjustable parameter  is decided by the 1-norm of the vector of ratio

10

parameter cij . Because 0  cij  1 , the absolute value term in the 1-norm of vector c

11

has been eliminated. By controlling the confidence level of uncertainty, the set U1I can

12

reach its maximum size when   Ji

13

covered all the possible values of uncertainties.

( cij  1, j  J i ), which means the set has

14

Definition 3.6 (Flexible Polyhedral Uncertainty Set II). The second kind of flexible

15

polyhedral uncertainty set is described using the 1-norm of newly-defined uncertain

16

vector as follows:

17

    U    J i     c 1   II 1

 jJ i

1 i  a ij   J i , cij  max , j  J i  cij a ij 

ij

(3.10)

18

For the set U1II , the ratio parameter controls the bound of each uncertain parameter

19

independently. When the set reaches its maximum size, all the uncertain values are 17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

covered by U1II . Once the confidence level is chosen, the considered range of each

2

uncertainty can be determined. However, for the set U1I , although the ratio parameters

3

are chosen independently, all the uncertainties are still controlled by the same adjustable

4

parameter and many needed possible values may not be contained in the set.

5

From the geometric view, the covered area of the proposed uncertainty sets are

6

different from the classical ones. Here, the uncertainty sets of two uncertainties 1 and

7

 2 are shown in Figure 3, where  ,  ,  are adjustable parameters for classical

8

uncertainty sets. For flexible uncertainty sets, once the confidence level for each

9

uncertainty is determined, the ratio parameter c1 and c2 can be calculated and the

10

flexible bounds of flexible uncertainty sets are also shown in Figure 3.

11 12

Figure 3. A geometric view of classical and flexible uncertainty sets I & II: (a) Box

13

uncertainty set; (b) Ellipsoidal uncertainty set; (c) Polyhedral uncertainty set.

14

When the concept of “confidence level” is introduced to the uncertainty, flexible

15

uncertainty sets I are still defined based on the classical ones, which remain many

16

features of classical sets clearly, such as the regular shape with the same bounds on

17

each dimension. As shown in Figure 3 , for flexible uncertainty set II, every uncertainty

18

can be considered at a different weight. Moreover, for the same c1 and c2 , flexible

19

uncertainty sets II are smaller than sets I and the bounds are more flexible. When all the 18


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1

ratio parameters are equal to one, the flexible uncertainty sets covered all the uncertain

2

scenarios. Then the sets become the same as classical uncertainty sets when the

3

adjustable parameter  (  ,  ) reaches its upper bound of the suggested range.

4

3.4 Motivating Example

5

For better illustrating the proposed uncertainty sets above, the numerical example in 17

6

Li, et al.

’s work is introduced in this section as Eq.(3.11), which motivated us to

7

compare the performance of flexible sets and classical ones. max s.t.

8

z  8 x1  12 x2 a 11 x1  a 12 x2  140 a 21 x  a 22 x  72 1

(3.11)

2

x1 , x2  0

9

To be consistent with the parameters in the original case, we define the above

10

continuous uncertainties as a 11  N 10, 0.11 , a 12  N  20, 0.44  , a 21  N  6, 0.04  ,

11

a 22  N  8, 0.071 . Thus the uncertainties’ mean value is equal to the defined nominal

12

value, and its standard deviation equals to a third of the range of the bounded

13

uncertainty.

14 15

According to the definition of bounded uncertainty with confidence level in Eq.(3.4), we regard the uncertainties in the following forms:

16

max a ij =aij1-  ij a ij , i  1, 2; j  1, 2

17

where  ij   cij , cij  , which means the uncertain interval can be determined by

18

confidence level of the uncertainty. For uncertainties with symmetric PDF, the nominal

19



1

2

value aij1  E  a ij  and it is easy to find the maximum range of uncertainty max

A

  a 

max ij

max  a 11     max  a 21

max a 12   1 2  . max a 22  0.6 0.8 

3

When the confidence level for uncertain parameter a 11 and a 12 is chosen as 95%,

4

90%, 85%, 80%, the corresponding transformed intervals are shown in Figure 4 (a)(b).

5

However, with the same confidence level, the adjustable parameter c11 and c12 are

6

different, which has been presented from the geometric view in Figure 4 (c).

7 1.5 Confidence level = 0.95 Confidence level = 0.90 Confidence level = 0.85 Confidence level = 0.80

1.6 1.4 Density

1.2

(0,1.332) (0,0.942)

1

1 (-0.638,0.666)

0.8 0.6

(0.638,0.666)

0.5

0.4

(0.902,0)

0.2 9

9.5

10 a11

10.5

11

11.5

(1.276,0)

0

(0,0)

(-1.276,0)

 12

0 8.5

(a)

-0.5

1 Confidence level = 0.95

0.8

Density

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 67

Box (Confidence = 0.95) Box (Confidence = 0.90) Box (Confidence = 0.85) Box (Confidence = 0.80) Ellipsoidal (Confidence = 0.95) Ellipsoidal (Confidence = 0.90) Ellipsoidal (Confidence = 0.85) Ellipsoidal (Confidence = 0.80) Polyhedral (Confidence = 0.95) Polyhedral (Confidence = 0.90) Polyhedral (Confidence = 0.85) Polyhedral (Confidence = 0.80)

-1

Confidence level = 0.80

0.6

(0.638,-0.666)

(-0.638,-0.666)

Confidence level = 0.90 Confidence level = 0.85

(0,-1.332)

0.4

-1.5

0.2 0 17.5

8

18

18.5

19

19.5

20 a12 (b)

20.5

21

21.5

22

22.5

-2 -1.5

-1

-0.5

0

0.5

1

1.5

2

 11 (c)

9

Figure 4. Geometric interpretation for flexible uncertainty sets (box, ellipsoidal and

10

polyhedral) under different confidence levels (95%, 90%, 85%, 80%). (a. Bounded

11

uncertainties transformed from a 11 ; b. Bounded uncertainties transformed from a 12 ;

12

c. Illustration of different kinds of flexible uncertainty sets.)

13

For the box, ellipsoidal and polyhedral uncertainty sets, the box set is the tightest and

14

belongs to the ellipsoidal and polyhedral set. When the confidence level is determined,

15

the percentage of uncertain scenarios lying within the uncertainty set is known. For 20


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1

example, as shown by the red dotted lined in Figure 4 (c), 95% uncertain values of a 11

2

and a 12 are covered inside the lines, and the shape of the uncertainty set is irregular.

3

In fact, if the confidence level for each uncertainty is different, the bound on each

4

dimension of the uncertainty set will become more flexible.

5

4. ROBUST FORMULATION INDUCED BY FLEXIBLE UNCERTAINTY

6

SETS

7

According to the definition of flexible uncertainty sets, the bounds of uncertainties

8

become changeable and more independent than the classical uncertainty sets. When

9

these kinds of uncertainty sets are introduced into the robust counterpart optimization,

10

the derivation of new formulations and the discussion of solutions are of great

11

importance. Because the newly-defined flexible uncertainty sets are still based on the

12

Euclidean norm of an uncertain vector, which is related to the random variable ij , the

13

derivation procedure of the equivalent robust formulation is similar to the procedure

14

proposed by Li, et al.

15

problem at first, and the proofs of equivalent formulation induced by the second kind

16

of flexible uncertainty set are illustrated in detail. For the robust mixed-integer linear

17

problems, the equivalent constraints are presented in supporting information.

17

. Here, we consider LHS uncertainty for the robust linear

18

Based on the bounded form of continuous uncertainty in Eq.(3.3), we define

19

max 1 i a ij = a ij and regard aij as the nominal value aij , which represents the median

20

value of the uncertain interval. For uncertainty with symmetric probability density

21

functions, the median value is equal to its mean value; for uncertainty with asymmetric

22

1 PDF, aij i refers to the middle of the uncertain interval, which ensures that the

23

transformed interval is symmetrical. 21



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Page 22 of 67

1

4.1 Robust Formulation Induced by Flexible Box Uncertainty Set

2

Property 4.1. If the set U is the flexible box uncertainty set UI , then the corresponding

3

robust counterpart constraint Eq.(2.4) is equivalent to the following constraint:  max   1  aij i x j     a ij u j   bi  j  jJi   u j  x j  u j , j  J i  cij  , j  J i   max j  

4

(4.1)

5

Property 4.2. If the set U is the flexible box uncertainty set U II , then the corresponding

6

robust counterpart constraint Eq.(2.4) is equivalent to the following constraint:

a

1 i ij

7

j

8

Proof.

9

  U    c

For

II 



the

1 i x j   a ij x j  bi , i

(4.2)

jJi

flexible

box

uncertainty

set

1 i a ij   ij     1    1, j  J i    ij  cij , cij  max , j  J i  , we define a ij   cij   





10

   CL1L ;01L  , p   0 L1 ;1 and K    L1; t   R L 1 

11

cardinality of the uncertainty set (i.e., L  J i ). Then the inner maximization problem

12

in Eq.(2.4) can be rewritten as

 t , where L is the

  max   ij a ij x j : P  p  K     jJi 

13 14



y   wi ; i   R L 1 and using the dual cone of K  :

Defining dual variable





15

K *   L1; t   R L 1  1  t , the conic dual of the inner maximization problem can be

16

formulated as

22


Page 23 of 67

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1 2 3

4

5 6 7 8 9

10

11 12

13

14 15




min  i : wij  cij a ij x j , j  J i , wi 1   i w ,



Since the above problem is a minimization problem, it can be further rewritten as the following equivalent formulation by replacing i with wi 1 

w jJ i

ij

,

  min   wij : wij  cij a ij x j , j  J i  w  jJi  Realizing that a ij  0 and cij  0 , we can reformulate the conic dual of the inner maximization problem as follows:

  min   wij : wij  cij a ij x j , j  J i    cij a ij x j w  jJi  jJi Replacing the original inner maximization problem with the above conic dual, then the following constraint is obtained:  aij x j   cij a ij x j  bi , i  j jJ i    1 i cij  a ij , i, j  J i max  a ij  max 1 When we regard aij i as the nominal value aij and define a ij = a ij , then the

constraints can be further transformed to the following constraints:  a1 i x   1ij i u  b , i a   ij j j i  jJ i  j  u j  x j  u j 

where u j is introduced as a positive variable. 4.2 Robust Formulation Induced by Flexible Ellipsoidal Uncertainty Set

23


(4.3)


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Page 24 of 67

1

Property 4.3. If the set U is the flexible ellipsoidal uncertainty set U2I , then the

2

corresponding robust counterpart constraint Eq.(2.4) is equivalent to the following

3

constraint:  1  aij i x j    j    cij2   jJ i

4

 jJ i



max a ij x j

2

 bi

(4.4)

5

Property 4.4. If the set U is the flexible ellipsoidal uncertainty set U 2II , then the

6


7

constraint:

a

1 i ij

8

xj 

j

9

10

Proof.   U    c

For



II 2

2

the

jJ i

flexible

   J i      

 ij   jJ i  cij

 x

1 i J i   a ij

2

2 j

 bi , i

ellipsoidal

uncertainty

1 i  a ij  J i , cij  max , j  J i  a ij  

2

   



(4.5)

we

define



11

 2  CL1 L ;01 L  , p2   0 L1 ;

12

the cardinality of the uncertainty set (i.e., L  J i ). Then the inner maximization

13

problem in Eq.(2.4) can be rewritten as

14

15 16

J i  and K 2   L1 ; t   R L 1  

,

set

2

 t , where L is

  max   ij a ij x j : P2  p2  K 2    jJi  Defining dual variable y   zi ; i   R L 1 and using the dual cone K 2* =K 2 , the conic dual of the inner maximization problem can be formulated as

24


Page 25 of 67

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1 2 3

min z,



J i  i : zij  cij a ij x j ,  j  J i , zi

2

 i



Since the above problem is a minimization problem, it can be further rewritten as the following equivalent formulation by replacing i with zi

2



z jJi

2 ij

,

4

  2 min  J i   zij2 : zij  cij a ij x j , j  J i  = J i   cij2 a ij x 2j z jJ i jJ i  

5

max Normally, we define a ij = a ij , thus the original constraints can be transformed as

  x  b , i

1 i  aij1 x j  Ji   a ij

6

j

jJi

2

2 j

i

7

4.3 Robust Formulation Induced by Flexible Polyhedral Uncertainty Set

8

Property 4.5. If the set U is the flexible polyhedral uncertainty set U1I , then the

9


10

constraint:

 aij1 i x j  pi  bi  j   max  pi  a ij x j , j  J i     cij jJ i 

11

(4.6)

12

Property 4.6. If the set U is the flexible polyhedral uncertainty set U1II , then the

13


14

constraint:

15

 aij1 i x j  J i  pi  bi , i  j  1  pi  a ij i x j ，j  J i 

25


(4.7)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

Proof.

For

flexible

2

      J i    U      c 1 II 1

polyhedral

uncertainty

1 i  a ij   J i , cij  max , j  J i  cij a ij 

ij

 jJ i

Page 26 of 67

,

we



set define



3

1   CL1 L ; 01 L  , p1   0 L 1 ; J i  and K1   L1 ; t   R L 1  1  t , where L is the

4

cardinality of the uncertainty set (i.e., L  J i ). Then the inner maximization problem

5

in Eq.(2.4) can be rewritten as

  max   ij a ij x j : P1  p1  K1    jJi 

6

7 8

Defining dual variable y   zi ; i   R L 1 and using the dual cone K1* =K , the conic dual of the inner maximization problem can be formulated as

10 11



min J i  i : zij  cij a ij x j , j  J i , zi

9

z,



 i



Since the above problem is a minimization problem, it can be further rewritten as the following equivalent formulation by replacing i with zi







 max cij a ij x j , j J i

12

min J i  max zij : zij  cij a ij x j , j  J i = J i  max cij a ij x j

13

max Normally, we define a ij = a ij , thus the original constraints can be transformed as

14

z

jJ i

a

1 i ij

j

15 16

17

18

jJ i

1 i x j  J i  max a ij x j  bi , i jJ i

Since the above problem is a minimization problem, we can introduce an auxiliary variable to replace and obtain the following equivalent description:  aij1 i x j  J i  pi  bi , i  j  1  pi  a ij i x j ，j  J i 

The constraints can be further transformed into an equivalent formulation as follows: 26


Page 27 of 67

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


 aij1 i x j  J i  pi  bi , i  j   1 i  pi  a ij x j ，j  J i   u j  x j  u j，j  J i 

1

(4.8)

2

Uncertainties in the above formulations induced by flexible uncertainty sets II have

3

been decoupled from each other, and the formulations induced by the box and

4

polyhedral uncertainty sets remain linear. Based on the above derivations, the robust

5

formulations for mixed-integer linear problems are concluded in supporting

6

information.

7

Motivating Example （ Continued ） . According to the definition of flexible

8

uncertainty sets and the above sets-induced formulations, the robust formulations of the

9

original constraints can be obtained as follows:

10

Table 2 Robust formulations of constraints under uncertainty for different kinds of

11

uncertainty sets. Robust constraints Type Classical Uncertainty Set 10 x1  20 x2    u1  2u2   140 6 x1  8 x2    0.6u1  0.8u2   72

Box

u1  x1  u1

x1 , x2  0

6 x1  8 x2  p2  72 p1  u1 , p1  2u2 p2  0.6u1 , p2  0.8u2

u1 , u2  0

10 x1  20 x2   x12  4 x22  140 6 x1  8 x2   0.36 x12  0.64 x22  72 x1 , x2  0 2 2   c112  c122  c21  c22

10 x1  20 x2  p1  140

u2  x2  u2

u1  x1  u1

  max c11 , c12 , c21 , c22 

6 x1  8 x2   0.36 x12  0.64 x22  72

u1  x1  u1

6 x1  8 x2    0.6u1  0.8u2   72

u1 , u2  0

10 x1  20 x2   x12  4 x22  140

Polyhedral

10 x1  20 x2    u1  2u2   140

u2  x2  u2

u2  x2  u2 u1 , u2  0

Ellipsoidal

Flexible Uncertainty Set I

10 x1  20 x2   p1  140 6 x1  8 x2   p2  72 p1  u1 , p1  2u2 p2  0.6u1 , p2  0.8u 2 u1  x1  u1  u 2  x2  u 2 u1 , u 2  0   c11  c12  c21  c22

27


Flexible Uncertainty Set II 10 x1  20 x2  c11u1  2c12u2  140 6 x1  8 x2  0.6c21u1  0.8c22u2  72 u1  x1  u1 u2  x2  u2 u1 , u2  0

10 x1  20 x2  4  c112 x12  4c122 x22   140 2 2 2 2 6 x1  8 x2  4  0.36c21 x1  0.64c22 x2   72

x1 , x2  0

10 x1  20 x2  4 p1  140 6 x1  8 x2  4 p2  72 p1  c11u1 , p1  2c12u2 p2  0.6c21u1 , p2  0.8c22u2 u1  x1  u1 u2  x2  u2 u1 , u2  0


1

For the formulations induced by flexible uncertainty sets, the uncertain parts of

2

constraints are controlled by the newly-defined parameters cij , i, j  J i , which are

3

decided by the confidence level of each continuous uncertainty. Here all the

4

uncertainties are considered with the same confidence level, and when the confidence

5

level changes, the objective values of different formulations are shown in Figure 5. 100 98 96 94 92

Objective

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 67

90 88 86

Classical Box Uncertainty Set (Ψ) Flexible Box Uncertainty Set Classical Ellipsoidal Uncertainty Set (Φ=Ψ*sqrt(L)) Flexible Ellipsoidal Uncertainty Set Classical Polyhedral Uncertainty Set (Γ=Ψ*L) Flexible Polyhedral Uncertainty Set

84 82 80 0

6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Adjustable Parameter (Ψ for Classical Ones; Confidence Level for Flexible Ones)

7

Figure 5. Objective values of different robust formulations.

8

If we regard the confidence level of continuous uncertainty as the adjustable

9

parameter for the robust formulation, the objective values for classical formulations are

10

more sensitive to the adjustable parameters  ,  ,  . In Figure 5, we find that when the

11

confidence level gets bigger, the objective values for flexible formulations get smaller

12

slowly. This means for the same “adjustable parameter”, the solutions of formulations

13

induced by classical uncertainty sets are more conservative than the flexible ones.

14

In this example, the probability density functions for all the uncertainties are Gaussian

15

functions (symmetric), and the corresponding parameters cij are always similar. As a

16

result, for flexible uncertainty sets I and II, the objective values are similar when the

17

confidence level changes, which has been merged into one kind. For the formulations 28


Page 29 of 67

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

induced by flexible uncertainty sets, it is obvious that uncertainties are with tightest

2

bounds when they are limited in the flexible box uncertainty set. The solution induced

3

by the flexible polyhedral uncertainty set is the most conservative among all the

4

solutions.

5

5. PROBABILISTIC BOUNDS FOR ROBUST FORMULATION

6

Based on the flexible uncertainty sets induced robust formulations, the conservatism

7

of the formulated model needs to be considered more carefully. When we compare the

8

solutions of different formulations, the tendency of changing conservatism can be found

9

through the falling or rising trend of the objective values. However, in some cases, when

10

the confidence level of uncertainty changes, the transformed formulations may become

11

infeasible, which means some constraints cannot be satisfied for some realizations of

12

uncertainties. Moreover, if the transformed model can satisfy all the scenarios of

13

uncertainties, the decision maker might allow for a certain degree of constraints’

14

violation and a proper adjustable parameter needs to be determined. Thus it is desirable

15

to analyze the influence of confidence levels on the probabilistic bounds for the robust

16

formulations induced by flexible uncertainty sets.

17

Here, for guiding decision makers to select proper confidence levels, we mainly focus

18

on calculating the probability of violation before solving the formulated model.

19

According to a priori probability bound proposed by Li, et al.

20

bounds can be derived as follows:

19

, the probabilistic

21

Lemma 5.1. When uncertainties are limited in the flexible uncertainty sets, if the

22

robust counterpart constraint is satisfied, the upper probability bound for the violation

23

of the original constraint can be obtained as

29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 67

    Pr  aij x j   ij a ij x j  bi   Pr   ij ij   jJ i  j   jJi 

1

2

where the parameters  and  ij are defined as follows:

3

(1) For the counterpart induced by flexible box uncertainty set II U II

  1,  cij1   ij  cij1 , j  J i ,

4

5

 

ij ij

1

(5.2)

J i ,  cij1   ij  cij1 , j  J i ,

c  jJ i

2 2 ij ij

1

(5.3)

(3) For the counterpart induced by flexible polyhedral uncertainty set II U1II   J i ,  cij1   ij  cij1 , j  J i

8 9

jJ i

(2) For the counterpart induced by flexible ellipsoidal uncertainty set II U 2II

6

7

c 

(5.1)

(5.4)

Proof. The proof can be seen in supporting information.

10

Different from the conclusion for classical uncertainty sets, the parameter  here

11

is decided by the type of flexible uncertainty sets and  ij is related to the confidence

12

level of each continuous uncertainty. Meanwhile, additional constraints for the two

13

parameters have been derived as shown in Eq.(5.2)-Eq.(5.4).

14

Lemma 5.2. If

  ij

jJi

are independent and subject to a symmetric probability

15

distribution, then the following probability of constraint violation holds for any   0

16

     2 k    ij ij         Pr  aij x j   ij a ij x j  bi   e     dFij     2k  !  jJ i k 0  jJ i   j  

30


(5.5)

Page 31 of 67

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2 3 4


where  and  ij are detailed in Lemma 5.1. Based on Lemma 5.1, the derivation of Lemma 5.2 is just the same as Li, et al. 19 has proved, and the only difference is the definition of the parameters  and  ij . However, for the independent uncertain variables

  ij

jJi

in flexible uncertainty

5

sets II, their symmetric bounds are decided by the confidence level of each uncertainty

6

a ij , i, j  Ji . Thus the probability bounds for the newly-transformed formulations in

7

section 4 are shown as Lemma 5.3.

8

Lemma 5.3. For independent and bounded uncertainties ij  jJ , if each of them is i

9

subject to a symmetric probability distributions supported on  1,1 for  i , j  J i ,

10

then the following bound on the probability of constraint violation holds for any   0

11

12

  2 ij2      Pr  aij x j   ij a ij x j  bi   exp  min        0  2 jJ i j J   j  i    

(5.6)

where  and  ij are detailed in Lemma 5.1.

13

Proof. See supporting information for the proof.

14

As shown in Lemma 5.3, once the confidence levels of uncertainties are determined,

15

the upper bound on the probability of constraints’ violation can be calculated from the

16

above analysis, which is illustrated in Theorem 5.1.

17

Theorem 5.1. For independent uncertain parameters ij 

18

bounded and symmetric probability distribution supported on

19

flexible box, ellipsoidal, polyhedral uncertainty sets induced robust counterparts,

31


jJi

, if they are subject to

 1,1 ,

then for the


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

    2    Pr  aij x j   ij aij x j  bi   exp    2J 2 jJ i   j i 

1

Page 32 of 67

 2 c  ij   jJ i 

(5.7)

2

where  represents a special constant for different uncertainty sets induced

3

formulations in Lemma 5.1.

4

Proof for Theorem 5.1.

5

On the basis of Lemma 5.1, for the flexible box, ellipsoidal and polyhedral

6

1 1 uncertainty sets induced robust counterparts, we have  cij  ij  cij , j  Ji , so that

7

 jJ i

8

1  J i and the following relationship holds: c

2 2 ij ij

2  2 ij2  2        ij  2   min     min          0 2    0  jJi  2   ij2 J i    jJ i   





 ij2 J i



9 10

11 12

    2 Ji

2 2

1

 jJ i

2 ij

    2 Ji

2 2

c jJ i

2 ij

Using the conclusion of Lemma 5.3, we have

    2     2 ij2     Pr  aij x j   ij aij x j  bi   exp  min       exp   2   0  2  jJ i j J  j  i      2 Ji

c jJi

2 ij

   

For the probability bounds in Eq.(5.7), the influence of uncertainties’ confidence level on the probability bounds can be easily concluded, which is:

13

(1) When the confidence levels are close to 1, nearly all the random scenarios need

14

to be considered and satisfied in the transformed model. If all the confidence

15

levels are equal to 1, the probability that the original constraint is violated reaches

16

    2  . its smallest upper bound, which is exp    2 Ji    32


Page 33 of 67

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

(2) When the confidence levels are close to 0, nearly no uncertainty is considered in

2

the model, and all the coefficients are close to their nominal values. If all the

3

confidence levels are equal to 0, the transformed model is the same as the

4

deterministic model. If the transformed model is feasible, the original model is

5

impossible to be infeasible.

6

In a word, the greater the confidence level is, the more uncertain values are considered

7

in the model, the original constraint is less likely to be violated, and the solution of the

8

transformed model is more conservative. Through analyzing the probability bounds and

9

the solution of transformed model, the above information can guide decision makers to

10

choose the proper type of flexible uncertainty sets and corresponding confidence levels

11

for different cases.

12

Motivating Example (Continued). According to the probability bounds in Eq.(5.7),

13

the probability bound of constraint violation for formulations in Table 2 are shown in

14

Table 3, and the calculated results are presented in Figure 6.

15

Table 3 Probability bounds of constraints’ violation for different kinds of

16

uncertainty sets. Probability (upper) bounds of constraints’ violation Type Classical Uncertainty Set

Flexible Uncertainty Set I

Box

 1  exp     2   8 

2  1  exp    max cij   i j ,  8 

 1  exp    cij2   32 jJi 

Ellipsoidal

 1  exp     2   8 

 1  exp    cij2   8 jJi 

 1  exp    cij2   8 jJi 

Polyhedral

 1  exp     2   8 

2  1   exp     cij    8  jJi    

 1  exp    cij2   2 jJi 





33


Flexible Uncertainty Set II


1

When all the uncertainties are with the same confidence level and their ratio

2

parameters cij are similar, their probability bounds are the same. As a result, flexible

3

uncertainty sets I and II are merged into one kind in this case. 1 0.9 0.8 0.7 Probability Bound

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 67

0.6 0.5 0.4 0.3 0.2 0.1 0 0

Classical Box Uncertainty Set (Ψ) Classical Ellipsoidal Uncertainty Set (Φ=Ψ*sqrt(L)) Classical Polyhedral Uncertainty Set (Γ=Ψ*L) Flexible Box Uncertainty Set Flexible Ellipsoidal Uncertainty Set Flexible Polyhedral Uncertainty Set

0.1

4 5

0.2

0.3 0.4 0.5 0.6 0.7 Adjustable Parameter (Ψ for Classical Ones; Confidence Level for Flexible Ones)

0.8

0.9

1

Figure 6. Probability bounds of constraints’ violation for different formulations.

6

In Figure 6, the probability bounds for classical formulations are tighter than flexible

7

ones. This is because when the adjustable parameter  equals to like 0.9, there are no

8

more than 90% uncertain values lying in the classical uncertainty sets, and maybe only

9

53% uncertain scenarios are concerned. As a result, the ratio parameter is smaller than

10

the adjustable parameter sometimes. For some uncertainties with other kinds of

11

probability density functions, whose random values mainly lie on the boundaries, the

12

ratio parameter may be larger than the adjustable parameter. Then the probability bound

13

of flexible formulations will be tighter conversely. In the flexible sets induced

14

formulations, we focus more on the exact degree of considering uncertainties rather

15

than the range only.

16

6. COMPUTATIONAL STUDIES

34


Page 35 of 67

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1

In this section, two examples are introduced for illustrating the proposed the

2

formulations and their probability bounds of constraints satisfaction. Compared with

3

classical uncertainty sets, the conservatism and the robustness of new formulations are

4

discussed in detail. Several rules for selecting uncertainty sets and choosing adjustable

5

parameters are also concluded.

6

6.1 Robust MILP Example

7

The proposed robust linear formulations can be extended to mixed-integer linear

8

problems (MILP), which are illustrated in supporting information. The traditional

9

State-Task Network (STN) case (Kondili, et al. 20) is chosen as the motivating example,

10

which has been widely used for analyzing the performance of different optimization

11

approaches (Shah, et al. 21, Ierapetritou and Floudas 22). Here, the discrete formulation

12

is cited in this section, and its state-task network structure is shown in Figure 7. Product 1 S8 40% 2h Heating

S1

(Task 1)

Feed A

1h

S4

40%

Int AB 60% 2h

Reaction 2 (Task 3)

Hot A Int BC

60%

Impure E

S6

S7 80%

2h S2

50%

Reaction 1

15 16 17

Feed C

50%

13

Separation (Task 5)

90% 1h

S9

Product 2

1h Reaction 3

(Task 2)

Feed B

14

10% 2h

S5

S3

(Task 4)

20%

Figure 7. Flowchart of the state-task network (STN) example. Based on the original formulation proposed by Kondili, et al.

20

, continuous

uncertainties are introduced to the utility constraints as

   i

jKi



Wi , j ,t   u ,i , j Bi , j ,t  PCu ,t

u ,i , j

35


u, t

(6.1)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

where Wi , j ,t is a binary variable designated the assignment of processing task i of

2

unit j at time period t ; Bi , j ,t represents the amount of material which starts

3

undergoing task i in unit j at the beginning of time period t ; PCu ,t represents the

4

 amount of utility u purchased at time period t ;  u ,i , j and  u ,i , j represent the

5

uncertain coefficients related to the invariant and variant part, which refers to the energy

6

consumption rate of equipment; Ki is the set of units which are capable of performing

7

task i .

8

Assuming that the consumed utility u in period t is in a linear relationship with

9

the total consumption of materials during period t , which can be regarded as two parts:

10

invariant part (related to the integer variable Wi , j ,t ) and variant part (related to the

11

positive continuous variable Bi , j ,t ). For the discussion of formulations induced by

12

different kinds of uncertainty sets, we assume that all the continuous uncertainties’

13

probability distributions are known as shown in Table 4. Here the consumption rate of

14

energy for the same unit is subject to the same probability distribution function.

15

Table 4 Specified distributions of uncertain parameters in utilities constraints.

Unit

 u,i, j

 u ,i , j

Unit 1

N(2, 0.02)

N(0.1, 0.02)

Unit 2

N(5, 0.01)

N(0.1, 0.01)

Unit 3

N(3, 0.01)

N(0.06, 0.01)

Unit 4

N(3, 0.02)

N(0.1, 0.02)

36


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1

According to the definitions of flexible uncertainty sets and corresponding robust

2

formulations (called “flexible ones”), the constraint in Eq.(6.1) can be transformed to

3

its equivalent forms. At first, compared with classical uncertainty sets induced

4

formulations (called “classical ones”), there are several differences between “classical

5

ones” and “flexible ones”, which are:

6

 Classical sets induced formulations only depend on one adjustable parameter 

7

(or  ,  ), while a series of independent adjustable parameters (refers to the

8

confidence level 1   i ) need to be determined for flexible sets induced

9

formulations;

10 11

 Classical ones control the influence of uncertain part on the whole model, while

flexible ones focus on the effect from each continuous uncertainty;

12

 Uncertainties in flexible formulations can be unbounded and bounded, while

13

classical formulations mainly used for solving problems under bounded

14

uncertainties.

15 16

Meanwhile, there is an important assumption for the two kinds of formulations, which is: uncertainties are regarded as independent of each other.

17

6.1.1 Comparison between classical and flexible sets induced formulations

18

Based on the above features and assumptions, here we implement two categories of

19

uncertainty sets (classical uncertainty sets and flexible uncertainty sets) induced

20

formulations on the classic STN case, and the solutions are shown in Figure 8. The

21

objective values change when the adjustable parameters and the type of uncertainty sets

22

are different, which are presented in Figure 9.

37



Scheduling Results

Equipment Heater

Reactor 1

Reactor 2

1

1

1

52

20

52

0

1.Heati ng

3

4

2

3

4

58

80

80

78

80

68.75

2

3

3

4

3

4

50

50

50

50

50

50

1

2

3

4

5

2.Reaction 1

3.Reaction 2

4.Reaction 3

5

5

5

80

50

118.75

6 Time (h)

1 2

Tasks

2

Still

7

8

9

10

11

5.Separati on

12

Figure 8. Scheduling results for the formulations when the utility uncertainty is

3

considered. 2000

1900

1800

1700

Objective

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 67

1600

1500

1400

1300

1200

1100 0

4 5 6

Classical Box Uncertainty Set (Ψ) Classical Ellipsoidal Uncertainty Set (Φ=Ψ*sqrt(L)) Classical Polyhedral Uncertainty Set (Γ=Ψ*L) Flexible Box Uncertainty Set I Flexible Ellipsoidal Uncertainty Set I Flexible Polyhedral Uncertainty Set I Flexible Box Uncertainty Set II Flexible Ellipsoidal Uncertainty Set II Flexible Polyhedral Uncertainty Set II

0.1

0.2

0.3

0.4

0.5 0.6 0.7 Adjustable Parameter (Ψ for Classical Ones; Confidence Level for Flexible Ones)

0.8

0.9

1

Figure 9. Objective values of different formulations (Classical ones v.s. Flexible ones).

7

For classical formulations, there is no strict limit for adjustable parameters, which

8

can be determined as a large constant but with a suggested range. When the parameter

9

equals to the upper bound of its suggested range, the uncertainty set covers all the

10

random scenarios. However, for flexible uncertainty sets, only when the adjustable

11

parameter is less than 1, the formulation makes sense, because the parameter refers to 38


Page 39 of 67

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1

the confidence level of each continuous uncertainty. Similarly, when the adjustable

2

parameters for flexible sets are equal to their upper bounds – one, the uncertainty sets

3

cover all the uncertain scenarios.

4

Within the suggested range of adjustable parameters, the objective values for classical

5

ones fall faster when the adjustable parameter gets bigger, which means for the same

6

value of the adjustable parameter, the classical sets induced formulations are more

7

conservative than flexible ones.

8

6.1.2 Comparison between flexible uncertainty sets I and II induced formulations

9

For flexible uncertainty sets induced formulations, when we focus on part of the

10

random scenarios of uncertainties, the flexible sets I induced formulations still rely on

11

the adjustable parameter which is indirectly calculated. The bounds for flexible sets I

12

on each dimension are still the same. As a result, we design two kinds of experiments

13

for analyzing the robustness and conservatism of flexible sets I and II induced

14

formulations:

15

Case 1: Only 75% of the uncertain scenarios of the variant part are considered, and

16

all the scenarios of the invariant part within the uncertainty sets are considered, which

17

is called “alpha case”. In this case, the model focuses more on the influence of static

18

energy consumption rate;

19

Case 2: Similarly, if only 75% of uncertain values of  u ,i , j are considered and all

20

the random values of  u ,i , j are considered, the model focuses more on the influence

21

of dynamic energy consumption rate, which is called “beta case”.

22

When we implement the two kinds of experiments on the flexible sets induced

23

formulations, the results of objective values (in red color) and probability bounds of

24

constraints’ satisfaction (in blue color) are combined and shown in Figure 10. 39



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2 3

Figure 10. Objective values and the probability bounds of constraints’ satisfaction for different formulations.

4

For flexible sets I induced formulations, no matter how many uncertain scenarios of

5

uncertainties are considered, the model will remain the same. Because the parameter

6

 (or  ,  ) for flexible sets I is calculated as the same value indirectly. For example,

7

in both “alpha case” and “beta case”, the adjustable parameter  in the flexible box

8

uncertainty set I induced formulation is still equal to max cu ,i , j , u , i, j  J i  . As a

9

result, for both cases, the objective values and probability bounds for flexible sets I

10

induced formulations remain the same.

11

In contrast, for flexible sets II induced formulations, “beta cases” show higher

12

objective values than “alpha cases”, which represents that the uncertainty in static

13

energy consumption rate has greater influence than the dynamic part. Thus we can infer

14

that decision makers need to reduce the running time of equipment and increase the

15

amount of processing materials properly. Similarly, more uncertain scenarios can be

16

generated for analyzing the influence of some other uncertainties. 40


Page 40 of 67

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1

According to the equations for calculating probability bounds in section 5, the

2

probability bounds for both cases are same. Obviously, polyhedral sets induced

3

formulations show tighter probability bounds of constraints’ satisfaction, and flexible

4

sets II induced formulations are with better performance than the sets I induced ones.

5

Compared with classical ones and flexible set I induced formulations, the flexible

6

uncertainty sets II induced formulations have been proved with the lowest conservative

7

solutions and tight probability bounds. For selecting the type of flexible uncertainty sets

8

II and the corresponding adjustable parameters properly, the combined results can help

9

us come to the following conclusions:

10 11

 The conservatism increases in the following order: box, ellipsoidal, polyhedral,

while the tightness of probability bounds shows an opposite order.

12

 Confidence levels can be chosen as several intervals, for example, [0, 0.50], [0.50,

13

0.85] and [0.85, 1.00]. For the first interval, the decision makers focus more on

14

the conservatism of model, and the uncertainties have little influence on the

15

objective; for the second interval, more uncertain scenarios are considered in the

16

model, and the falling or rising trends of solutions and objective values are very

17

valuable for making decisions; for the third interval, uncertainties are considered

18

at the highest weight in the model, and the transformed model needs to satisfy

19

most of the uncertain scenarios.

20

 Based on the three intervals of confidence levels and corresponding motivations,

21

the uncertainty sets for this problem can be chosen according to the rules like a)

22

box and ellipsoidal sets for the first interval; b) ellipsoidal and polyhedral sets

23

for the second interval; c) polyhedral sets for the third interval.

41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

In fact, the new formulations induced by flexible uncertainty sets help decision

2

makers consider independent continuous uncertainties at different weight. Meanwhile,

3

the new formulations provide more space for the discussion of robustness and

4

conservatism of complex optimization problems. On one hand, the flexible sets have

5

similar features of classical sets; on the other hand, the “confidence level” of uncertainty

6

makes the discussion more meaningful for real-world optimization problems.

7

6.2 Process Industry Example

8

For better illustrating the application of the new robust optimization approach in real-

9

world problems, the model of the integrated production and utility system of an ethylene

10

plant is introduced in this section. The diagram of the production process is shown in

11

Figure 11.

12 13 14

Figure 11. The diagram of the integrated production and utility system of an ethylene plant.

15

Four kinds of crude oils (ETHA, NAP, AGO, HVGO) and five kinds of energy

16

materials (fuel, steam, water, electricity, wind) are considered in the production. For 42


Page 42 of 67

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1

better recycling the intermediate products, part of the fuel gas is generated from the

2

separation section and the other fuel resources (natural gas and fuel gas) need to be

3

bought from other suppliers. Through the cracking furnaces, cooling section,

4

compression section, separation section, nine kinds of products can be obtained.

5

The mathematical model has been illustrated in detail in Zhang’s work, which can

6

also be seen in Supporting Information. The objective is to maximize the profit of

7

selling products, where the penalty of the cost of crude oils, energy resources and

8

operations is also considered. In our research, the uncertainties in the fuel gas

9

consumption rate of some furnaces are introduced to the energy balance constraints,

10 11

such as

  a f

12

f ,r



FC f ,r ,t  b f ,r DS f ,r ,t  c f ,r  FPPfuel ,t  PC fuel ,t

t

(6.2)

r

where FC f ,r ,t and DS f ,r ,t represent the amount of raw material and dilution

13

steam consumed by furnace f during period t when processing crude oil

14

 f ,r represents the uncertain fuel gas consumption rate of furnace f respectively; a

15

when processing raw material

16

for calculating the consumption of fuel gas by furnace f when processing material

17

Thus the meaning of the constraint is: during each period t , the consumption of fuel

18

gas by all the furnaces needs to be less than the sum of purchased fuel PC fuel ,t and

19

self-produced fuel FPPfuel ,t .

r ; b f ,r ,

r

,

c f ,r represent coefficients of the linear model r.

20

Through 80 days’ historical data, 10637 valuable records of running furnaces are

21

obtained, from which the kernel-based estimation results of the consumption rate for

22

several furnaces are shown in Figure 12. 43



Estimation of Consumption Rate when Processing NAP BA106


0.115

0.10

0.11

0.12 0.13 Consumption Rate (c)

0.14

0.10

0.140

0.145 0.150 Consumption Rate (b)

0.155


0.13

Density

Estimation of Consumption Rate when Processing HVGO BA107

0.0

0.3

Density


0.160

0.050

0.055 Consumption Rate (d)

0.060

0.115

0.120 0.125 0.130 Consumption Rate (d)

0.135

Density

0.0

0.2


0.0 0.2 0.4

Estimation of Consumption Rate when Processing NAP BA108 Density

0.15

0.0

Density

0.0 0.2 0.4


0.135

0.2

Density

0.120

0.8

0.100 0.105 0.110 Consumption Rate (a)

0.4

0.095

0.0

Density 0.090

0.00 0.15

0.2 0.0

Density

0.4


0.055

0.060


0.075

0.080

0.08

0.09

0.10 Consumption Rate (c)

0.11


0.112

0.114

1 2

0.3

Density

0.0

0.3

Density

0.6


0.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 67

0.116

0.118 0.120 Consumption Rate (d)

0.122

0.124

0.090

0.095

0.100 Consumption Rate (d)

0.105

0.110

Figure 12. Kernel-based estimation of fuel gas consumption rate for BA105-BA110

3

when processing NAP or HVGO.

4

As shown in Figure 12, furnaces BA105-BA108 processed NAP and HVGO during

5

the 80 days’ period, while BA109 and BA110 only processed one kind of crude oil.

6

Thus the probability density function of the consumption rate for the above furnaces

7

can be introduced to the uncertainties in Eq.(6.2). Then the robust counterpart of the

8

constraint can be obtained as Eq.(6.3).

9

  a f

10

r

f ,r

  FC f ,r ,t  b f ,r DS f ,r ,t  c f ,r   max     f ,r a f ,r FC f ,r ,t   FPPfuel ,t  PC fuel ,t  U  f FU r 

t

(6.3)

11

According to the robust formulations derived in section 4, when the flexible

12

uncertainty sets are chosen as box, ellipsoidal and polyhedral, the corresponding

13

formulations are presented as Eq.(6.4)-Eq.(6.6), respectively. Here, we mainly focus on

14

the flexible uncertainty sets II induced formulations.

44


Page 45 of 67

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1


  a

1 f ,r

f

FC f , r ,t  b f , r DS f , r ,t  c f ,r  

r

 c

f FU

5

t

(6.4)

  a

1 f ,r

f

4

max a f ,r FC f ,r ,t  FPPfuel ,t  PC fuel ,t

r

2

3

f ,r

r

FC f ,r ,t  b f ,r DS f ,r ,t  c f ,r  

FU 

  c

f FU

f ,r

max a f , r FC f ,r ,t

r



2

 FPPfuel ,t  PC fuel ,t

t

(6.5)   a1f,r FC f ,r ,t  b f ,r DS f ,r ,t  c f ,r   FU  zt  FPPfuel ,t  PC fuel ,t  f r  max  zt   c f ,r a f ,r FC f ,r ,t t , f  FU  r

t

(6.6)

6

For optimizing the scheduling of fuel gas and crude oils in 10 days (period = 10 days),

7

the new robust models are implemented in GAMS 24.1. When flexible box set II is

8

chosen as the uncertainty set of all uncertainties, the model has 650 discrete variables

9

and 10067 continuous variables. When the type of uncertainty set is changed, the

10

amount of variables would be a little different, such as there are 10072 continuous

11

variables for polyhedral set induced formulation. Then the mathematical model is

12

solved on a desktop computer with the following specifications: Dell Precision T5610

13

with Intel Xeon CPU E5-2609 v2 at 2.350GHz (total four threads), 16GB of RAM, and

14

running Windows 7 Enterprise. The local MIP solver is CPLEX 12.5 and DICOPT is

15

chosen as the MINLP solver. The R ks package is used for kernel smoothing estimation

16

of the uncertainties. Because the box and polyhedral set induced formulations remain

17

linear, the corresponding MILP problem can be solved within 5 minutes, while the

18

calculation time for ellipsoidal set induced models (MINLP problem) is about 28

19

minutes. When the confidence level changes, the optimization results of fuel gas

20

consumption in 10 days are presented in Figure 13.

45



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2

Figure 13. Consumption of fuel gas in 10 days for all the furnaces.

3

In Figure 13, the solid lines represent the rising trend of the total amount of consumed

4

fuel gas when the confidence level changes from 0 to 100%. The flexible polyhedral

5

uncertainty set induced result appears to be the most conservative, where the

6

consumption ranges from 9612.46t to 10347.82t. This means the uncertainty in

7

consumption rate of BA105-BA110 could cause 7%~8% of fluctuations in the fuel

8

gas’s consumption, which is similar to the exact performance in the real world.

9

Similarly, the results of ellipsoidal and box set induced models shows 4%~5% and

10

3%~3.5% fluctuations, respectively.

11

Different from the classical robust optimization approach, the proposed formulation

12

makes it possible to discuss the influence of any individual uncertainty. Here, according

13

to the daily working condition and the type of processing materials, the furnaces with

14

uncertainties (BA105-BA110) can be divided into two groups: Group A (BA105,

15

BA107, BA109), Group B (BA106, BA108, BA110). Then two cases are generated:

16

Case 1: 100% of uncertain scenarios in the consumption rate of Group A furnaces

17

should be satisfied, furnaces in Group B are regarded as stable and with no uncertainties. 46


Page 46 of 67

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1 2


Case 2: On the contrary, no uncertainty of Group A furnaces is considered, and all the uncertain values of consumption rate for Group B furnaces are considered.

3

The previous result of the polyhedral induced model (the solid line) can be taken as

4

a reference for the comparison between Case 1 and Case 2, because all the uncertainties

5

in both Group A and Group B have been considered in the previous case. Thus when

6

we choose polyhedral set as the uncertainty set, the dotted lines in Figure 13 have shown

7

that Furnace Group A contributes more to the fluctuation of the fuel gas consumption,

8

especially when the confidence level changes from 20% to 25% and from 90 to 100%.

9

And this conclusion is consistent with the observation in the energy daily of the ethylene

10

plant, which is that furnaces in Group A consumed more fuel gas than Group B and

11

with bigger fluctuations. Through our research, the percentage and amount of

12

fluctuations can be calculated and predicted, which can help decision makers to

13

distribute energy resources better and to store enough fuel gas for certain furnace in

14

advance.

15

In a word, for real-world scheduling problems under continuous uncertainties, if the

16

distribution of uncertain values can be estimated, the flexible uncertainty set can be

17

introduced to describe the uncertainty. Moreover, the decision maker can decide the

18

percentage of considered scenarios for any uncertainty. By choosing a suitable type of

19

flexible uncertainty set and selecting a proper combination of uncertain scenarios, the

20

real world performance can be captured through the proposed robust optimization

21

approach. If more uncertainties could be considered in the model, the approach can be

22

applied as scheduling tool, which is configurable and extensible.

23

7. CONCLUSION

47



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

For optimization problems under independent and continuous uncertainties, new

2

robust formulations induced by flexible uncertainty sets are proposed based on classical

3

box, ellipsoidal and polyhedral set induced formulations. Uncertainties are considered

4

at different weights, and the bounds of new uncertainty sets become more flexible than

5

classical sets. Two kinds of flexible uncertainty sets and corresponding formulations

6

are proposed. Flexible uncertainty sets I are regarded as a transition from classical ones

7

to real flexible ones, which remain many features of classical sets. The conservatism

8

and probability guarantees on constraints’ satisfaction are compared through several

9

experiments on different set-induced formulations. Robust formulations induced by

10

flexible uncertainty sets II are proved least conservative and with tight probability

11

bounds. Several rules for selecting uncertainty sets and choosing adjustable parameters

12

are concluded. For real-world optimization problems, the new formulations can also be

13

applied for those models under uncertainties with asymmetric probability density

14

functions, such as yield uncertainties and operational parameters (temperature, pressure,

15

flowrate). At last, the probability bounds and formulations induced by combined

16

flexible uncertainty sets will also be studied in the future.

48


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1


SUPPORTING INFORMATION

2

1. Proofs for Lemma 5.1-5.3;

3

2. Robust MILP formulations induced by flexible uncertainty sets;

4

3. The deterministic optimization model of the industrial example for a real-world

5

ethylene plant.

6 7

The Supporting Information is available free of charge on the ACS Publication website at http://pubs.acs.org/.

8 9

AUTHOR INFORMATION

10

Corresponding Author

11

*Tel: 086-87953145. Email: [email protected].

12 13

ACKNOWLEDGEMENTS

14

The authors gratefully acknowledge financial support from the National High

15

Technology R&D Program of China (2014AA041805) and Projects of International

16

Cooperation and Exchanges NSFC (61320106009).

17 18

REFERENCES

19

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20

challenges. Computers & Chemical Engineering 2008, 32 (4), 715-727.

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kernel density estimation. Annals of Operations Research 2004, 132 (1-4), 47-68.

4

3. Charnes, A.; Cooper, W. W., Chance-constrained programming. Management

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science 1959, 6 (1), 73-79.

6

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9

5. Jiang, R.; Guan, Y., Data-driven chance constrained stochastic program.

Cooper, W. W.; Deng, H.; Huang, Z.; Li, S. X., Chance constrained programming

10

Mathematical Programming 2012, 1-37.

11

6. Calfa, B. A.; Grossmann, I. E.; Agarwal, A.; Bury, S. J.; Wassick, J. M., Data-

12

driven individual and joint chance-constrained optimization via kernel smoothing.

13

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1


TABLE OF CONTENTS (TOC) GRAPHIC

Flexible Uncertainty Sets Induced Robust Formulations

Construction of Flexible Uncertainty Sets (FUS) Uncertainty Set (Interval)

Calculation of Probabilistic Bounds for Robust Formulation

Confidence Level

Online Sampling and Estimation

2

Robust Solutions with Confidence Levels

Real-world Production Process (with continuous uncertainties)

3

53


Apply for


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Geometric view of typical uncertainty sets: (a) Box uncertainty set; (b) Ellipsoidal uncertainty set; (c) Polyhedral uncertainty set. Figure 1 178x56mm (300 x 300 DPI)


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Bounded uncertainties transformed from continuous uncertainty. Figure 2 176x80mm (300 x 300 DPI)



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Geometric view of typical and flexible uncertainty sets Figure 3 190x58mm (300 x 300 DPI)


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Figure 4-Geometric interpretation for flexible uncertainty sets Figure 4 444x198mm (96 x 96 DPI)



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Figure 5-Objective values of different robust formulations. Figure 5 444x198mm (96 x 96 DPI)


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Probability bounds of constraints’ violation for different formulations. Figure 6 444x240mm (96 x 96 DPI)



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Flowchart of the state-task network (STN) example. Figure 7 201x97mm (300 x 300 DPI)


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Scheduling results for the formulations when the utility uncertainty is considered. Figure 8 139x60mm (300 x 300 DPI)



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Objective values of different formulations Figure 9 444x240mm (96 x 96 DPI)


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Objective values and the probability bound of constraints’ satisfaction for different formulations. Figure 10 100x57mm (300 x 300 DPI)



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The diagram of the integrated production and utility system of an ethylene plant. Figure 11 424x254mm (300 x 300 DPI)


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Kernel-based estimation of fuel gas consumption rate for BA105-BA110 when processing NAP or HVGO. Figure 12 379x223mm (96 x 96 DPI)



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Consumption of fuel gas in 10 days for all the furnaces. Figure 13 180x97mm (150 x 150 DPI)


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Table of Contents Graphic TOC 157x102mm (300 x 300 DPI)


Optimization under Continuous Uncertainty - ACS Publications

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