Process Systems Engineering, 3. Mathematical Programming ...folk.ntnu.no/skoge/prost/proceedings/ullmann-2012... · Process Systems Engineering, 3. Mathematical Programming (Optimization)

Process Systems Engineering, 3. MathematicalProgramming (Optimization)

LUKE E. K. ACHENIE, Virginia Polytechnic Institute and State University, Department

of Chemical Engineering, Blacksburg, Virginia, United States 24061

ANGELO LUCIA, University of Rhode Island, Department of Chemical Engineering,

Kingston, Rhode Island, United States 02811

URMILA DIWEKAR, Vishwamitra Research Institute, Center for Uncertain Systems:

Tools for Optimization and Management, Clarendon Hills, Illinois, United States 60514

GUENNADI OSTROVSKY, Karpov Institute of Physical Chemistry, Moscow, Russia

1. Introduction. . . . . . . . . . . . . . . . . . . . . 12. General Optimization Concepts. . . . . . 23. Global Terrain Method . . . . . . . . . . . . 83.1. Simple Example of the Global Terrain

Method . . . . . . . . . . . . . . . . . . . . . . 103.2. General Optimization Problems . . . 123.3. Advanced Techniques . . . . . . . . . . . 133.4. Applications and Extensions of the

Terrain Method . . . . . . . . . . . . . . . . 13

4. Optimization under Uncertaintyin Process Systems Engineering . . . . . 13

4.1. Methods and Algorithms . . . . . . . . . 154.2. Uncertainty Analysis and Sampling . 164.3. Optimization Algorithms . . . . . . . . . 184.4. Applications . . . . . . . . . . . . . . . . . . . 194.5. Future Directions. . . . . . . . . . . . . . . 23

1. Introduction

According to Wikipedia, optimization isdefined as: ‘‘In mathematics and computerscience, optimization, or mathematical pro-gramming, refers to choosing the best elementfrom some set of available alternatives. In thesimplest case, this means solving problems inwhich one seeks to minimize or maximize a realfunction by systematically choosing the valuesof real or integer variables from within anallowed set . . ..’’ Optimization has well over10 major subfields, each of which has severalresearchers/adherents. For this article, the abovedefinition which is quite good and pertains toseveral disciplines including process systemswill be adopted. In process systems a systematicand thorough study of a chemical process tounderstand the relationship among the possiblymany process variables is done. Although someof the relationships are observable, for mostprocesses the understanding of the relationshipsis rather limited. Various mathematical formu-lae (commonly referred to as process models)have been developed to mimic the observed

relationships. With these process models, oneis equipped to devise ways to systematicallyimprove the process using process optimizationconcepts. This is done by appropriately settingvalues of a subset of the process variables atlevels that would make another subset of pro-cess variables attain their desired values. Thischapter discusses process systems optimization.

A process systems optimization model [1–3]involves the minimization or maximization of afunction while keeping constraints withinacceptable limits. A fairly general optimizationmodel is given as

minx;u

jðz; y; uÞ

wiðz; y; uÞ ¼ 0 i ¼ 1; . . . ;m

cjðz; y; uÞ � 0 j ¼ 1; . . . ; p

z 2 Rnz y 2 f�n1; . . . ; n2gny u 2 Rnu ð1Þ

Here, z and y are vectors of continuous/discrete valued process variables, respectively.In several process systems models, each

� 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim10.1002/14356007.o22_o07

member of y tends to take the value of 0 or 1(y 2 f0; 1gm2 ) indicating the absence or pres-ence of a process unit. The vector of decisionvariables are given by u. Examples of processvariables are z ¼ ½concentration; temperature�Tand u ¼ ½feedconcentration; reactorsize�T . It isstraightforward to convert amaximization prob-lem into a minimization problem (and vice-versa). Specifically minw jðwÞ is equivalent tomaxwð�jðwÞÞ. If minimization is considered,there is no loss of generality.

The function jðz; y; uÞ is more commonlyreferred to as the performance objective (e.g.,profit); it is a measure of how good the design is.Sometimes there are several competing perfor-mance objectives as follows: fj1; . . .; jpg; insuch a case a new objective function is formu-lated from the available set, for example, byusing a weighted average of the set. A moregeneral combination of the individual objec-tives leads to a convolution operator [4]. Theset fwiðz; y; uÞg includes material and energybalances while the set fcjðz; y; uÞg may repre-sent process constraints such as safety, environ-mental limitations, and process specifications.Sometimes the model is more convenientlyexpressed as

minx

jðxÞ

wiðxÞ ¼ 0 i ¼ 1; . . . ;m

cjðxÞ � 0 j ¼ 1; . . . ; p

xL � x � xU ð2Þ

where x ¼ ½z; y; u� and the superscripts ½L;U�denote lower and upper bounds which are alsocalled box-constraints.

The strategy for the solution of the optimi-zation model may either result in a locallyoptimal solution or a globally optimal solution.

2. General Optimization Concepts

To be a viable optimization problem at the veryleast one degree of freedom is required, in otherwords nzþny > mþp (Eqs. 1, 2) and [5] for an indepth exposition of conditions under whichsolutions exist. For now it is assumed the opti-mization model has a solution.

There are many process optimization formu-lations that are multimodal, i.e., lead to multiplelocal optima. Sometimes one is interested infinding the best optima and global optimizationalgorithms have to be employed [6, 7]. Themethods that are discussed in this section arelocal optimization methods. A crude but practi-cal way to find or get close to global optima(using local optimization algorithms) is to runthe algorithm for multiple initial starting pointsand keeping the best solution. A good collectionof mostly free optimization software can befound at ! Mathematics in Chemical Engi-neering, Chap. 10 and the following web sites:

Local optimization: http://www.mat.univie.ac.at/�neum/glopt/software_l.html [8]

Global optimization: http://www.mat.uni-vie.ac.at/�neum/glopt/software_g.html [9]

Unconstrained Optimization. For uncon-strained optimization the equality and inequali-ty constraint sets fwig; fcjg and the box-constraints are all absent. For simplicity allvariables are assumed to be continuous (i.e.,the discrete variable y is absent). The explicitconsideration of discrete variables leads to com-binatorial optimization [10] and the uncon-strained optimization problem becomes

minx

jðxÞ ð3Þ

Several solution approaches have beendeveloped [11]. The necessary conditions foroptimality is obtained by setting the derivativeequal to zero as follows

!jðxÞ ¼ f ðxÞ ¼ 0 f 2 Rn ð4Þ

Gradient based iterative methods for solvingEquation (4) determine a new improved point as

xjþ1 ¼ xjþajsj ¼ xjþpj ð5Þ

where xj is the current point and aj determinesthe step length along the search direction sj,given as

sj ¼ �Gjgj

gj ¼!f ðxjÞ ð6Þ

Here, gj is the gradient of f ðxÞ at xj. Thechoice of the matrix Gj depends on the solutionstrategy. Common to most solution approachesfor unconstrained minimization is the followinggeneric algorithm 1:

2 Process Systems Engineering, 3. Mathematical Programming (Optimization)

. Step 1: Set j ¼ 0. Give an initial guess x0 andG0

. Step 2: If j > 0 determine Gj

. Step 3: Find the search directionsj ¼ �Gjgj gj ¼!f ðxjÞ

. Step 4: Perform a line search along sj; that is,determine aj such that f ðxjþajsjÞ < f ðxjÞ

. Step 5: STOP with the solution if k!fk � e

. Step 6: Set j ¼ jþ1. Go to Step 2

Line search methods are described in detailin [12]. If Gj ¼ I (the identity matrix) thenalgorithm 1 is the steepest descent method. IfGj ¼ ðHðxjÞÞ�1 (where HðxÞ is the Hessianof f ðxÞ), then the algorithm is the Newtonmethod.

The steepest descent method only requiresthe gradient of f ðxÞ and has a linear rate ofconvergence close to the solution. On the otherhand the Newton method has quadratic conver-gence rate close to the solution and is thereforefaster close to the solution of Equation (3).However, it requires second derivatives (Hes-sian) of f ðxÞ. To avoid the computational burdenof calculating the Hessian, quasi-Newton meth-ods were developed, in which some approxima-tion of the Hessian (or inverse Hessian) isemployed. Quasi-Newton methods have super-linear convergence rate close to the solution andrequire only first derivative information. Final-ly, steepest descent has global convergence (forconvex, differentiable functions) in the sensethat the method converges from any startingpoint as long as first derivatives can be calcu-lated with enough accuracy. In contrast theNewton method does not have global conver-gence for a number of reasons; one of these is thepossibility of HðxÞ becoming singular.

The main distinguishing feature of any of theabove methods is the technique for formingGðxÞ.

Newton Method. Consider a set ofequations described by the vector quantityhðxÞ ¼ ½h1ðxÞ; . . .; hmðxÞ�T ¼ 0, where the ex-pression from Equation (4) is a special case ofthis. For hðxÞ ¼ 0, the Jacobian is defined as

J ¼

qh1qx1

� � � qh1qxn

..

.} ..

.

qhmqx1

� � � qhmqxn

26666664

37777775

ð7Þ

Subsequently the update steps in the Newtonmethod can be written as

xjþ1 ¼ xjþajsj 0 � aj � 1

Jjsj ¼ �hj hj � hðxjÞ ð8Þ

In order to speed up the method in the earlystages of iteration and at the same time improveconvergence of the method close to the solu-tion, a good rule of thumb is to initially pick thestep length aj to be 1 and systematically reduceits value. The step length is based on

khjþ1k < khjk

khjk �ffiffiffiffiffiffiffiffiffiffiffiffiffiXni¼1

h2ij

sð9Þ

where hij is the i-th component of hj. In avicinity of the solution the algorithm has qua-dratic convergence; in other words if x* is thesolution of Equation (4), then

kxiþ1�xk � ckxi�xk2:

The Newton method has the followingproperties:

. The method requires a good initial approxi-mation x0. If the latter is bad the methodoften converges slowly or does not convergeat all

. It is necessary to solve at each iteration the setof linear equations in Equation (8) for thesearch direction, thus exacting a significantcomputational overhead

. There is a need to calculate the Jacobian ateach iteration

Step 3 leads to significant computationaloverhead at each iteration because quite often,the difference approximation is the only viableway to obtain estimates of the derivatives in theJacobian matrix; thus ðnþ1Þ evaluations of hðxÞare needed; on the other hand quasi-Newtonmethods have less computational burden at eachiteration since they do not require the explicitevaluation of the Jacobian.

Quasi-Newton Method. In the quasi-Newtonmethod xjþ1 is assumed to be known andEquation (7) is used to determine Jj. In Table 1differences in the Newton and quasi-Newtonmethods are shown.

Process Systems Engineering, 3. Mathematical Programming (Optimization) 3

The update steps in the quasi-Newtonmethod can be written as

sj ¼ �Gjhj ð10Þ

The updated Gj can be computed using theBroyden–Fletcher Goldfarb–Shanno (BFGS)[13] formula

Gjþ1 ¼ Gjþyjy

Tj

sTj yj�Gjsjs

Tj Gj

sTj Gjsj

yj ¼ hjþ1�hj ð11Þ

Constrained Optimization. Methods forsolving Equation (2) are considered. Let D bethe feasible region of the problem, thusD ¼ fx : fðxÞ ¼ 0;cðxÞ � 0g. The Lagrangefunction is defined as

L ¼ f ðxÞ�Xmi¼1

lifiþXpj¼1

mjcj ð12Þ

The Kuhn–Tucker necessary conditionsfor the minimum of L [12, 13] are given asfollows

qLqxi¼ 0 ð13Þ

mj � 0 j ¼ 1; . . . ; p ð14Þ

mjcj ¼ 0 j ¼ 1; . . .; p ð15Þ

wiðxÞ ¼ 0 i ¼ 1; . . .;m ð16Þ

cjðxÞ � 0 j ¼ 1; . . .; p ð17Þ

Let fx; f �g (where f ¼ f ðx�Þ) be the solutionof Equation (2). The equality in Equation (15)implies that

. If mj > 0 then cjðxÞ ¼ 0

. If cjðxÞ < 0 then mj ¼ 0.

Methods for solving the nonlinear programin Equation (2) are listed below:

1. Successive solution of unconstrained opti-mization methods (penalty methods [14],Langrange multipliers methods [15], andaugmented Lagrangian methods [16]

2. Successive solution of quadratic program-ming subproblems; this method is morecommonly referred to as sequential quadrat-ic programming (SQP) [1, 17] and! Math-ematics in Chemical Engineering, Section10.2

3. Successive solution of nonlinear opti-mization problems with linear constraints[18]

Langrange Multiplier Method. For consid-ering the Lagrange multiplier method [15] theduality function is introduced

hðl;mÞ ¼ minx

Lðx;l;mÞ ð18Þ

where hðl;mÞ gives a lower bound ofthe optimal value of the objective function inEquation (2). Therefore

hðl;mÞ � f ðxÞ ð19Þ

The following relation also holds

hðl;mÞ � Lðx;l;mÞ 8x 2 X; 8l; 8m > 0 ð20Þ

hence, in particular

hðl;mÞ � Lðx; l;mÞ ð21Þ

Under some convexity conditions the opti-mal objective function value from Equation (2)satisfies

f ¼ maxl;m�0

hðl;mÞ ¼ maxl;m�0

minx

Lðx; l;mÞ ð22Þ

Thus the constrained optimization problemis reduced to a two-level optimization proce-dure. At the first level the unconstrained opti-mization problem in Equation (18) is solved andat the second level a problemwith simple boundconstraints is solved as follows

maxl;m

hðl;mÞm � 0

ð23Þ

If Equation (18) has a single solution then thederivatives of hðl;mÞ are calculated as

qhqli¼ qL

qli¼ �fi xð Þ

qhqmj

¼ qLqmj

¼ cj x l;mð Þð Þ ð24Þ

Table 1. Comparison of Newton and quasi-Newton methods

xjþ1 Jj Goal

Newton unknown known determine xjþ1Quasi-Newton known unknown determine Jj


Penalty Method. The outer penalty meth-od [14] may be used for solving the problemsstated in Equation (2); the following auxiliaryfunction results

Fðx; ji; g jÞ ¼ fþXmi¼1

jif2iþ

Xpj¼1

g jQðcjÞ ð25Þ

where ji 1; g j 1 and QðcjÞ is given by

QðcjÞ ¼0; if cj � 0

c2j if cj > 0:

(

Looking for the minimum of Fðx; ji; g jÞwithrespect to x one can show that under certainconditions the solution of the unconstrainedproblem

minx

Fðx; ji; g jÞ ð26Þ

will tend to the solution of Equation (2) asji!¥ and g j!¥ [14]. However, for large jiand g i the function Fðx; ji; g jÞ is ill condi-tioned [14]. To partially avoid this problem,minimization of Fðx; ji; g jÞ for relatively smallvalues of the penalty coefficients fji; g jg iscarried out. Subsequently, the penalty coeffi-cients are systematically increased and minimi-zation of Fðx; ji; g jÞ is repeated. Also, similar toof the Lagrange multiplier method, the proce-dure for solving Equation (2) is two-level. Spe-cifically, minimization of F with respect to xcorresponds to the first (lower) level and theincrement in the coefficients fji; g jg corre-sponds to the second (upper) level.

TheModified Lagrange FunctionMethod.For simplicity the case when inequality con-straints are absent is considered and the modi-fied Lagrange function is defined as

Fðx; l; jÞ ¼ LþjXmi¼1

w2i ð27Þ

where L is the original Lagrange function

L ¼ fþXmi¼1

liwi

Recalling that fx; f �g is the minimizer ofEquation (2) let l� be the corresponding vectorof Lagrange multipliers. At x� the necessarycondition for optimality of Equation (2) is givenby

rL � rfþXmi¼1

lirwi ¼ 0 ð28Þ

Note the following

rF ¼ rfþXmj¼1ðljþ2jfjÞrfj ð29Þ

Since at x* constraints of Equation (5) aremet then according to Equation (28)

rF ¼ rL ¼ 0 ð30Þ

One can show that for a sufficiently large andfinite j, theminimumofFwill correspond to theminimum of Equation (2) [16]. In this case,again, the constrained optimization problem inEquation (2) is reduced to a two-level procedurein which a sequence of unconstrained optimiza-tion problems is solved. Here, to the lower levelminimization of F corresponds

minx

Fðx; lðiÞ; jÞ ð31Þ

where lðiÞ is the i-th approximation of theLagrange multipliers and xi� is the solution ofthe problem. On the second level we mustchange l in order to obtain the solution ofEquation (2) which is done as follows:

First minimization of F for fixed l is per-formed. At the minimum point the followingcondition is met

rF ¼ 0 ð32Þ

Since at the solution of Equation (2),Equation (28) must be satisfied, then the follow-ing approximation for the Lagrange multiplierscan be used:

lðiþ1Þj ¼ l

ðiÞj þ2jwi*

j

wi*j ¼ wjðxi*Þ: ð33Þ

The modified Lagrange function methoddoes not have the drawbacks of either the pen-alty method or the method of Lagrange multi-pliers. In contrast to the method of Lagrangemultipliers it does not require convexity ofF inthe vicinity of the solution of Equation (2).Moreover, in contrast to the penalty methodthere is no need to systematically increase thepenalty coefficients towards infinity. In otherwords, in general F is not ill conditioned.

Sequential Quadratic Programming.Some text books also refer to sequentialquadratic programming (SQP) as successivequadratic programming [1]. Recalling that the


Lagrange function for Equation (2) is

L ¼ f ðxÞ�lTfþmTc ð34Þ

the following definitions are needed

xiðx; l;mÞ ¼qLqxi

; i ¼ 1; . . . ; n

wiðxÞ; i ¼ ðnþ1Þ; . . . ; ðnþmÞ

8>><>>:

The vector function x ¼ ðxi;x2 . . . xnþmÞcan be written as

xðx; l;mÞ ¼ rf�JT1 lþJT2 mfðxÞ

(ð35Þ

where J1 and J2 are Jacobian matrices

J1 ¼ qðwi; . . .;wmÞqðx1; . . .; xnÞ ¼

qw1

qx1; . . . ;

qw1

qxn

..

. ...

qwm

qx1; . . . ;

qwm

qxn

0BBBBBB@

1CCCCCCA

ð36Þ

Example:

form ¼ 2 and n ¼ 3;

J1 ¼ qðw1;w2Þqðx1; x2; x3Þ ¼

qw1

qx1

qw1

qx2

qw1

qx3qw2

qx1

qw2

qx2

qw2

qx3

0BB@

1CCA

J2 ¼qðc1; . . .;cpÞqðx1; . . .; xnÞ ¼

qc1

qx1; . . . ;

qc1

qxn

..

. ...

qcp

qx1; . . . ;

qcp

qxn

0BBBBBB@

1CCCCCCA

ð37Þ

Using theKarush–Kuhn–Tucker (KKT) nec-essary optimization conditions [19] and !Mathematics in Chemical Engineering, Section10.2 one obtains

xðx; l;mÞ ¼ 0

diagðcÞm ¼ 0

cðxÞ � 0

m � 0 ð38Þ

Here, m ¼ ðm1; . . . ;mpÞT , diag(c) is a diag-onal matrix with elements fc1; . . . ;cpg. Thusfor solving the problem in Equation (2), it isnecessary to find the solution of a system of (nþm þ p) nonlinear equations and 2p inequalitieswith respect to (n þ m þ p) variables fx; l;mg.

If Equation (2) does not contain inequalitiesthen theKKT conditions are reduced to a systemof nonlinear equations with (n þ m) variablesfx; lg which can be solved using, for example,the Newton method. In general, the presence ofthe inequalities complicates the problem.

Now an extension of Newton’s method forsolving a set of nonlinear equations is consid-ered. Let fx�; l�;m�g be the solution of Equa-tion (38) and f�x; �l; �mg be values at some itera-tion. Furthermore, the point f�x; �l; �mg is suffi-ciently close to fx�; l�;m�g such that for smallvalues fDx;Dl;Dmgx� ¼ �xþDx; l� ¼ �lþDl; m� ¼ �mþDm ð39Þ

Let g ¼ ðx1; . . . ; xnþm; c1; . . . ;cpÞ andz ¼ ðx1; . . . ; xn; l1; . . . ; lm; m1; . . . ;mpÞthen the Jacobian matrix is of the form

J ¼ qgqz¼ qðx1; . . . ; xnþm; c1; . . . ;cpÞ

qðx1; . . . ; xn; l1; . . . ; lm; m1; . . . ;mpÞð40Þ

In the matrix J, the first n þ m rows corre-spond to the functions x1,. . .,xn þ m and the lastrows with indices (n þ m þ 1, . . ., n þ m þ p)correspond to the functions c1; . . . ;cp. The firstn columns correspond to x, the following mcolumns correspond to l, and the last p columnscorrespond tom. Thematrix J can be representedin the following block form

J ¼

qðx1; . . . ; xnÞqðx1; . . . ; xnÞ

qðx1; . . . ; xnÞqðl1; . . . ; lmÞ

qðx1; . . . ;xnÞqðm1; . . . ;mpÞ

qðxnþ1; . . . ; xnþmÞqðx1; . . . ; xnÞ

qðxnþ1; . . . ; xnþmÞqðl1; . . . ; lmÞ

qðxnþ1; . . . ; xnþmÞqðm1; . . . ;mpÞ

qðc1; . . . ;cpÞqðx1; . . . ; xnÞ

qðc1; . . . ;cpÞqðl1; . . . ; lmÞ

qðc1; . . . ;cpÞqðm1; . . . ;mpÞ

0BBBBBBBB@

1CCCCCCCCA

ð41Þ

Using the fact that f and c do not depend onfl;mg, J takes the form

J ¼G �JT1 þJT2J1 0 0

J2 0 0

0B@

1CA ð42Þ

where G is the Hessian (with respect to x) of L(G ¼ r2L). Now substitute fx�; l�;m�g fromEquation (39) into the system Equation (38) andcarrying out a second-order Taylor seriesexpansion of the left-hand side of the system


Equation (38), this system is reduced to

J

DxDlDm

0B@

1CA ¼

�rLð�x; �l; �mÞ�fð�xÞ��cð�xÞ

0B@

1CA ð43Þ

and

diagðcþdcÞðmþDxÞ ¼ 0

�mþdm � 0 ð44Þ

where dc ¼ J2Dx.In order to determine fDx;Dl;Dmg it is

necessary to solve the above system of linearinequalities. One approach to solving the prob-lem is to consider the following quadraticprogramming problem:

miny

gT yþ 1

2yTGy

� �

J1yþ�f ¼ 0

J2yþ�c � 0 ð45Þ

where �w ¼ wð�xÞ; �c ¼ cð�xÞ; g ¼ rf . TheLagrange function for this problem is

L_ ¼ gTyþ 1

2yTGy�vT ðJ1yþ�wÞþuT ðJ2yþ�cÞ

where n and u are Lagrange multipliers corre-sponding to the equality and inequality con-straints. The relevant KKT conditionsare

ry L_ ¼ gþGy�JT1 vþJT2 u ¼ 0

J1yþ�w ¼ 0

J2yþ�c � 0

diagðJ2yþ�cÞu ¼ 0

u � 0 ð46Þ

Substituting the quantities y ¼ Dx;v ¼ �lþDl; u ¼ �mþDm into Equation (46), oneobtains Equation (43). Thus fDx;Dl;Dmg canbe obtained by solving the quadratic program-ming (QP) problem to obtain a new point in thespace of x and so on. It should be noted that sucha procedure could diverge just like the Newtonmethod. Therefore, a modification similar to the

modified Newton method can be used. Specifi-cally, starting from f�x; �l; �mg one can do a linesearch in the space of the variables fx;l;mgalong the direction p ¼ fDx;Dl;Dmg. There areother line search strategies, e.g., based on aminimization of the norm of Lðx; l;mÞ. TheArmijo line search has been used extensively [1]with great success.

In the following an SQP algorithm for Equa-tion (4) is outlined:

. Step 0: Initialize z, iteration counter i 0, andother initializations

. Step 1: At the i-th iteration form the LagrangefunctionLðiÞ ¼ f ðxÞ�ðli�1ÞTfðxÞþðmi�1ÞTcðxÞwhere li�1;mi�1 are Lagrange multipliers

. Step 2: Find the Jacobian for f and c, thegradient rf and the Hessian r2LðiÞ

. Step 3: Solve the QP in Equation (45) to yielda new search direction pi

. Step 4: Do a line search along pi and findnext iteration point zi ¼ zi�1þaipi

(zi � fxi; li;mig), where ai is the searchlength

. Step 5: Check the termination criterion. If theKarush–Kuhn–Tucker error is less than thedesired tolerance e, STOP with the optimalsolution

. Step 6: Set i iþ1 and go to Step 1

The drawbacks of this variant of the SQPalgorithm consist in:

1. At each search point the Hessian (G � r2L)needs to be evaluated

2. r2L may not be positive semi-definite,which means the search direction may notbe a descent direction (i.e. will not lead to theminimum of Equation (4))

To overcome the first drawback one canemploy a quasi-Newton approximation of theHessian, given by the BFGS relations

Bi ¼ Bi�1þ yiyTi

yTi si�Bi�1sisTi Bi�1

sTi Bi�1sið47Þ

where i is an iteration counter, Bi is an approxi-mation of r2L. That is employing G � Bi in-stead of G � r2L. Here yi ¼ ðrLi�rLi�1Þ,si ¼ ðxi�xi�1Þ, and Bi is positive definite if


Bi�1 is positive definite [17] leading to

yTi si ¼ sTi yi > 0 ð48Þ

Consider, for example, the quadratic func-tion y ¼ zTAzþbTzþc. In this case yi ¼ Az andsTi yi ¼ sTi Asi. If the quadratic function is posi-tive definite then A> 0 and Equation (26) holdsfor all search points. Since r2L can be indefi-nite, one cannot ensure satisfaction of Equa-tion (48) at all search points. Therefore, insteadof the transformation Equation (47), one canemploy the following transformation [17]

Bi ¼ Bi�1þwiwTi

wTi si�Bi�1sisTi Bi�1

sTi Bi�1sið49Þ

where

w ¼ uyiþð1�uÞBi�1si

u ¼1 if sTi yi � 0:2 siBi�1si

0:8sTi Bi�1si

sTi Bi�1si�sTi yiif sTi yi < 0:2sTi Bi�1si

8><>: ð50Þ

It should be noted that Bi obtained fromEquation (49) will be positive definite if Bi�1is positive definite.

3. Global Terrain Method

The global terrain method [20, 21] is a deter-ministic method of global optimization. Thebasic idea behind the terrain method is to intel-ligently explore the terrain of an objective func-tion by following valleys. The terrain methoddoes this by moving up and down the objectivefunction surface using well known numericalmethods for each of its basic tasks (i.e., movingdownhill, moving uphill, accelerating conver-gence to singular points) and by using eigenval-ue and eigenvector information to decide whatto do next. The novel aspects of the globalterrain method include (1) the use of well estab-lished building blocks (numerical methods suchas Newton’s method, a trust region strategy, anSQP method, etc.) assembled in a unique man-ner, and (2) a novel SQP corrector formulationand algorithm for returning iterates to valleys.

The terrain method assumes that the objec-tive function, j, is a twice continuously differ-entiable of unknown variables z 2 Rn with agradient given by g ¼ gðzÞ and aHessianmatrixor matrix of second partial derivatives denotedbyH ¼ HðzÞ. It is also assumed that the feasible

region is defined by upper and lower bounds onvariables of the form

zLi � zi � zUi i ¼ 1; . . .; n ð51Þ

where zLi and zUi are the lower and upper bounds

on the ith variable, respectively. Other con-straints (e.g., linear mass balance constraints,etc.) are handled using projection orelimination.

It is easiest to view the terrain method byconsidering the nonlinear least squares problem.Let j ¼ jðzÞ and J ¼ JðzÞ denote the Jacobianmatrix of the function j; thus the nonlinear leastsquares objective function is jTj. The appropri-ate global optimization problem then becomesthat of finding all stationary and singular pointsof

jTj subject to zLi � zi � zUi i ¼ 1; . . .; n ð52Þ

The corresponding gradient of jTj is

g ¼ gðzÞ ¼ JTj ð53Þ

The corresponding Hessian matrix, HðzÞ, ofj is given by

H ¼ HðzÞ ¼ JTJþX

jir2ji ð54Þ

wherer2ji are the element Hessian matrices ofthe component functions, ji.

The global terrain method is comprised offive basic steps:

1. Downhill movement2. Uphill movement3. Second-order acceleration to singular points4. Eigenvalue–eigenvector decomposition5. Termination criterion

Downhill Movement. Downhill movementin the terrain method is accomplished using atrust region strategy [22] and given by

D ¼ �bDNþðb�1Þg ð55Þ

where DN ¼ J�1j is the Newton step and whereb e ½0; 1� is determined by the following simplerules. If kDNk � R, then b ¼ 1, where R is thetrust region radius. If kDNk > R and kgk � R,then b ¼ 0. Otherwise, b is the unique value on[0,1] that satisfies kDk ¼ R. The new iterate isaccepted if it reduces kjk. Otherwise, the newiterate is rejected, the trust region radius isreduced and the calculations are repeated untila reduction in kgk occurs. Furthermore, if


during downhill movement, kgk=kDNk � §,then quadratic acceleration is used. (See Eq. 59below).

Downhill movement is terminated wheneither kjk � e or kgk � e where e is a conver-gence tolerance and can result in convergence toa minimum, saddle point, or singular point ofjTj.

Uphill Movement. Uphill movement con-sists of predictor–corrector steps. Uphill move-ment uses Newton predictor steps which candrift from the valley because of approximation.To correct for this drift, intermittent SQP cor-rector steps are used to force iterates back to thevalley.

Predictor Steps: Uphill predictor steps areNewton steps, DzN , given by

DzN ¼ apJ�1j ð56Þ

where ap is a step size such that 0 < ap � 1,which can either remain fixed or be adjustedusing available information. Uphill Newtonsteps tend to follow valleys reasonably well butdo drift some. Therefore, corrector steps areused intermittently to return iterates to thecurrent valley and are invoked when the angletest

u ¼ 57:295 arccos ½ðDTNvÞ=ðkDNkkvkÞ� � Q ð57Þ

is satisfied, where v is the current estimate of theeigenvector associated with the smallest posi-tive eigenvalue of H and Q is 5 degrees.

Corrector Steps: Corrector steps are comput-ed by solving the following nonlinear program-ming problem

gTg subject to jTj ¼ L ð58Þ

where L denotes the current value of the objec-tive function contour (or level set).

Second-Order Acceleration. Accelerationis needed to converge quickly to singular pointsand for this second-order Newton steps, givenby

DzN2 ¼ �H�1g ð59Þ

are used.

Eigenvalue–Eigenvector Decomposition.After a stationary point of the objective function

is found, the terrain method uses an eigen-decomposition of the Hessian matrix of theobjective function to characterize the stationarypoint as a minimum, saddle point, or singularpoint and to decide what to do next. If H has allnonnegative eigenvalues, then the current sta-tionary point is a minimum and uphill move-ment is indicated. Uphill movement from a localminimum or global minimum of the least-squares function always takes place along adirection of smallest positive curvature. If, onthe other hand, H has one or more negativeeigenvalues, then the current stationary point isa saddle point and downhill exploration of theobjective function landscape is indicated next.Downhill movement from a saddle point toeither a singular point of lower norm or asolution always takes place along an eigen-direction, n, of negative curvature.

However, it is important to understand thatonly a few eigenvalues and eigenvectors areneeded. For these calculations we use the in-verse power method together with incompletefactorization to compute a few eigenvalues andeigenvectors. Also we never actually formmatrix products like HTH to avoid roundingerrors.

Termination. Termination is based on thenumber of times the terrain method strikes theboundary of the feasible region. Collisions witha boundary of the feasible region are used tosignal an end to the usefulness of exploration ina particular eigen-direction. LUCIA andYANG [20, 21] base their termination criterionon something they call limited connectedness.This means that stationary points are really onlyconnected to neighboring stationary pointsalong specific eigen-directions. It is assumedthat the number of important connections be-tween stationary points along valleys and ridgesis limited to four or less and is related to domi-nant geometric distortions (i.e.,þ/- the smallestpositive eigen-direction and þ/- the most nega-tive eigen-direction) caused by the strongest‘attractions’ between neighboring stationarypoints. This makes it possible to dynamicallycatalogue connections in a set, C, and to con-clude that all of the important connectionsbetween stationary points have been exploredwhenC is empty. Thus termination occurs whenC is the empty set.


3.1. Simple Example of the GlobalTerrain Method

A two-dimensional example of a nonlinearcontinuous stirred tank reactor (CSTR) takenfrom [23] is used to illustrate the basic featuresof the global terrain method. Figure 1 shows thecontours of the least squares objective functionand the stationary points and their connected-ness for this example. There is a low lying valleythat runs diagonally across the bottom of thefigure.

Problem Statement. The equations that de-scribe the behavior of the CSTR are the mass-balance equation

j1ðx; TÞ ¼ 120x�75kð1�xÞ ¼ 0 ð60Þ

and the energy-balance equation

j2ðx; TÞ ¼ �xð873�TÞþ11ðT�300Þ ¼ 0 ð61Þ

where x denotes the conversion, T is thereactor temperature in Kelvin and k, thereaction rate constant, which is given byk ¼ 0:12 exp ½12581ðT� 298Þ=ð298TÞ�. Thebounds on conversion and temperature for thisexample are

0 � x � 1 and 300 � T � 400 ð62Þ

Problem and Solver Attributes. All first andsecond partial derivatives are calculated analyt-ically. The equations are solved to an accuracyof jTj � 10�16. The maximum number ofdownhill steps for each downhill subproblemis set to 50. The initial trust region radius isD ¼ 10 and downhill Newton steps are adjustedautomatically using the rule

kDzNk < 0:1D; then ad ¼ minð1; 2adÞ ð63Þ

where ad is the downhill Newton step size.Otherwise, ad remains the same. The step sizefor uphill Newton predictor steps is fixed atap ¼ 0:05 and the maximum number of uphillNewton predictor steps is 10. After 10 predictorsteps, corrector steps are used to force iteratesback to the valley if the angle test indicates drift.The maximum number of corrector steps percorrector subproblem is 50. Acceleration occursif either the ratio of the Newton step to thegradient step is less than 10�6 or the Newtonstep reverses itself during uphill movement.The maximum number of acceleration steps isset to 50.

Problem Solution. Sufficient numerical de-tails for all subproblems are given so that read-ers interested in implementing the global terrainmethod can validate their implementation ofindividual components (downhill trust region,

Figure 1. Connectedness of stationary points for a nonlinear CSTRa) Local minimum; b) Saddle point; c) Solution


second-order acceleration, uphill predictor–cor-rector steps, eigenvalue–eigenvector computa-tions, etc.) of the terrain method. The startingvalue for the unknown variables is x ¼ 0:2 andT ¼ 328 K.

Downhill Steps (First Set). The initial set ofdownhill steps for this example is shown inTable 2.

All steps in Table 2 are Newton steps andnote that the value of jTj is monotonicallydecreasing.

Acceleration occurs after 20 downhill stepsbecause the ratio of the Newton step to thegradient step becomes small. Table 3 shows thesecond-order Newton acceleration steps con-verge to a local minimum of jTj. The localminimum isz� ¼ ðx; TÞ ¼ ð0:094595; 304:677 KÞ and indi-cated by the fact that gTg � 10�8. Note jTj andthus j is not zero! However, gTg is approxi-mately zero and this implies that g ¼ JTj ¼ 0.Since j „ 0, this means JT and therefore J issingular.

The eigenvalues and eigenvectors of theHessian matrix, H, at this local minimum areall positive and shown in Table 4.

Uphill Steps (First Set). Uphill stepare initiated in the eigen-direction

Dz ¼ ð�0:019607;�0:999808Þ and strike theboundary of the feasible region atz ¼ ð0:038036; 300 KÞ in 6 iterations. Uphillsteps from the initial stationary point are subse-quently initiated in the eigen-directionv ¼ ð�0:019607;�0:999808Þ and take 30 up-hill predictors step followed by 9 correctoriterations to get back to the valley. This is thenfollowed 5 uphill steps where Newton predictorsteps go through a maximum in jTj and triggeracceleration again, this time to a saddle point.

The acceleration steps to the saddle point areshown in Table 5. Note that norm reductioncannot be imposed on acceleration steps. Thesaddle point is z� ¼ ð0:774548; 331:507 KÞ.The eigenvalues and eigenvectors of H at thesaddle point are given in Table 6.

The negative eigenvalue of H indicates thatthe stationary point is a saddle point and that the

Table 2. First set of downhill steps for CSTR example

Iteration x T, K xTx

0 0.2 328.0 1.40874 1051 0.214034 327.795 1.27280 105

2 0.239982 327.370 1.03586 105

3 0.283752 326.468 6.77424 104

4 0.341531 324.443 2.73817 104

5 0.350459 322.000 1.12087 104

6 0.331081 319.336 4.62678 103

7 0.296777 316.622 1.92990 103

8 0.254796 313.927 8.19848 102

9 0.207496 311.195 3.61077 102

10 0.151637 308.161 1.73688 102

11 0.063339 303.541 1.21009 102

12 0.063339 303.541 1.21009 102

13 0.068223 303.791 1.19814 102

14 0.073105 304.041 1.18948 102

15 0.077984 304.290 1.18417 102

16 0.079702 304.079 1.18313 102

17 0.080100 304.399 1.18295 102

18 0.080182 304.403 1.18292 102

19 0.080198 304.404 1.18291 102

20 0.080201 304.404 1.18291 102

Table 3. Second-order acceleration to local minimum of CSTR

example


0 0.080201 304.404 1.18291 102

1 0.081526 304.426 1.12931 102

2 0.083945 304.466 1.04701 102

3 0.087935 304.537 9.53293 101

4 0.093037 304.638 9.02564 101

5 0.094578 304.676 9.00427 101

6 0.094595 304.677 9.00427 101

Table 4. Eigenvalues and eigenvectors of H at local minimum

Eigenvalues Eigenvectors

2.10713 (0.019607, 0.999808)

3.43487 105 (0.999808, –0.019607)

Table 5. Second-order acceleration to saddle point for CSTR

example


0 0.755706 331.709 5.71861 103

1 0.806247 334.907 9.89472 103

2 0.804901 334.682 9.55795 103

3 0.802153 334.253 9.02562 103

4 0.796517 333.474 8.38298 103

5 0.785435 332.254 7.99690 103

6 0.777929 331.711 7.98447 103

7 0.774669 331.512 7.98500 103

8 0.774548 331.507 7.98501 103

9 0.774548 331.507 7.98501 103


next move should be downhill from the saddle.Also because the location of the first stationarypoint is known, it is straightforward to deter-mine that the correct movement downhill is in adirection away from the local minimum.

Downhill Steps (Second Set). Next theglobal terrain method initiates downhill move-ment in the direction v ¼ ð0:026170; 0:999658Þscaled by 1/l. Table 7 shows the steps of thesecond downhill subproblem.

All steps in Table 7 are Newton steps.Also note that jTj decreases monotonicallyand the asymptotic rate of convergenceis quadratic to the global minimum

z� ¼ ð0:963868; 346:164 KÞ. The eigenvaluesand eigenvectors of H at the global minimumare shown in Table 8.

The eigenvalues in Table 8 show that the lastentry in Table 7 is a global minimum of jTj andthat the next exploration should be uphill ex-ploration of the least squares objective functionsurface.

Uphill Steps (Second Set). Uphill move-ment from the globalminimum is initiated in theeigen-direction v ¼ ð0:0041241; 0:999991Þ.The global terrain uses an additional 291 func-tion and gradient evaluations, taking 10 uphillpredictor steps and then correcting along thevalley before it strikes the boundary atz ¼ ð0:999752; 400:0 KÞ and terminates be-cause it has encountered the boundary twice.

Table 9 gives a summary of the computa-tional work needed by the global terrain methodon the example problem measured in terms ofiterations and function and gradient evaluationsfor the downhill, uphill (predictor and correctorsteps), acceleration, and eigenvalue–eigenvec-tor computations. Also shown is the number offunction and gradient evaluations need to gofrom a stationary point to a boundary of thefeasible region.

The time required for all of the computationsshown in Table 9, including input and output, is0.16 s on a 3.6 GHz Dell High Precision Work-station using the Lahey–Fijitsu LF95 compiler.

3.2. General Optimization Problems

Up to this point the discussion has focused onnonlinear least squares problems. To use theglobal terrain method for any general objectivefunction, j, it is only necessary to recognize thatthe equations to be solved become

gðxÞ ¼ 0 ð64Þ

and therefore j should be replaced with g, J of jwith H of j, and r2ji with r2Gi in the

Table 6. Eigenvalues and eigenvectors of H at saddle point


�92.7333 (0.026170, 0.999658)

8.7427 105 (0.999658, �0.026170)

Table 7. Second set of downhill steps for CSTR example


0 0.785408 331.922 7.96902 103

1 0.842049 334.605 7.13659 103

2 0.866098 336.005 6.28044 103

3 0.880198 336.934 5.60146 103

4 0.890527 337.676 5.01492 103

5 0.898670 338.305 4.49846 103

6 0.905347 338.853 4.03999 103

7 0.910963 339.341 3.63120E 103

8 0.915770 339.781 3.26572 103

9 0.919942 340.180 2.93837 103

10 0.923599 340.546 2.64479 103

11 0.926833 340.882 2.38125 103

12 0.929713 341.193 2.14450 103

13 0.932293 341.481 1.93168 103

14 0.934615 341.749 1.74030 103

15 0.938816 342.248 1.40007 103

16 0.942257 342.682 1.12769 103

17 0.945116 343.062 9.09132 102

18 0.947518 343.396 7.33456 102

19 0.949555 343.691 5.92067 102

20 0.951295 343.952 4.78159 102

21 0.954287 344.416 3.02492 102

22 0.956483 344.778 1.91955 102

23 0.958128 345.063 1.2207110224 0.950378 345.288 7.77474 101

25 0.961299 345.644 2.74419 101

26 0.962368 345.854 9.77472 100

27 0.963596 346.103 3.66418 10-1

28 0.963869 346.164 1.10322 10-5

29 0.963868 346.164 4.01405 10-16

30 0.963868 346.164 1.23355 10-25

Table 8. Eigenvalues and eigenvectors of H at the global minimum


98.2769 (0.0041241, 0.999991)

1.13079 107 (0.999991, �0.0041241)


right-hand sides of all equations presented [24].Finite difference derivatives can be used inplace of analytical derivatives and often leadto successful convergence.

3.3. Advanced Techniques

Advanced techniques such as those that handleintegral path bifurcations and nondifferentiabil-ity of the objective function can be found in [24,25]. Integral path bifurcations can be tangent orpitchfork bifurcations and techniques for deal-ing with this behavior monitor and exploitGauss curvature. Nondifferentiability of theobjective function can occur in process engi-neering problems in which model switching canoccur (e.g., in phase stability computationswhere different phases are modeled using dif-ferent phase models).

3.4. Applications and Extensions ofthe Terrain Method

There are many process engineering examplesdescribed in [24, 25] which include finding

1. The glass temperature of a polymer mixture2. Equilibrium structures to nanostructured

materials3. Roots to the SAFT equation4. All azeotropes of a binary mixture5. Equilibrium phase compositions and pres-

sure for a retrograde flash6. The temperature and conversion for nonadi-

abatic CSTRs with isola7. All solutions to heterogeneous distillation

problems8. Phase split and phase stability calculations

nonideal mixtures9. Molecular conformation of small molecules

More recentlyLUCIA et al. [26] have extendedthe global terrain method by combining it withlogarithmic barrier functions to solve heat andmass transfer in a catalyst pellet modeled usingcollocation over finite elements.

4. Optimization under Uncertaintyin Process Systems Engineering

The deterministic optimization literature clas-sifies problems as linear programming (LP),nonlinear programming (NLP), integer pro-gramming (IP), mixed integer LP (MILP), andmixed integer NLP (MINLP), depending on thedecision variables, objectives, and constraints! Mathematics in Chemical Engineering, Sec-tion 10.1. However, the future cannot be per-fectly forecast but instead should be consideredrandomor uncertain. Optimization under uncer-tainty refers to this branch of optimizationwhere there are uncertainties involved in thedata or the model, and is popularly known asstochastic programming or stochastic optimiza-tion problems. In this terminology, stochasticrefers to the randomness, and programmingrefers to the mathematical programming tech-niques such as LP, NLP, IP,MILP, andMINLP.In discrete (IP, MILP, MINLP) optimization,there are probabilistic techniques like simulatedannealing and genetic algorithms; these tech-niques are sometimes referred to as the stochas-tic optimization techniques because of the prob-abilistic nature of the method. In general, how-ever, stochastic programming and stochasticoptimization involve optimal decision makingunder uncertainties. Parametric programmingwhich is a method based on sensitivity analysisalso involves consideration of uncertainties andprovides how optimal decisions and designsvary with uncertainties.

Table 9. Summary of computational work for global terrain method on example problem

Stationary point Value Iterations (function and gradient evaluations)

x, T, K downhill predictor corrector acceleration eigens

Local minimum (0.094595, 304.677) 20 (20) n.a. n.a. 6 (6) 8 (8)

Saddle (0.774548, 331.507) n.a. 35 (35) 9 (36) 9 (9) 8 (8)

Global minimum (0.963868, 346.164) 30 (30) n.a. n.a. n.a. 8 (8)

Boundary 6 (6) 164 21 (119) n.a. 8 (8)

n.a. ¼ not applicable.


The need for including uncertainty in com-plex decision models arose early in the historyof mathematical programming. The first modelforms, involving action followed byobservationand reaction (or recourse), appeared in the1950s [27, 28]. The commonly used exampleof a recourse problem is the news vendor prob-lem described below [29]. The news vendor ornews boy problem has a rich history that hasbeen traced back to the economist EDGE-

WORTH [30], who applied a variance to a bankcash-flow, problem. However, it was not untilthe 1950s that this problem, likemany otherOR/MS models seeded by the war effort, became atopic of serious and extensive study byacademicians [31].

Example: The simplest form of a stochasticprogrammay be the news vendor (also known asthe newsboy) problem. In the news vendorproblem, the vendor must determine how manypapers (x) to buy now at the cost of c cents for ademand which is uncertain. The selling price issp cents per paper. For a specific problem,whose weekly demand is shown below, the costof each paper is c¼ 20 cents and the selling priceis sp ¼ 25 cents. Suppose we need to solve theproblem assuming the news vendor knows thedemand uncertainties (Table 10) but does notknow the demand curve for the coming week(Table 11) a priori. Assume no salvage value s¼0, so that any papers bought in excess of demandare simply discarded with no return.

Solution: In this problem, the question is howmany papers the vendor must buy (x) to maxi-mize the profit. Let r be the effective sales andwbe the excess which is going to be thrown away.This problem falls under the category of sto-chastic programming with recourse where thereis action (x), followed by observation (profit),and reaction (or recourse) (r and w). The firstidea to solve this problem is to find the averagedemand and the optimal supply x correspondingto this demand. Since the average demand fromthe Table 10 is 70 papers, x ¼ 70 should be thesolution. However, with this solution wheresupply is 70 papers per day, the news vendorwill be making a loss of 50 cents per week. Thisprobably is not the optimal solution. Can we dobetter? For that one needs to propagate theuncertainty in the demand to see the effect ofuncertainty on the objective function and thenfind the optimum value of x. The above infor-mation can be transformed for daily profit asfollows.

Profit ¼ �cxþ 5

7sp d1þ 1

7spxþ 1

7sp x ¼ �cxþ 5

7sp d1þ 2

7sp x

if d1 � x � d2

ð65Þ

or

Profit ¼ �cxþ 5

7sp d1þ 1

7sp d2þ 1

7sp x if d2 � x � d3 ð66Þ

Notice that the problem represents two equa-tions for the objective function (Eqs. 65 and 66);this results in an objective function that is adiscontinuous function and is therefore no lon-ger a LP. The optimal solution to this problem isx ¼ d1 ¼ 50 providing the news vendor with anoptimum profit of 1750 cents per week.

The difference between taking the averagevalue of the uncertain variable as the solution ascompared to using stochastic analysis (propa-gating the uncertainties through the model andfinding the effect on the objective function asabove) is defined as the value of stochasticsolution, VSS. If we take the average value ofthe demand, i.e., x ¼ 70 as the solution, weobtain a loss of 50 cents per week, therefore, thevalue of stochastic solution, VSS, is 1750 –(�50) ¼ 1800 cents per week.

Now consider the case, where the vendorknows the exact demand (Table 11) a priori.This is the perfect information problem where

Table 10. Uncertainties in demand

j Demand, dj Probability, Pj

1 50 5/7

2 100 1/7

3 140 1/7

Table 11. Weekly demand

Day Demand

Monday 50

Tuesday 50

Wednesday 50

Thursday 50

Friday 50

Saturday 100

Sunday 140


the solution xi for each day i has to be found. Theproblem could be described in terms of xi.

Maximizexi

profit ¼ �cxiþsalesðr;w; diÞ

subject to

salesðr;w; diÞ ¼ sp riþswi

ri ¼ minðxi; diÞ ¼xi; if xi � di

di; if xi > di

(

wi ¼ maxðxi�di; 0Þ ¼0; if xi � di

xi�di; if xi > di

(ð67Þ

Here each problem (for each i) need to besolved separately, leading to the followingdecisions shown in Table 12.

One can see that the difference between thetwo values, (1) when the news vendor has theperfect information (2450 cents per week) and(2) when he does not have the perfect informa-tion (1750 cents per week) but can represent itusing probabilistic functions, is the expectedvalue of perfect information, EVPI. The latteris 700 cents per week for this problem.

Maximizing expected profit given that thereare uncertainties associated with demand andsupply is a classic problem in any manufactur-ing industry including the chemical indus-try [29]. However, decision making in chemicalmanufacturing is not restricted to just supplyand demand cost minimization problems butinvolves models that are linear, nonlinear, andmixed integer in nature.

4.1. Methods and Algorithms

The literature on optimization under uncertain-ties very often divides the problems into

categories such as ‘‘wait and see’’, ‘‘here andnow’’, and ‘‘chance constrained optimiza-tion’’ [32, 33]. In ‘‘wait and see’’ one waits untilan observation is made on the random elements,and then solves the (deterministic problem).This is the ‘‘wait and see’’ problem of MADANS-

KY [34], originally called ‘‘stochastic program-ming’’ [35], is not in a sense one of decisionanalysis. In decision making, the decisions haveto be made ‘‘here and now’’ about the activitylevels. The ‘‘here and now’’ problem involvesoptimization over some probabilistic measure—usually the expected value.By this definition,chance constrained optimization problems canbe included in this particular category of opti-mization under uncertainty. Chance constrainedoptimization involves constraints which are notexpected to be always satisfied; only in a pro-portion of cases, or ‘‘with given probabilities’’.‘‘Parametric programming’’ is based on sensi-tivity analysis of optimal solution and is similarto the ‘‘wait and see’’ problems. These variouscategories require different methods for obtain-ing their solutions. It should be noted that manyproblems have both here and now, and wait andsee problems embedded in them.

Here and Now Problems. The ‘‘here andnow’’ problems require that the objective func-tion and constraints be expressed in terms ofsome probabilistic representation (e.g., ex-pected value, variance, fractiles, or most likelyvalues). For example, in chance constrainedprogramming, the objective function is ex-pressed in terms of expected value, while theconstraints are expressed in terms of fractiles(probability of constraint violation), and inTaguchi’s off-line quality control method[36, 37], the objective is to minimize variance.These problems can be classified as ‘‘here andnow’’ problems.

Is it possible to propagate the uncertaintyusing moments of input uncertainties (such asmean and variance) thereby obtaining a deter-ministic representation of the problem? This isthe basis of the chance constrained program-ming method, developed very early in the his-tory of optimization under uncertainty, princi-pally byCHARNES andCOOPER [38]. In the chanceconstrained programming (CCP) method, someof the constraints likely need not hold as wasassumed in earlier problems. CCPs can be

Table 12. Supply and profit

Supply Profit

50 250

50 250

50 250

50 250

50 250

100 500

140 700


represented as follows:

Optimize jðx; uÞ ¼ P1ðjðx; uÞÞ ¼ Eðzðx; uÞÞ ð68Þ

PðgðxÞ � uÞ � a ð69Þ

In the above formulation, Equation (69) is thechance constraint. In chance constraint formu-lation, this constraint (or constraints) is (are)converted into a deterministic equivalent underthe assumption that the distribution of the un-certain variables, u, is a stable distribution [39].Normal, Cauchy, Uniform, and Chi-square areall stable distributions that allow the conversionof probabilistic constraints into deterministicones. The deterministic constraints are in termsof moments of the uncertain variable u (inputuncertainties).

MARANAS et al. [40, 41] have used chanceconstraint programming to solve (1) a chemicalsynthesis problem of polymer design and (2) ametabolic pathways synthesis problem in bio-chemical reaction engineering.

Wait and See Problems. In contrast to ‘‘hereand now’’ problems, which yield optimal solu-tions that achieve a given level of confidence,‘‘wait and see’’ problems involve a category offormulations that shows the effect of uncertain-ty on optimum design. A ‘‘wait and see’’ prob-lem involves deterministic optimal decisions ateach scenario or random sample, equivalent tosolving several deterministic optimization pro-blems. Parametric programming is one way tosolve the ‘‘wait and see’’ problems [42, 43].

The difference between the ‘‘here and now’’and ‘‘wait and see’’ solutions is the EVPI. Theconcept of EVPI was first developed in thecontext of decision analysis and can be foundin classical references [44]. In the above prob-lem, there was action (x), followed by observa-tion (profit), and reaction (or recourse) (r andw).Recourse problems with multiple stages (simi-lar to the multiperiod problems in chemicalengineering) involve decisions that are takenbefore the uncertainty realization (here andnow) and recourse actions which can be takenwhen information is disclosed (wait and see).These problems can be solved using decompo-sition methods.

As can be seen from the above description,both ‘‘here and now’’ and ‘‘wait and see’’ pro-blems require the representation of uncertainties

in the probabilistic space and then the propaga-tion of these uncertainties through the model toobtain the probabilistic representation of theoutput.

4.2. Uncertainty Analysis andSampling

The probabilistic or stochastic modeling [45]iterative procedure involves:

1. Uncertainty quantification which involvesspecifying the uncertainty in key input para-meters in terms of probability distributions

2. Sampling the distribution of the specifiedparameter in an iterative fashion

3. Propagating the effects of uncertaintiesthrough the model and applying statisticaltechniques to analyze the results

Uncertainty Characterization and Quanti-fication. In general, uncertainties can be

characterized and quantified in terms of proba-bilistic distributions. The type of distributionchosen for an uncertain variable reflects theamount of information that is available. Forexample, the uniform and log-uniform distribu-tions represent an equal likelihood of a valuelying anywhere within a specified range, oneither a linear or logarithmic scale, respectively.Further, a normal (Gaussian) distribution re-flects a symmetric but varying probability ofa parameter value being above or below themean value. In contrast, lognormal and sometriangular distributions are skewed such thatthere is a higher probability of values lying onone side of the median than the other. A betadistribution provides a wide range of shapes andis a very flexible means of representing vari-ability over a fixed range. Modified forms ofthese distributions, uniform* and log-uniform*,allow several intervals of the range to be distin-guished. Finally, in some special cases, user-specified distributions can be used to representany arbitrary characterization of uncertainty,including chance distribution (i.e., fixed proba-bilities of discrete values).

It is easier to assume the upper and lowerbound of uncertain variables and, hence, uni-form distribution is the first step towards uncer-tainty quantification. Most of the papers in


chemical engineering use this simplistic ap-proach and use upper and lower bounds ofuncertain variables. Few studies [46–48] iden-tified most likely values and use triangulardistributions. Extensive data which are obtainedfrom DECHEMA and represent realisticquantification of uncertainties related toUNIFAC parameters are used by GANI andDIWEKAR [49, 50].

Interval methods are also proposed [52–54]for handling uncertainties where there is noinformation about uncertainties. For intervalmethods, interval mathematics is used to prop-agate the uncertainty through themodel. For thispurpose, the use of probability boxes (P-boxmethods) instead of just intervals is pro-posed [54]. However, these methods are not yetapplicable to large scale or black box models.

Sampling Techniques. One of the mostwidely used techniques for sampling from aprobability distribution is theMonte Carlo sam-pling technique, which is based on a pseudo-random generator used to approximate a uni-form distribution (i.e., having equal probabilityin the range from 0 to 1). The specific values foreach input variable are selected by inversetransformation over the cumulative probabilitydistribution. Crude Monte Carlo methods canresult in large error bounds (confidence inter-vals) and variance. Variance reduction techni-ques are statistical procedures designed to re-duce the variance in the Monte Carlo esti-mates [55]. Importance sampling, latin hyper-cube sampling (LHS, [56, 57]), descriptivesampling [58], and Hammersley sequence sam-pling (HSS) [59–61] are examples of variancereduction technique.

Importance Sampling. In importanceMon-teCarlo sampling, the goal is to replace a sampleusing the distribution of u with one that uses analternative distribution that places more weightin the areas of importance. Obviously such adistribution function is problem dependent andis dificult to find. One of the examples ofimportance sampling in chemical engineeringis the Metropolis criterion used in molecularsimulations [62].

The following two sampling methods pro-vide a generalized approach to improve thecomputational efficiency of sampling.

Latin Hypercube Sampling. The main ad-vantage of Monte Carlo methods lies in the factthat the results from anyMonteCarlo simulationcan be treated using classical statistical meth-ods; thus results can be presented in the form ofhistograms, and methods of statistical estima-tion and inference are applicable. Nevertheless,in most applications, the actual relationshipbetween successive points in a sample has nophysical significance; hence, the randomness/independence for approximating a uniform dis-tribution is not critical [63]. Moreover, the errorof approximating a distribution by a finite sam-ple depends on the equidistribution properties ofthe sample used for U(0,1) rather than its ran-domness. Once it is apparent that the uniformityproperties are central to the design of samplingtechniques, constrained or stratified samplingtechniques become appealing [64].

LHS [56, 57, 65] is one form of stratifiedsampling that can yield more precise estimatesof the distribution function. In LHS, the range ofeach uncertain parameter Xi is subdivided intononoverlapping intervals of equal probability.One value from each interval is selected atrandom with respect to the probability distribu-tion in the interval. The n values thus obtainedfor X1 are paired in a random manner (i.e.,equally likely combinations) with n values ofX2. These values are then combined with valuesof X3 to form n-triplets, and so on, until n k-tuplets are formed. Inmedian LHS (MLHS) thisvalue is chosen as the mid-point of the interval.MLHS is similar to the descriptive samplingdescribed by [58]. In chemical engineering LHSis used in early stochastic modeling and optimi-zation frameworks [45, 47, 66].

The main drawback of this stratificationscheme is that, it is uniform in one dimensionand does not provide uniformity properties in k-dimensions. Sampling based on quadrature, cu-bature techniques [67], or collocation techni-ques [68] face similar drawback. These sam-pling techniques perform better for lower di-mensional uncertainties. Therefore, many ofthese sampling techniques use correlations totransform the integral into one or two dimen-sions. However, this transformation is possibleonly for limited distribution functions when theuncertain variables are tightly correlated. Forhighly correlated samples similar to what hasbeen observed in thermodynamic phase


equilibria, a sampling technique based on con-fidence region estimates [69] can be used.

Hammersley Sequence Sampling. HSS,based on Hammersley points, was developedby [59, 60]; the technique uses an optimaldesign scheme for placing the n points on a k-dimensional hypercube. This scheme ensuresthat the sample set is more representative of thepopulation, showing uniformity properties inmultidimensions, unlikeMonteCarlo, LHS, andMLHS techniques. It is clearly observed thatHSS shows greater uniformity than other strati-fied techniques such as LHS, which are uniformonly along a single dimension and do not guar-antee a homogeneous distribution of points overthe multivariate probability space. Similar be-havior is observed for correlated samples also.The implementation of correlation structures isbased on the use of rank correlations [57, 65].Some of the new variants of HSS include theHSS2 and latin hypercube Hammersley sam-pling (LHHS) [61, 70, 71].

4.3. Optimization Algorithms

Numerical optimization techniques constitute afundamental part of theoretical and practicalscience and engineering as can be seen fromthe number of papers published each year inliterature. Although in real world systems un-certainties cannot be ignored, the literature onoptimization under uncertainty methods and/orapplications is sparse as compared to any otheroptimization area. This can be attributed to thefact that in optimization under uncertainty area,one has to pay attention to uncertainty analysisas well as optimization.

Decomposition Algorithms. Even a linearproblem in optimization under uncertainty (e.g.,the news boy problem) results in nonlinearities(discontinuous function) due to the probabilisticfunctional. A decomposition method is a com-mon approach to solve such problems. In de-composition approach, the problem is decom-posed into a master problem and a subproblem.The master problem is a simplified (usually)linear approximation of the complete problemand this approximation is derived from a num-ber of subproblems. First proposed for mixed

integer problems, Bender’s decomposition [72]forms the basis for decomposition algorithmsproposed for deterministic mixed integer pro-gramming and stochastic linear as well asmixedinteger programming problems [73, 74]. Thetwo main algorithms commonly used in thestochastic programming literature for stochasticlinear (and mixed integer) programming withfixed recourse are the L-shaped [75–77] and thestochastic decomposition methods [78, 79]. TheL-shapedmethod is used when the uncertaintiesare described by discrete distribution. On theother hand, the stochastic decomposition meth-od uses sampling when random variables arerepresented by continuous distribution func-tions. Although chemical engineering applica-tions involve a large number of stochastic linearprogramming (and stochastic MILP) problems,researchers in chemical engineering have notexploited these generalized algorithms. On theother hand, they have used specific structure of aparticular problem (e.g., flexibility) to derivedifferent decomposition schemes and/orbounds [80–83]. The domain (flexibility) spe-cific nature of the problem restricts the applica-bility of these methods.

Stochastic Nonlinear Programming. Sinceengineering models are usually nonlinear, anduncertainties are not restricted to uniform ornormal distributions, the algorithms and ap-proaches presented above have restrictions thatlimit their application potential to large-scaleproblems in the various domains of chemicalengineering. In order to develop a more generalapproach the better optimization of nonlinearuncertain systems (BONUS) algorithmwas pro-posed [84, 85], which uses sampling and re-places the inner model evaluation loop with areweighting scheme. Deterministic NLP pro-blems use quasi-Newton’s methods and requirecalculation of derivatives with respect to eachdecision variable, at each optimization iteration.These methods are widely used and the codesaremade robust over the years. BONUS extendsthese algorithms to stochastic NLP problems.BONUS provides significant reductions inmod-el iterations.

Discrete Optimization under Uncertainty.Many chemical engineering applications in-volve discrete decision variables (mixed integer


problems). Although decomposition methodshave been used to solve these problems, theapplicability of these algorithms to solve pro-blems of real world scale is very limited in [86].This is mainly due to the fact that most of thesemethods rely heavily on convexity conditionsand simplified approximations of the probabi-listic functions containing uncertainties. Var-iants of probabilistic methods like the stochasticannealing algorithm [87] show great promise inthis regard. In the stochastic annealing algo-rithm, the optimizer not only obtains the deci-sion variables but also the number of samplesrequired for the stochastic model. Furthermore,it provides the trade-off between accuracy andefficiency by selecting an increased number ofsamples as one approaches the optimum. Sto-chastic annealing and its recent variantshave been used to solve real world problemssuch as:

. Nuclear waste-management problem [88]

. Computer-aided molecular design, and poly-mer design under uncertainty [89]

. Greener solvent selection [50]

. Methylene chloride process synthesis [90]

New variants of genetic algorithms based onthe theory used in stochastic annealing, weredeveloped and applied to solvent selection andrecycling problems [91–93].

Parametric programming is based on thesensitivity analysis theory, distinguishing fromthe latter in the targets. Sensitivity analysisprovides solutions in the neighborhood of thenominal value of the varying parameters,whereas parametric programming provides acomplete map of the optimal solution in thespace of the varying parameters. Theory andalgorithms for the solution of a wide range ofparametric programming problems have beenreported in the literature [43].

4.4. Applications

Problems in chemical engineering are linked tothe life cycle of the process that extends fromraw material selection (chemical synthesis),process development (process synthesis anddesign) through the planning and management

of the production process. Process design startswith chemical synthesis where the chemicalpathway from reactants to the product is definedat the laboratory scale. Process synthesis trans-lates the chemical synthesis to a chemical pro-cess. It involves decisions about process unitoperations and connections.

In general, process design activities start atthis level and process simulation is the last stepin computer-aided process design which pre-dicts the behavior of the process if it was con-structed. Incorporation of pollution prevention,operability, and other concepts into design anddevelopment at the initial stages lead to pro-cesses that are less cost intensive, thereby re-ducing the technical and economic risks. There-fore, process synthesis remains an importantstep in analyzing and designing environmental-ly benign processes. Furthermore, researchershave realized the importance of including thechemical synthesis in the process design, andmethods for solvent selection, molecular designare gaining importance [50]. Most of the currentsystems analysis approaches focus on the simu-lation step. This is because several simulationprograms (like the ASPEN simulator [94]) andmodels are available to describe these basicphenomena. However, this is the last step indecision making, as it predicts the behavior ofthe plant (or strategy) if it was constructed (orimplemented). As the envelope extends to in-clude process synthesis, chemical synthesis, ormanagement and planning considerations, theability to include uncertainties become impor-tant. Unlike the traditional process designwhereengineers are looking for low cost options,objectives of optimization in greener and robustdesign is not restricted to maximizing profit butincludes several other criteria like reliability,flexibility, operability, controllability, environ-mental and ecological impacts, safety, and qual-ity need to be considered at the different stagesof analysis.

Stochastic Objectives and Constraints Theformulation of objective function is one of thecrucial steps in the application of optimizationto a practical problem [95].

The earliest work related optimization underuncertainty in chemical engineering used the‘‘here and now’’ formulation where expectedvalue of cost is used as the objective function for


a given level of risk [96–98]. Soon researchersrealized the importance of including operabilityconsiderations like feasibility and flexibili-ty [99–102] in the optimal design problem.Flexibility is concerned with the problem ofensuring feasible steady-state operation over avariety of operating conditions. On the otherhand, reliability is concerned with the probabil-ity of normal operation given that failures canoccur, while safety is concerned with hazardsthat are consequence of failure. The other as-pects of operability are controllability whichdeals with the concept of quality and stabilityof the dynamic response of the system.

Earlier approaches to obtaining flexible de-sign dealt with finding overdesign factors [98].By introducing the concept of flexibility index, awell-structured optimization problem that couldbe easily converted to a mini–max deterministicproblem, was provided [102]. Flexibility indexis defined as a scalar matrix whose value for anyfixed design characterizes is equal to the size ofthe region of feasible operation in the space ofuncertain parameters. The basic assumptionbehind the definition of flexibility index is thatthe uncertainty parameters vary within upperand lower boundswith equal probability (equiv-alent to uniform distribution). These parametersare assumed to vary independently. Geometri-cally, this approach corresponds to inscribingwithin the feasible region a hyper-rectanglewhich is centered at the nominal point. The sizeof the feasible region is then characterized bythe lengths of the sides of the rectangle, which inturn define the lower and upper bounds of theparameter. This hyper-rectangle then definesthe actual ranges for each uncertain parameteroverwhich the feasible operation can be guaran-teed. The maximum hyper-rectangle that canexpand around the nominal parameter pointtouches the boundary at a vertex of the hyper-rectangle. This concept was extended further tostochastic flexibility expressing it in terms ofprobability of feasible operation. The design for‘‘optimal flexibility’’ falls under the category of‘‘here and now’’ problem.On the other hand, theflexibility index or stochastic flexibility withmultiperiod optimization problems involve‘‘wait and see’’ decisions and solution of adeterministic optimization problem for eachscenario so that one gets a probabilistic repre-sentation of optimal solutions.

Many researchers use flexibility synony-mous to uncertainty analysis. For example,solution techniques for optimization under un-certainty was categorized into three categoriesby [42]:

1. Multiperiod2. Stochastic programming3. Parametric programming [103]

It should be noted that these categories arerelated to the optimal flexibility and flexibilityindex problems and cannot be generalized forall optimization under uncertainty problems inengineering or operations research. Further, theflexibility is one of the concepts in this field andis still an abstract measure [104]. The problemof flexibility is a well structured problem ame-nable to solutions using deterministic ap-proaches with decomposition [80–82] or usingnew approximation to derive the probabilisticfunctions describing the effect of uncertain-ties [105]. Further, the concept of flexibility isalso extended to include reliability and control-lability in a similar fashion, demanding similarapproaches. However, controllability also dealswith the concept of quality control.

The aim of a control system is to keep theprocess output specifications on target, despitechanges in the process input. In such an ap-proach, the control engineer is often presentedwith difficult control problems that may requireextensive and expensive modifications to bothprocess and control system hardware to obtainsatisfactory performance of the control system.Furthermore, the effectiveness of the controlsystem is highly dependent upon the nominalvalues of the operating variables and the me-chanical design which are set by the designer ofthe processing unit. Parameter design method-ology is an off-line quality control method,popularized by TAGUCHI [36], for designingproducts and processes that are robust to un-controllable variation at the design stage. Inparameter design the objective is to find settingsof the product or process design parameterswhich minimize an average quadratic loss func-tion defined as the average standard deviation ofthe response from a target value. The rapidgrowth of interest in the Taguchi approach overthe last few years led to a great expansion in thenumber of published case studies relating to


different areas of industrial activities [106].Although the popularity of the Taguchi ap-proach seems to be pervasive in all engineeringbranches; application of this procedure to chem-ical industries was not widely reported until1990. BOUDRIGA [107] presented one of the firstsystematic studies of using different statisticalapproaches to the problem of off-line qualitycontrol for chemical processes. DIWEKAR andRUBIN [37] posed this problem as a stochasticoptimization problem where the objective is tominimize output variance from the specifiednominal value. This approach is the basis ofrobust design. A similar approach (objectivefunction) was used for dynamic models [108].However, this approach is derived from thefinancial literature using the mean-variance op-timization models for portfolio optimization[109].

In view of growing environmental concerns,there is a critical need for designing large chem-ical processes with environmental considera-tions. Earlier design under uncertainty stud-ies [46, 47] focused on design for the environ-ment, included environmental considerations asprobabilistic constraints in terms of risk ofexceeding the specified emission limits. Thesepapers considered the end-of-pipe treatmentslike new environmental control technology de-signs. Nowadays, industries are practicing theart of pollution prevention, which involves fun-damental changes in the processes to minimizethe formation of pollutants, as opposed to pol-lution control, involving end-of-pipe treatmentof process emissions. This alsomeans instead ofincluding environmental considerations as con-straints, they should be included as objectiveslike minimum environmental impacts. Currentmethodologies, such as the generalized wastereduction algorithm (WAR), provide a first steptowards evaluating impacts such as ozone de-pletion, global warming and acid rain poten-tials [110, 111]. However, environmental im-pacts must also be weighted and balancedagainst other concerns, such as their cost andlong-term sustainability. These multiple, oftenconflicting, goals pose a challenging and com-plex optimization problem, requiring multi-objective optimization under uncertainty[90, 93, 94, 110, 112, 113]. Environmentalconsiderations are also translated into specificobjectives for problems like expected value of

solvent selectivity in the solvent selection prob-lem [49, 96], and minimizing expected value of‘‘glass’’ formed in vitrification of nuclear wasteproblems [88].

Chemical Synthesis (! Process SystemsEngineering 4. Process and Product Syn-thesis, Design, Analysis, Chap. 6) Chemi-

cal synthesis involves search for moleculespossessing desired physical, chemical, biologi-cal, and health properties for easy manufactureof products. Group contributionmethods rely onexperimental data and theoretic formulations toassign numerical values to chemical groups,which form the basis building blocks for com-puter-aided chemical synthesis. By combiningthese building blocks, it is possible to determinea wide range of characteristics for any givenchemical. The reverse approach, computer-aided molecular design (CAMD)! MolecularModeling uses group contribution techniques todetermine physical characteristics by generat-ing test molecules using primary buildingblocks. CAMD uses a set of groups as a startingpoint. These groups are uniquely designed togenerate all possible molecules by exploring allcombinations. The properties of each group and/or the interaction parameters between groupscan be theoretically calculated, experimentallyobtained, or statistically regressed. There arethree main CAMD approaches: generation-and-test, mathematical optimization, and combina-torial optimization approaches. All methodolo-gies for CAMDare exposed to uncertainties thatarise from experimental errors, imperfect theo-ries or models and their parameters, improperknowledge, or ignorance of systems. In addi-tion, available group parameters may not bepresent, and current group contribution models(GCM) cannot estimate all necessary properties.

A technique for designing polymers underuncertainty was developed by [40]. This ap-proach uses MINLP formulations, where thechance constraints represent the probability ofmeeting the target values. Following traditionalchance constrained programming techniques, adeterministic equivalent is obtained. Two for-mulations are presented: (1) stochastic propertymatching identifies molecules that meet allcharacteristics with a given probability a, (2)stochastic property optimization determines themolecules that have a maximum value for a


given property with probability awhile all otherprobabilities are met with given probability. Ageneralized framework to solve the same poly-mer design problem based onHSS sampling andstochastic annealing was presented [89]. Theframework allowed them to study:

. Impact of uncertainties

. Effect of various probability distributionsincluding stable and nonstable distributions(as CCP does not allow nonstable distribu-tions), mixed distributions, and

. Different forms of objective functions

The framework presented a set of solutions tochoose from instead of a single optimal solution,providing flexibility to the designer.

SINHA et al. [115] proposed that locally opti-mal solutions to the traditional CAMD problemare potential sources of uncertainties. To bypassthis, a global optimization algorithm is pre-sented. The CAMD is modeled as a MILP, anda case study is presented. Solvent selectionusing a CAMD approach that selects environ-mentally benign molecules through group com-bination tools have been analyzed by [49, 50,91]. This is the first study where the uncertain-ties in group contribution methods are system-atically characterized using available literatureand experimental data. New variants of stochas-tic annealing are developed and used to findenvironmentally benign solvents. They alsocoupled process synthesis along with chemicalsynthesis to obtain environmentally friendlyand cost effective solutions [92, 116].

Process Synthesis and Design ((! ProcessSystems Engineering 4. Process and Prod-uct Synthesis, Design, Analysis, Chap. 2))

Process synthesis translates the chemical syn-thesis to a chemical process. It involves deci-sions about process unit operations and connec-tions. In general, process design activities startat this level and process simulation, which isshown to be the last step in computer-aidedprocess design, predicts the behavior of theprocess if it was constructed.

One of the main goals in synthesis problemsis to establish methodologies for selecting opti-mal flowsheet configurations. Approaches toprocess synthesis problems essentially fall un-der the following areas:

1. Thermodynamic approach2. Evolutionary methods3. Hierarchical approach, based on intuition

and engineering judgment4. Optimization approach based on mathemat-

ical programming techniques

The optimization approach to process syn-thesis involves (a) formulation of a conceptualflowsheet incorporating all the alternative pro-cess configurations (superstructure) and (b)identification of an optimal design configurationbased on optimal structural topology and theoptimal parameter level settings for a system tomeet specified performance and cost objectives.Once the superstructure is known, combinato-rial optimization methods like MINLP algo-rithms can be used to solve the synthesis prob-lem. The first step in the solution of the processsynthesis problem is to develop the superstruc-ture containing all alternative designs to beconsidered for the optimal solution. The designof new processes is, however, complicated bythe fact that technical and economic uncertain-ties arise, which lead to uncertainties in theprediction of plant performance and overallplant economics. An example where such tech-nical and economic uncertainties occur and arenot treated or characterized rigorously, is in thedesign of integrated environmental control pro-cesses for advanced power systems [45]. Sincethe conceptual design of any chemical processinvolves the identification of possible flowsheetconfigurations, design methods must also ad-dress the issues of process synthesis underuncertainty, as it has important implications onprocess viability, and other quality measuressuch as controllability, safety, and environmen-tal compliance.

The literature in the area of process synthesisand process design under uncertainty have beenconcentrated on two focused application areas:

1. Pollution prevention by design2. Designing for flexibility

Pollution Prevention by Design. The earli-er papers in design under uncertainty with pol-lution prevention focus dealt with integratedenvironmental control systems for coal-basedpower systems. Thework continued and extend-ed to address synthesis problems in this


area [45–47]. Nuclear waste management poseda very hard synthesis problem [117]. This is alarge-scale, real world problem related to theenvironmental restoration of Hanford site. Con-version of high-level radioactive waste intoglass is crucial to the disposal of toxic wastedumps generated over 40–50 years at the Han-ford nuclear waste site. The procedure essen-tially consists of mixing the sources of wastesinto blends, to which appropriate glass formers(frit) is added to make glass. The objective is tomaximize the amount of wastes per glass log, bykeeping the amount of frit added to a minimum.Processibility and durability conditions requirethat certain restrictions on crystallinity, solubil-ity and glass properties are met. Increasing thenumber of wastes, increases the combinatorialsize of the problem. The combinatorial, non-convex nature of the problem was hard to solveeven for the deterministic optimization meth-ods. Uncertainties associated with the tank con-tents and models caused further problems anddemanded new algorithms [88]. The new sto-chastic annealing algorithm provided optimaland robust solution to this problem in the face ofuncertainties with reasonable computationaltime. A multiobjective extension of this prob-lem to include policy aspect was possible due tothese new algorithms. This new algorithm wasalso used for methylene chloride processsynthesis [90].

Designing for Flexibility (! Process Sys-tems Engineering 4. Process and ProductSynthesis, Design, Analysis, Chap. 3; !Process Systems Engineering, 5. ProcessDynamics, Control, Monitoring, and Iden-tification). Process flexibility is an area that

received significant attention, as it ensures thatprocesses are operational and safe when ex-posed to variations in operating conditions. Thestudies include distillation network design, heatexchanger network synthesis [82], reactor net-work synthesis [105], and batch processingplant design and operation for wastetreatment [118].

Management, Scheduling, and Planning(! Process Systems Engineering 8. PlantOperation, Integration, Planning, Schedul-ing and Supply Chain) Most problems in

management, scheduling, and planning include

combinatorics (discrete choices and decisions)and uncertainties [104, 119]. These problemsbelong to batch processing due to time depen-dent nature of these chemical processes. Batchprocessing is generally used in high-valueadded, low-volume speciality chemicals andpharmaceuticals. Uncertainties abandon inbatch plant operation. Although batch proces-sing is facedwith all kinds of uncertainties,mostof the literature in this area deals with demanduncertainties.

In general the scheduling problem is to de-termine a time-based assignment of tasks toequipment so that no process constraints areviolated. The problems in the domain of processscheduling and planning can be convenientlydescribed by resource–task–equipment net-work. The process design and retrofit problemsadd longer time horizons to the schedulingproblems and include decisions regarding addi-tions of equipments. Supply chain managementproblems extend the scheduling problem in thespatial dimension and consider the coordinatedmanagement of multiple facilities and the ship-ment of materials through an associated trans-portation network. The product and researchpipeline management problem has much over-lap both with supply chain management andprocess scheduling problems. These problemsare closely related to pharmaceutical industrieswhere new drugs and products are inventedregularly. Obviously these problems have greatdeal of uncertainty. Research management ingeneral is also related to prioritization andreducing uncertainties. The chemical engineer-ing literature is concentrated in problems inbatch scheduling and planning including designand retrofit problems [104, 119]. Supply chainmanagement problems are rare [120], and re-search management problems have only recent-ly being studied [48, 122–124].

4.5. Future Directions

Research in all aspects of optimization willcontinue at a fast pace. However, the solutionstrategies for optimization models with smoothfunctions, continuous derivatives and continu-ous-valued optimization variables appears to bematuring especially with regard to locallyoptimal solutions. Evolutionary methods (e.g.,


genetic algorithms, simulated annealing,particle swarm optimization, etc.) for globaloptimization have made a lot of inroads intoengineering applications for two main reasons:(a) they are conceptually easy to understand andeasy to program into a computer working code,(b) they are capable of handling a mixture ofcontinuous-valued and discrete-valued optimi-zation variables, and (c) they are capable ofhandling nonsmooth functions, for example,those that appear in material, energy and mo-mentum balances. However, these evolutionarymethods have the drawback of not being able toprove that they have reached a globally optimalsolution and that their theoretical foundation isnot as well-developed as deterministic methods.The challenge is for researchers in deterministicoptimization to reach out to users of evolution-ary methods, for example, researchers in thebiological and life-sciences areas.

Research into optimization under uncertain-tywill continue to grow at a rapid pace. Formostproblems in life sciences, physical sciences, andengineering, the presence of uncertainty is theoverwhelming norm and researchers can nolonger pretend uncertainty does not matter oris insignificant. Multiobjective optimization inwhich several conflicting criteria need to beoptimized is another growth area. In the currentage when we have to build sustainable systemsthat have to maximize profit, minimize resourceuse, do minimal harm to the environment, makepolicymakers happy and get the politics toworkout right, multiobjective optimization will haveto be the modus operandi.

Interval and P-box methods will also befocus of future articles. In this keyword, we didnot treat time-dependent uncertainties whichhave been recently studied for batch processdesign [125–128] and sustainability [129] ‘‘seealso the special issues on ‘‘energy and sustain-ability’’ (Computers and Chemical Engineer-ing, 35(8), 2011); ‘‘energy systems engineer-ing’’ (Computers and Chemical Engineering, 35(9), 2011)’’. These areas present new trends inoptimization under uncertainty.

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