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Doklady Chemistry, Vol. 376, Nos. 1–3, 2001, pp. 38–41. Translated from Doklady Akademii Nauk, Vol. 376, No. 2, 2001, pp. 215–218.Original Russian Text Copyright © 2001 by Ostrovskii, Volin.

A necessary requirement in chemical engineeringprocess design is the fulfillment of certain technologi-cal and service constraints. The satisfaction of thisrequirement is substantially complicated by the factthat one has to apply inaccurate mathematical models,whose parameters

t

i

can take any values from an uncer-tainty range

T

. This gives rise to the important problemof finding the optimum overdesign factors, which cancompensate for the model inaccuracies, i.e., to theproblem of designing a flexible chemical engineeringprocess. A detailed analysis shows that, under theuncertainty of mathematical model parameters, oneneeds to distinguish between “hard” and “soft” con-straints, and in practice, it is possible for some of theconstraints to be hard, and the others to be soft. At thesame time, the existing statements of the optimizationproblem under uncertainty take no combined accountof hard and soft constraints; and when only soft con-straints are imposed, a single-stage problem is usuallysolved, which reflects no specificity of the problemstatement at the process design stage. In this paper, weintroduced new statements that enable one to moreaccurately estimate the overdesign factors under softand, particularly, mixed constraints.

At known values of the parameter

t

, the optimizationproblem in design has the form

(1)

(2)

where

J

= (1, 2, …,

m

),

d

is the vector of design vari-ables (equipment parameters), and

z

is the vector ofcontrol variables.

In statement (1)–(2), inequalities (2) constitute themathematical formulation of the above design con-straints. On the basis of this problem, the optimizationproblem under uncertainty is stated, whose main ele-ment is the condition of flexibility of the chemical engi-

f d z t, ,( ),d z,

min

g j d z t, ,( ) 0, j J ,∈≤

neering process. The following key factors affect theformulation of the flexibility condition and the optimi-zation under uncertainty.

(1)

Degree of uncertainty at the design stage.

Twovariants are possible: (a) our knowledge about theuncertain parameters is restricted only to the fact thatthey take their values from range

T

; (b) in addition, wealso know the probability distribution functions for theuncertain parameters.

(2)

Degree of uncertainty at the operation stage.

The formulation of the optimization problem dependson the possibilities of the experimental data acquisitionsystem (the presence of the corresponding sensors andtheir accuracy), which governs the completeness andthe accuracy of the experimental information availableat the operation stage. According to the accuracy of thedetermination of the uncertain parameters at the opera-tion stage, they can be classified into three groups [1].The first of these groups comprises parameters whosevalues can be found accurately enough, the second con-sists of parameters whose values cannot be refined (theuncertainty range remains the same as that at the designstage), and the third contains parameters whose valuescan be refined but will retain some residual uncertainty.

(3)

Method of ensuring the flexibility of thechemical engineering process.

There can be caseswhen (a) both design and control variables are avail-able, (b) only design variables are available, and (c)only control variables are available.

(4)

Type of constraint.

Constraints can be hard andsoft. Under no circumstances must hard constraints beviolated. Soft constraints should be met within a givenprobability. For example, safety and environmentalconstraints are hard, and constraints on the productquality and the output may often be assigned to softconstraints. Most of actual problems involve con-straints of both types. However, the main existing prob-lem statements contain constrains of only one of theabove types. Among the problems with such statementsare the chance-constrained model [2], the two-stageoptimization problem (the stochastic programmingmodels with recourse) [3, 4], and the discrete version ofthe latter problem [3, 5]. This can lead to overestima-tion of the overdesign factors. Indeed, let us consider a

CHEMICAL TECHNOLOGY

Optimization of Chemical Engineering Processesunder Uncertainty under Hard and Soft Constraints

G. M. Ostrovskii and Yu. M. Volin

Presented by Academician I.I. Moiseev August 1, 2000

Received July 26, 2000

Karpov Research Institute of Physical Chemistry,Russian State Scientific Center, ul. Vorontsovo pole 10,Moscow, 103064 Russia

DOKLADY CHEMISTRY

Vol. 376

Nos. 1–3

2001

OPTIMIZATION OF CHEMICAL ENGINEERING PROCESSES 39

chance-constrained problem in its ordinary statement,when the

j

th constraint should be satisfied with proba-bility

α

j

(

j

= 1, 2, …,

m

)

:

(3)

(4)

where

E

t

f

(

d

,

z

,

t

)

is the mathematical expectation ofthe quantity

f

(

d

,

z

,

t

),

and

µ

(

t

)

is the probability densityfunction of the parameter

t

.In this statement, one can solve the problem under

hard and soft constraints, putting

α

j

= 1 for hard con-straints. However, in this case, the overdesign factorswill be overestimated, since model (3)–(5) enables noadjustment of the control variables to meet the con-straints for each set of the uncertain parameters. This isa common drawback of using single-stage optimizationmodels (to which model (3)–(4) belongs) to solvedesign problems.

Let us now consider the two-stage optimizationproblem [1, 3, 4]

(5)

(6)

If we solve problem (5)–(6), this will mean thatproblem (6) can be solved for all

t

∈

T

; i.e., we havefound such

d

that allows the satisfaction of all the con-straints at all

t

∈

T

. The discrete version of the two-stage optimization problem also requires the strict ful-fillment of all the constraints, since it involves the con-straint [1, 3]

(7)

and this means that all the constraints (both hard andsoft) should be obeyed with the probability 1. There-fore, these approaches also result in overestimation ofthe overdesign factors when they contain constraintsthat can be considered soft.

Below, we will give several new variants of thestatement of the optimization problems involving softconstraints, which cause no overestimation of the over-design factors. The examination will be performed interms of the stochastic flexibility [6]. For the first threevariants, we will consider that all the uncertain param-eters belong to the first group; i.e., that their values atthe operation stage can be found accurately. In thefourth variant, there will be parameters from twogroups (the second group contains parameters whosevalues cannot be refined at the operation stage). The

Et f d z t, ,( ) ,d z,

min

Pr g j d z t, ,( ) 0≤[ ] α j, j≥ 1 2 … m,, , ,=

Pr g j d z t, ,( ) 0≤[ ] µ t( ) td

Ω j d z,( )∫ ,=

Ω j d z,( ) t T : g j d z t, ,( )∈ 0≤ ,=

Et f * d t,( ) ,d

min

f * d t,( ) f d z t, ,( )/g j d z t, ,( ) 0,≤z

min=

j 1 … m., ,=

F1 d( ) h d t,( )t T∈max g j d z t, ,( ) 0,≤

j J∈max

zmin

t T∈max= =

variants to be put forward will be a generalization oftwo-stage problem (5)–(6) to the case when the prob-lem involves soft constraints. Let us impose therequirement (which is natural from the practical stand-point) that the initial statement should contain (as asubproblem) the formulation of the optimization prob-lem that is to be solved during the control of the processat the operation stage. This requirement is particularlysubstantial for the fourth variant.

First variant. Let us assume that all the constraintsare soft and, in the aggregate, should be met with aprobability of at least α. We should formulate an opti-mization criterion and constraints of the problem. Theoptimization criterion can be constructed in the follow-ing manner. Let Ω be a set of the t values at which theproblem constraints are satisfied, and the stochasticflexibility [6] Pr[t ∈ Ω ] ≥ α. In the optimization crite-rion, for each t ∈ Ω , the variable z should be chosenfrom the condition that f(d, z, t) is minimum under theconstraint h(d, t) ≤ 0; and at t ∉ Ω , it should be selectedfrom the condition of minimization of the sum of thefunction f(d, z, t) and the penalty for violation of theconstraints gj(d, z, t) ≤ 0:

(8)

where c is a penalty coefficient (c > 0).The optimization problem can now be written as

(9)

(10)

(11)

is the stochastic flexibility [6].Note that, if there is such d that F1(d) ≤ 0, then, as

α 1, Ω T, and in the limit (at α = 1), I2(d) = 0,the problem goes over into ordinary two-stage optimi-zation problem (5)–(6).

Second variant. Let us assume that the problemcontains constraints belonging to two groups. The firstof these groups comprises hard constraints (with thesubscripts j ∈ J1 = (1, …, m1)), and the second consistsof soft constraints (with the subscripts j ∈ J2 = (m1 + 1,…, m)). The soft constraints should be fulfilled with aprobability of at least α. This problem has been dis-cussed previously [7] and has been analyzed with theuse of a penalty function [8]. But in those investiga-tions, a given probability of satisfaction of the soft con-

f d z t, ,( ) f d z t, ,( ) cmax g j d z t, ,( ),0j J∈

max( ),+=

I d( )d

min I1 d( ) I2 d( )+( ),d

min=

I1 d( )

= f d z t, ,( )/g j d z t, ,( ) 0 j J∈,≤z

min( )µ t( ) td ,

Ω∫

I2 d( ) f d z t, ,( )µ t( )dt,z

min

T \Ω∫=

Pr t Ω∈[ ] α ,≥

Ω Ω d( ) t T : g j d z t, ,( ) 0≤j J∈

maxz

min∈ .= =

Pr t Ω∈[ ]

40

DOKLADY CHEMISTRY Vol. 376 Nos. 1–3 2001

OSTROVSKII, VOLIN

straints was not guaranteed, and the problem statementwas inaccurate. An accurate statement of this problemcan be obtained by modifying the formulation of thefirst variant and has the following form:

(12)

(13)

where Ω is given by relation (11).In the discrete formulation of the problem (when the

integral is replaced by a weighted sum), an additionalconstraint should be imposed:

(14)

(see [1] for details).In the obtained solution, all the constraints are

obeyed with a probability of at least α. If there is suchd that F1(d) ≤ 0, then, as α 1, Ω T, and in thelimit (at α = 1), I2(d) = 0, and the problem transformsinto two-stage optimization problem (5)–(6). Problem(9)–(10) can be regarded as its limiting case (when J1 =

, where is the empty set).Third variant. Let us assume that the problem

involves constraints belonging to two groups. Con-straints from the first of these groups (with the sub-scripts j ∈ J1) should be met with a probability of atleast α1, and constraints from the second group (withthe subscripts j ∈ J2 = J\ J1) should be fulfilled with aprobability of at least α2, with α1 > α2. This case is morecomplex than the one when the first groups containshard constraints. But in this case, one can also proposean accurate problem statement in a manner close to thatused above. We will use a somewhat modified problemformulation. Namely, we require that all the constraintsshould be satisfied with the probability α2. This modi-fication somewhat simplifies the problem from thecomputational standpoint and appears justified in thepractical aspect. Let us introduce the following sets:

Ω1 is the set where all the constraints from the firstgroup are obeyed, and Ω is the set where all the con-

I d( )d

min I1 d( ) I2 d( )+( ),d

min=

I1 d( )

= f d z t, ,( )/g j d z t, ,( ) 0 j J∈,≤z

min( )µ t( ) td ,

Ω∫

I2 d( )

= f d z t, ,( )/g j d z t, ,( ) 0 j J1∈,≤z

min( )µ t( ) td ,

T \Ω∫

Pr t Ω∈[ ] α ,≥

F1 d( ) g j d z t , ,( ) 0 ≤ j J

1

∈

max z

min t T

∈

max=

Ω1 Ω1 d( ) t T : g j d z t , ,( ) 0 ≤ j J

1

∈

max z

min ∈ ,= =

Ω Ω

d

( )

t T

: g j d z t , ,( ) 0 ≤ j J

∈

max z

min ∈ .= =

straints are met. Apparently,

Ω

⊆ Ω

1

⊆

T

. The optimi-zation problem can now be stated as follows:

(15)

(16)

One can readily see that, in the limit

α

1

1

, prob-lem (15)–(16) goes over into problem (12)–(13); and as

α

1

α

2

, it is reduced to problem (9)–(10).

Fourth variant.

Let us assume that the conditionsof the second variants are met, and that the vector

t

con-sists of the subvectors

t

1

and

t

2

of parameters belongingto the first and second groups, respectively. Let

t

1

∈

T

1

and

t

2

∈

T

2

.At a fixed moment of time at the operation stage, the

t

1

value is known, and

t

2

can take any value from therange

T

2

.The optimization problem (under the constraint

imposed prior to the description of the first variant) canbe formulated in the following form:

(17)

(18)

(19)

I d( )d

min I1 d( ) I2 d( )+( ) I3 d( ),+d

min=

I1 d( )

= f d z t, ,( )/g j d z t, ,( ) 0 j J∈,≤z

min( )µ t( ) td ,

Ω∫

I2 d( )

= f d z t, ,( )/g j d z t, ,( ) 0 j J1∈,≤z

min( )µ t( ) td ,

Ω1\Ω∫

I3 d( ) f d z t, ,( )µ t( )dt,z

min

T \Ω1

∫=

Pr t Ω∈[ ] α2, Pr t Ω1∈[ ] α1.≥≥

I d( )d

min I1 d( ) I2 d( )+( ),d

min=

I1 d( )

= Et2 f d z t1 t2, , ,( ) / g j

t2

T2∈

max d z t1 t2, , ,( ) 0,≤z

min(Ω1

∫j 1 … m )µ1 t1( )dt1,, ,=

I2 d( )

= Et2 f d z t1 t2, , ,( ) / g j

t2

T2∈

max d z t1 t2, , ,( ) 0,≤z

min(

T1\Ω

1

∫j 1 … m1 )µ1 t1( )dt1,, ,=

Pr t1 Ω1∈[ ] α ,≥

Ω1 t1 T1: g j d z t, ,( ) 0≤j J∈

maxt2

T2∈

maxz

min∈

.=

DOKLADY CHEMISTRY Vol. 376 Nos. 1–3 2001

OPTIMIZATION OF CHEMICAL ENGINEERING PROCESSES 41

In statement (17)–(19), such t1 values are, however,possible when the soft constraints are not satisfied evenin a probabilistic sense. Therefore, for this variant, weintroduce one more statement, in which, at the opera-tion stage, the hard constraints are always fulfilled, andthe soft constraints at each t1 should be obeyed with aprobability of at least α. Let

(20)

(21)

(22)

The optimization problem can then be stated as

(23)

Statement (23) is conservative, but it describes wellthe situation occurring frequently in practice when thevector t1 varies slowly and can be identified accuratelyenough, and the vector t2 changes rapidly, and is diffi-cult, if not impossible, to identify reliably at the opera-tion stage.

In discrete variants of problems (17)–(19) and (20)–(23), the inequality

should be added.

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2000, vol. 370, no. 6, pp. 773–776.2. Bernardo, F.P. and Saraiva, P.M., AhE J., 1998, vol. 4,

no. 9, pp. 2007–2017.3. Halemane, K.P. and Grossmann, I.E., AIChE J., 1983,

vol. 29, no. 3, pp. 425–433.4. Shapiro de-Mello, T.H., Math. Progr. Ser. A, 1998,

vol. 81, pp. 301–305.5. Ostrovskii, G.M. and Volin, Yu.M., Dokl. Akad. Nauk,

1994, vol. 339, no. 6, pp. 782–784.6. Straub, D.A. and Grossmann, I.E., Comput. Chem. Eng.,

1993, vol. 17, no. 4, pp. 339–354.7. Ierapetritous, M.G. and Pistikopoulos, E.N., Ind. Eng.

Chem. Res., 1994, vol. 33, no. 8, pp. 1930–1942.8. Wellons, H.S. and Reklaitis, G.V., Comput. Chem. Eng.,

1989, vol. 13, no. 1/2, pp. 115–126.

f * d t1,( ) Et2 f d z t1 t2, , ,( ) ,

zmin=

g jt2

T2∈

max d z t1 t2, , ,( ) 0, j≤ 1 … m1,, ,=

Prt2 g j d z t1 t2, , ,( ) 0 j = m1 1 … m, ,+,≤[ ] α .≥

I d( )d

min f * d t1,( )µ1 t1( ) t1.d

T1

∫dmin=

F1 d( ) g j d z t1 t2, , ,( ) 0≤j J1∈max

t2

T2∈

minz

mint1

T1∈

max=