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0PTI«177.ATT0N STRATEGIES FOR FLEXIBLE CHEMICAL PROCESSES

by

I . E . G r o s s i i i H n n , K . P . I f o l 3 r r r j n « & R . E . S v - z a n a y

December, 1932

DRC-05-33 -82

OPTIMIZATION STRATEGIES FOR FLEXIBLE CHEMICAL PROCESSES

I. E. Grossmann, K. P. Halemane and R. E. Swaney

Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, U.S.A.

Abstract. The objective of this paper is to give an overview of theoptimization strategies that are required when designing chemical processesin which the existence of regions of feasible steady-state operation mustbe ensured in the face of parameter variations. Two major areas areconsidered: optimal design with a fixed degree of flexibility, and designwith optimal degree of flexibility. For the first area the problems ofmultiperiod design, and design under uncertainty are analyzed. For thesecond area the problem of deriving an index of flexibility in the contextof multiobjective optimization is discussed. As shown in the paper, themajor challenge in these problems lies in the development of efficientsolution procedures for large scale nonlinear programs which are eitherhighly structured, or otherwise involve an infinite number of constraints.

INTRODUCTION

Flexibility is one of the main concerns in the

design of chemical plants. The reason is that

for a design to be useful in practice it is

essential that the plant be able to satisfy

specifications and constraints despite variations

that may occur in parameter values during

operation. For example, in practice it is quite

likely that the amount and quality of the

feedstreams to the process will vary during

operation. This aspect will be particularly

critical when the plant has to process alternate

feedstocks as is commonly the case in many

chemical processes (see for instance Draaisma

and Mol. 1977; Rhoe and de Blingiers. 1979).

Other examples of changes that often occur

during plant operation include variations in the

ambient temperature, deactivation of catalysts,

fouling of heat exchangers, and wearout of

mechanical equipment such as pumps and

compressors. Therefore, it is clear that at the

design stage some degree of flexibility must be

introduced to ensure that the plant will be able

to cope with uncertain parameters during

operation.

Current address: Westinghouse R6J) Center,1310 Beulah Road, Pittsburgh, U.S.A.

The usual approach that is used in practice is to

design and optimize chemical plants for nominal

values of the parameters. Since considerable

uncertainties in these values often exist,

empirical overdesign factors are used to provide

for flexibility in the operation of the chemical

•plant (Rudd and Watson, 1968>. However, it is

clear that with this approach not much insight

can be obtained as to the actual degree of

flexibility that is being achieved in the design.

Also, with this approach, it becomes difficult to

justify on economic grounds the extent of the

overdesign.

In the context of the theory of chemical process

design, the need for a rational method of

designing flexible chemical plants stems from

the fact that there is still a substantial gap

between the designs that are obtained with

currently available computer-aids and the designs

that are actually implemented in practice. The

major reason for this gap is that the computer-

aids do not explicitly account for operability

considerations at the design stage. This would

involve handling simultaneously the aspects of

flexibility, controllability, reliability and safety

of the chemical plant. It should be noted that

although some of these aspects are quite

similar, they actually correspond to different

UNIVERSITY UB^AWESCARNEGIE-MELLOW UNIVERSITY

PnTSBURGH, PENNSYLVANIA 15213

Multiperiod Design Problem

One way to introduce flexibility in a chemicalplant is to design it for a specified number Nof different operating conditions (Grossmann andSargent, 1979). For example, a plant may bespecified to process a variety of feedstocks,produce different products, or operate atdifferent levels of capacity. The goal is then toensure that the plant will be able to meet thespecifications for N successive periods ofoperation, requiring at the same time that theplant be designed and operated so as tooptimize a given objective function, which istypically a combination of the investment andoperating costs.

Very little has been discussed in the literatureabout deterministic multiperiod problems.Loonkar and Robinson (1970), Sparrow, Forderand Rippin (1975), Oi, Itoh and Muchi (1979),Suhami (1981), Suhami and Man (1982). Knopf,Okos and Reklaitis (1980), Takamatsu, Hashimotoand Hasebe (1981). discuss design proceduresthat are applicable only to batch/semicontinuousprocesses. This important class of problems inwhich scheduling of operations is one of themain issues will be covered in detail in thisSymposium by Professor Rippin (1982).Grossmann and Sargent (1979) give a generalformulation for designing multipurpose chemicalplants which can also be applied to problemsdescribed by the deterministic multiperiodmodel. As shown below, this involves thesolution of a large nonlinear program, whereinthe main computational difficulty is due to thelarge number of decision variables involved.

In order to formulate the multiperiod designproblem it is assumed that the plant issubjected to piecewise operating conditions in Nsuccessive time periods. Also, dynamic effectsare neglected, since the lengths of the transientsare considered to be much smaller than the timeperiods for the successive steady states. Theoptimal design problem is then given by thefollowing multiperiod nonlinear program

N

min C°(d) Ci(d.zi.xi.ti)

s.t. h'fd.z'.x'.f) * 0 1

g'td.z'.x'.t1) * oj

(1)

whered is the vector of design variables representing

equipment sizes

z' is the vector of control variables in period i

x1 is the vector of state variables in period i

t1 is the length of time for each period i

h' is the vector of equations in period i

g' is the vector of inequalities in period i

r is the vector of inequalities that involvevariables of all periods

N is the number of periods

It should be noted that the vector of designvariables d remains fixed throughout the periodsof operation as it represents the sizes of theunits. Also, the control variables z1 representthe degrees of freedom in the operation of theplant, and therefore, they correspond tovariables that can be manipulated directly orindirectly in the operation of the plant.

Since the dimension of h1 is the same as for x1,the decision variables for the problem in (1) aregiven by the design variables d, the controlvariables z1. i=1,2,...N. and the lengths of periodst\ i=1,2,...N. The main difficulty that arises inthe solution of this multiperiod problem is the

, fact that the number of decision variables canbecome rather large as the number of periods Nincreases. This implies that the computationalburden that would be required by using currentnonlinear programming algorithms could becomeexcessive, and also that the numerical solutioncould be very difficult to obtain. However, itshould be noted that problem (1) has a veryspecial structure. Firstly, the objective functionis separable in the design variables and in the Nperiods of operation. Secondly, since thevariables x1, z\ t1, are associated with thecorresponding period i, the constraints have abordered block-diagonal structure, where thecoupling variables are given by the vector d.and the coupling constraints by the vectorr. Therefore, it is clear that an efficientoptimization algorithm ought to take advantageof this structure in order to reduce thecomputational requirements. As will be shown inthe next section algorithms can be developedthat accomplish this goal.

r(d.z1 .-zN .x\.-x2 .t \- .t1^

Projection-Restriction Strategy

Recentlyj Grossmann and Halemane (1980) havedeveloped a very efficient decompositionscheme based on a projection-restrictionstrategy for solving an important particular caseof problem (1). Namely, if one assumes that thelengths of the periods t' are specified by thedesigner and that the vector of inequalities ronly involves the design variables d, themultipenod design problem is given by

N

( 2 )

i = 1.N

min C * C°(d) • > C'(d,z'.x')d,z\z2....zN tT

s.t. h^d.z1,^) »0 1

g'td.z'.x1) ^ 0 J

r(d) * 0

Note that problem (2) also has a block-diagonalstructure, but it involves coupling in thevariables d only and no coupling in theconstraints. Although at first sight this problemappears to be too specific, it turns out to beone of the underlying formulations for solvingdesign problems under uncertainty as will beshown later in the paper. Furthermore, theformulation in (2) still has a wide applicability indeterministic multiperiod design problems.

.1 m1 .2 ,2 , 3 , 3 ,4 .4

Fig. 1. Block-diagonal structure of theconstraints in problem (2) .

The decomposition technique proposed byGrossmann and Halemane (1980) exploits twobasic features in this design problem. The firstone is the block-diagonal structure in theconstraints which is shown in Fig. 1. Since theobjective function C is separable in the Nperiods, this implies that if the vector d isfixed, the optimization problem decomposes in Nuncoupled subproblems. each having as decision

variables the control variables z1, i=1,2,...N. Thesecond feature that is exploited, and which isstrictly heuristic in nature, is that many of theinequality constraints become active at thesolution. Clearly, this feature cannot beexpected to hold necessarily for any arbitrarymathematical problem. However, in the contextof multiperiod chemical plant design thiscondition seems to hold true in general. Themain reason for this is that cost functions tendto be monotonic. and therefore, the optimalsolutions commonly lie at the boundary of thefeasible region (Westerberg and Debrosse, 1973).Another important reason is that in theformulation of multiperiod problems it isnecessary to treat most of the output variablesof the process in the form of inequalities inorder to introduce a positive number of degreesof freedom (see Grossmann and Sargent, 1979).Since these output variables (e.g. productionrates, purity specifications, target temperaturesand pressures) are normally fixed for the singleperiod problem, they will have a high tendencyto become active at the solution. In fact, theobservation that many inequalities do becomeactive at the solution has been confirmednumerically in a number of example problems(see Grossmann and Halemane, 1980).

The main steps involved in the projection-restriction strategy, which is based on someideas proposed by Grigoriadis (1971) and Ritter(1973) for linearly constrained problems, are as

• follows:

Step 1 - Find a feasible point d, z1, x1. i = 1,2,...N.for problem (2).

Step 2 -

(Projection) Fixing the values of the vector d,solve the N subproblems

minz1

s.t. h'fd^x1) = 0

g'fd.z'.x1) £ 0

(3)

1,2,...N

Step 3 -

(Restriction) (a) For each subprobiem i, convertthe n^ inequality constraints g' that are activein Step 2 into equalities and define

h1

gj, • g| «=1.2....N (4)

where h1. g1 are the redefined sets of equalityand inequality constraints, and gj are the sets ofinequality constraints that are not active in Step2.

(b) Eliminate n' variables z'A from the vector

z1 * I I , so as to define

V ! - x«'

( 5 )

. i*1.2.~N

where z^ is the redefined vector of controlvariables which results from eliminating thevector z^ of n^ elements, and x1 is theexpanded vector of state variables.

Step 4 - Solve the restricted problem:

N

minimize C = C°(d) * T " C'(d,z' ,x')-fc-* R R

d 'W"2R ( 6 )

r(d) £ 0

Step 5 - Return to Step 2 and iterate until nofurther changes occur in the values of thevariables d and in the active set of constraints.

Note that in Step 4 the projection-restrictionstrategy really consists in solving problem (2)simultaneously for all variables, but in generalwith a much smaller number of decisionvariables, since many of these get eliminated bythe active constraints determined in Step 2.Clearly, the effectiveness of this strategy reliesheavily on the number of inequality constraintsthat actually become active at the solution.

Also, it should be noted that for effectiveimplementation of this procedure it is necessaryto find an initial feasible point in Step 1efficiently, and to ensure nonsingularity in theredefined system of equation h' in Step 3. Forthe first point Grossmann and Halemane (1980)have suggested an alternate optimization schemeof design and control variables in which the sumof squares of violation of constraints isminimized. For the second point, they performan analysis on the reduced jacobian of the

system of equations to determine its maximumrank. This scheme allows one to incorporateonly those active inequalities that lead to a non-singular system of equations (see Halemane andGrossmann. 1981b).

The results that have been obtained with theprojection-restriction strategy are extremelyencouraging. Grossmann and Halemane (1980) andAvidan (1982) have found that with the proposedprojection-restriction strategy the computer timevaries only linearly in the number of periods.This is in great contrast with the polynomialincrease (second to third degree) of computertime that is experienced when no decompositionis performed. The reductions in computer timethat have been obtained in an example of areactor with cooler and an example of aflowsheet that involves a reactor, compressorand two separators are of at least one order ofmagnitude.

In order to substantiate the claims ofcomputational efficiency theoretically,Grossmann and Halemane (1980) have developeda model that represents the computation timerequired with the decomposition scheme andwithout it. With this model it was found that ifall the control variables are eliminated in therestriction step, one can indeed prove that thecomputer time has to be linear in the number ofperiods. For the case when not all the controlvariables are eliminated it is not possible toobtain a theoretical solution in closed form.However, it is possible to perform a parametricstudy which shows the interesting result that asthe number of periods increases, the fraction ofcontrol variables that must be eliminated toensure reduction in computer time with theprojection-restriction strategy decreases

exponentially. In other words, as long as somefraction of the inequalities becomes active atthe solution (which is almost always the case)very substantial savings in computer time canbe achieved with the projection-restrictionstrategy.

Recently Avidan (1982) has implemented theprojection-restriction strategy in the generalpurpose computer package FLEXPACK using asthe optimizer the variable-metric projectionmethod by Sargent and Murtagh (1973). Oneinteresting point that emerged from his workwas the fact that further reductions of computertime in the decomposition strategy are possibleif the vector of design variables d is partitioned

in the capacity variables d (e.g. volumesvessels, power of compressors) and in the fixeddesign variables df (areas of exchange, numberof plates) as discussed in Grossmann and

Sargent (1979). Since the variables d aredefined by expressions of the form d . - max{c }, where c is the i'th capacity required forperiod i, the standard procedure is to replacethem by inequalities to avoid discontinuousderivatives. However, since in problem (2) thelengths of the time periods are fixed it is verycommon that a given period i will define thebottleneck for a given capacity variable d .Since by solving the projection step the periodswhere the bottlenecks occur can be identified,one can replace each design variable d by thedominant capacity, which in turn reduces theproblem size in the restriction step, particularlyif the majority of the design variables arecapacity variables. However, it is clear that thisprocedure will only work if the periods wherebottlenecks occur remain the same in therestriction step, and therefore caution should beexercised when using this procedure.

Future directions. It is clear that the next step inthe area of multiperiod design problems wouldbe to derive a decomposition strategy forproblem (1) which involves the couplingconstraints r. This would be an importantdevelopment since that structure would fit themultiproduct batch plant design problem(Grossmann and Sargent, 1979). It is interestingto note that if only a few variables occur in theconstraint r, for instance the lengths of periodst. one could still apply the projection-restrictionstrategy if those variables are treated as designvariables. However, this would increase thenumber of decision variables in the restrictionstep, and therefore, an extension of theprojection-restriction strategy for this casewould seem to be worth exploring.

Design under Uncertainty

In chemical plant design there are usually anumber of parameters for which there isconsiderable uncertainty in their actual values.For instance, these parameters can correspond tointernal process parameters such as transfercoefficients, reaction constants, efficiencies orphysical properties. In addition, the uncertainparameters can also be external to the processsuch as specifications in the feedstreams. utilitystreams, environmental conditions or economiccost data.

The general form of the problem of designunder uncertainty is given by

min C(d,z,x,0)d.z

s.t. h(d,z,x.0) * 0

g(d,z,x.0) £ 0

( 7 )

where d,z,x, are the vectors of design, controland state variables, and 6 is a vector of pparameters for which there is significantuncertainty in their values.

There have been several approaches to theproblem of design under uncertainty reported inthe literature. They differ from eacn other interms of problem formulation as well assolution strategies, since in principle theproblem of design under uncertainty is not well-defined. Some authors consider the probabilitydistribution of the parameters as either knownor predictable, and minimize the expected valueof cost. Another approach consists intransforming the problem into a deterministicone, assuming that the parameters vary withinbounded ranges of values that are soecified bythe designer or by a statistical analysis. As itwill be shown in the next section, this latterapproach can be regarded as a generalization ofthe multiperiod problem if the control variablesare allowed to be adjusted for the differentparameter realizations in order ro achievefeasible operation. However, it is worthwhile tofirst present a brief review on the extensiveprevious work that has been published on theproblem of design under uncertainty.

In the earlier work, the stochastic approach wasthe one that was most frequently used. Forinstance, Kittrel and Watson (1966) assume thatprobability distribution functions of theparameters are available, and propose to selectthe decision variables in the design so as tominimize the expected value of cost. Wen andChang (1968) define the 'relative sensitivity' ofthe cost as the fractional change in the costfunction from its nominal value. In selecting theoptimal design they minimize either the expectedvalue or the maximal probable value of thisrelative sensitivity. Weisman and Holzman(1972) incorporate a penalty in the cost functioninvolving the probability of violation forindividual constraints, and perform anunconstrained minimization of the expected valueof the cost. Although they made an attempt tominimize the probable violation of the

constraints, their formulation does not ensure alower limit on the probability of failure of anygiven constraint, which in fact could be achievedby using the formulation suggested by Charnesand Cooper (1959) for chance constrainedoptimization. Lashmet and Szczepanski (1974)apply Monte Carlo simulation for determiningoverdesign factors for distillation columns. Theyperform a series of statistical experiments bychoosing random values of the parameters(within the specified range), and in each casedetermine the number of stages in the columnneeded to meet the specifications. From thesedata the overdesign factor is determined as theadditional number of stages corresponding to90% cumulative distribution over that fornominal design. Freeman and Gaddy (1975) definedependability as the fraction of time that theprocess can meet the specifications, and use itas a criterion for selecting the optimum design.They perform a stochastic simulation anddetermine the optimum values of the decisionvariables for different values of the parameterschosen randomly. The expected value of thecost corresponding to any given dependabilitylevel that minimizes the expected cost is chosenfor the optimum design.

It is to be noted that in the above methods nodistinction is made between the two types ofdecision variables in the problem of optimalprocess design. The design variables,representing for example the sizes of equipment,get their values assigned in the design stage,and remain unaltered during the operation of theplant. The control variables represent thevariables of the plant that can be adjusted inthe operation after the plant is installed. Forany given design, the optimal plant operationitself can be considered as a means for meetingthe specifications while minimizing the operatingcost. Therefore, it requires an appropriate choicefor the values of control variables depending onthe values of the parameters being realized. In arealistic strategy for optimal process designunder uncertainty, it is important to incorporatethis basic difference between design and controlvariables into the mathematical formulation.

The following researchers have made adistinction between design and control variables.Watanabe, Nishimura and Matsubara (1973) applythe concept of statistical decision theory by

- considering the problem of optimal processdesign as a two-person statistical game betweenNature and the designer. They minimize a utility

function which is a convex combination of theexpected value and the maximum probable valueof cost. That is. they follow a strategy which isintermediate between minimax strategy (maximalprobable value of cost is minimized) and Bayes'strategy (expected value of cost is minimized).Nishida, Ichikawa and Tazaki (1974) proposed aminimax strategy wherein the maximum value ofthe cost function, as obtained at the worstparameter value in a specified range, isminimized by selecting the appropriate design.They view the design strategy as a game,wherein the uncertainty in parameter values isconsidered to result in the maximization of cost,whereas the objective of the designer is tominimize it. It is important to note that thedesign they come up with actually correspondsto the optimum solution for a particular (namely,the economically worst) value of the parameters,and therefore the design, cannot be claimed tobe optimal in an overall sense when theparameters do take on different values. Also,the feasibility of this design for other values ofthe parameters cannot be guaranteed, since thisaspect is not explicitly considered in theproblem formulation.

Takamatsu, Hashimoto and Shioya (1973)assumed that the parameters vary withinspecified bounds, and minimize the deviation ofthe objective function from its value at thenominal solution, while satisfying the constraintslinearized around their nominal values. Theyevaluate each constraint with a separateparameter value that would result in the worstviolation of that constraint. In this way theyseek a design that would meet the specificationswith a single and common operating conditionfor all the bounded parameter values. Thisapproach will tend to give conservative designssince no advantage is taken of the fact that,depending on the values of the parameters beingrealized, the operation of the plant can bemanipulated in order to satisfy thespecifications in the most economical way.Dittmar and Hartmann (1976) use a similarapproach as Takamatsu, Hashimoto and Shioya(1973). but instead suggest the use of the sameextreme value of the parameter for all theconstraints. They determine the design marginfor each of these extreme values and select thelargest design margin so obtained.

Avriel and Wilde (1969) discuss differentstrategies such as two-stage there-and-now).wait-and-see, and permanently-feasible programs

when applied to design problems that

correspond to geometric programs. In a two-

stage stochastic program, the designer selects

values for the design variables (first stage), then

observes the actual realization of the uncertain

parameters. and accordingly chooses the

appropriate values for the control variables

(second stage). While selecting the values for

design variables in the first stage, it is essential

to ensure feasibility of the second stage sub-

program, namely that values of control variables

can be chosen to satisfy the constraints. The

objective is to minimize the expected value of

cost while selecting a feasible and optimal

design, which appears to be one of the most

suitable representations for the problem of

chemical process design under uncertainty. In the

wait-and-see strategy, the designer waits for an

observation of the uncertain parameters and then

chooses the optimal values for both design and

control variables. Here, each new value of the

parameters results in a corresponding optimal

design; or in other words, all decision variables

are treated as control variables. In the

permanently-feasible program. the designer

selects (a single set of) values for both design

and control variables which will be feasible for

every possible realization of the uncertain

parameters. Unlike the wait-and-see strategy,

here the values of neither design nor control

variables change with the variations in the

values of the uncertain parameters. That is, in

the permanently-feasible program, all decision

variables are treated as design variables. Avriel

and Wilde (1969) suggest a procedure for

obtaining the optimal design, which consists of

bounding the objective function value that would

be obtained at the solution of the two-stage

program by solving the wait-and-see program

and permanently feasible program. However, they

restrict their approach only to geometric

programming formulations. Malik and Hughes

(1979) apply a similar approach for general

process design problems, although there is no

guarantee on the feasibility of the design. Also,

the stochastic programming method they

propose, based on Monte Carlo simulation,

requires great computation effort. Johns,

Marketos and Rippin (1976) outline a design

strategy in which parameter uncertainty is

considered in addition to the possibility of

expansions through a two-stage multiperiod

formulation. However, they did not address the

problem of deriving an efficient solution

"procedure for handling these problems.

Among the more general purpose formulations

for optimization problems with uncertainty,

Friedman and Reklaitis (1975a,b) deal with the

case of linear programming problems that have

uncertainties in the coefficients of the

constraints. They show that this formulation can

be applied to problems such as planning future

operation policies for large interacting systems,

production scheduling, resource allocation and

determining optimum blending schemes. They

incorporate the required flexibility in their

system by allowing for possible future additive

corrections on the current decisions, and

optimize the system by applying an appropriate

cost-for-correction in the objective function. It

is interesting to see that they were able to

identify the need for different corrections for

different outcomes of the uncertain coefficients

in order to make the problem feasible, and

devise a procedure to achieve this in their

computations. One obvious drawback with their

approach is that it is applicable only to linear

systems. A second limitation with their approach

is that it cannot be applied directly to the

problem of optimal design of flexible chemical

plants, because in this case no additive

corrections can be applied on the design

variables. Instead, it is only the control

variables that can be manipulated, so as to meet

the specifications in spite of the variations in

the values of the uncertain parameters. Kilikas

and Hutchison (1980) use a linear process model

wherein the coefficients are considered to be

varying within specified bounds. Their approach,

however, would tend to select the optimum

value of their decision variables so as to

satisfy their constraints for only one set of

parameter values.

With any of the suggested approaches for

dealing with parameter uncertainties in process

design, one is always faced with the question

of whether the designed plant can in fact be

guaranteed to operate and satisfy specifications

for the entire range of parameter values

involved. This question, along with the fact that

the problem of optimal process design under

uncertainty is not well-defined, requires a

systematic procedure in formulating as well as

solving such design problems. First of all, it is

to be noted that apart from minimizing the cost,

the main concern of the design engineer is to

ensure feasible steady state operation of the

plant for every value of the parameters witnin

specified bounds. Grossmann and Sargent (1973)

propose a formulation that tries to incorporate

this objective. They approximate the expectedvalue of the cost by a weighted average of afinite number of terms, assuming discreteprobabilities for a finite set of parametervalues. They select the optimum design byminimizing this expected cost subject tomaximizing each of the individual inequalityconstraints with respect to the parameters. Intheir solution procedure a small set of extremevalues of parameters is selected by analyzingthe signs of the gradients of each of theindividual inequality constraints. and theoptimization is performed for this set ofparameter values, in the form of a multiperioddesign problem. However, t.u.eir approach cannotalways guarantee that the extreme values thathave been selected wi l l ensure feasibility ofoperation for all the other parameter values. Inthe next section the formulation proposed byHalemane and Grossmann (1981c) is presented.This formulation is an extension of the work byGrossmann and Sargent (1978), and rigorouslyensures feasible operation for the specified setof bounded parameter values.

Two-Stage Proqramminq Formulation

Assuming that bounded values of the uncertainparameters are specified in problem (7), theregion T that is defined to contain all possiblevalues of these parameters is given by

{$ | 6L * 6 $ 6U) (8)

where 0L and 9U represent given lower andupper bounds on 6. Of course the parameterscould also be dependent, in which case theywould typically be related by linear constraints.However, for the sake of simplicity in thepresentation they wi l l be assumed to beindependent.

In order to derive the mathematical formulationit is convenient to consider the design strategyused by Halemane and Grossmann (1981c) asbeing composed of two stages: an operatingstage and a design stage.

I. Operating stage: Assuming that a givendesign d has been selected, it is considered thatthe plant wi l l be operated optimally whilesatisfying the constraints of the process for allpossible realizations of the parameters inT. Hence, the objective in this stage is to select

"Tor every realization 0€T, a control z which isboth optimal and feasible.

Clearly, for the given design d and for any

value of 6. the state variables can be expressed

as an implicit function of the control z from the

system of equations of the process.

h(d,z.x,0) * 0 =» x = x(d.z,0) (9)

Since the control variable z should be selectedso as to satisfy the specifications given by thevector of inequality constraints,

g(d.z.x,0) = g(d,z,x(d.z,0),0) = f(d.z,0) < 0 (10)

the optimal operation of the plant thatminimizes the cost while satisfying theconstraints wil l be given by the nonlinearprogram

min C(d,z.0)z

s.t. fid.z.0) <> 0(11)

The solution to this problem defines the costfunction C"(d.0) which corresponds to theoptimal operation of the plant for fixed valuesof d.8. Furthermore, if the optimization isperformed for every realization #€T, the averagecost of operation wi l l be given by the expectedvalue E {C#(d,0)K

II. Design Stage: In order to achieve the basic

objective of feasible operation in the region of

parameters T, the design variable d must be

chosen so as to ensure that for every value of

d the control variable z in the operating stage

can indeed be selected to satisfy the constraints

in (11). Note that an improper selection of d can

lead to infeasible operation for some realization

of 6, in which case no selection of the control

z wi l l exist so as to satisfy the inequality

constraints in (11). Furthermore, in order to

achieve the optimal design, the design variable d

must be selected so as to minimize the

expected value of the optimal cost function

C*(d.#) over the entire region T.

This strategy for dealing with uncertainties indesign can be interpreted qualitatively in thefol lowing way. In stage II , the designer selectsa design such that if in stage I the operatorproperly adjusts the controls depending on therealization of the parameter values, feasible andoptimal operation of the plant can be achievedwithin the specified range of parameter values.

Note that the assumption made here is thatessentially perfect control of the plant can beachieved, since for instance no noise isassumed in the measurement of the parameters.Although this could clearly be regarded as alimitation of the strategy, it is considered thatat the design stage including more detailedinformation on the control scheme would makethe problem virtually unmanageable. Despite thelimitation, it is clear that the strong point of thestrategy is that it does recognize explicitly thatchemical plants can be adjusted during operationto achieve feasibility.

The strategy as stated above can be expressedmathematically as the two-stage programmingproblem,

minimize E { min C(d,z.£) | f(d,z,0) £ 0}d del z

(12)s.t. V *€T < Az ( V i€J I f (d.z.0) < 0 ~" '

where J = {1,2 m} is the index set for thecomponents of the vector of constraintfunctions f. The constraint in (12) is denoted asthe feasibility constraint, because the existenceof a feasible region of operation in the region Tcan be ensured if and only if this constraint issatisfied. In fact, this logical constraint statesthat for every point 0ET, in the space ofparameters, there must exist at least one valuefor the vector z of control variables that givesrise to non-positive values for all the individualconstraint functions. Qualitatively, this meansthat irrespective of the actual values taken bythe parameters, the proposed plant of design dthe plant can be operated to satisfy thespecifications.

It is interesting to note that since there is aninfinite number of possible realizations for thevalues of the parameters 6. and since theoptimal operation of the plant is implicitlydependent on 6. the overall number of decisionvariables involved in problem (12) is infinite.This is because for every value of 6 an optimalvalue of the control variables z is being chosen.Also, note that the feasibility constraintrepresents an infinite set of constraints sincethe inequalities in (10) are defined for theinfinite set of values #€T. Therefore, problem(12) corresponds to a two-stage nonlinearinfinite program.

A first step in simplification to make problem(12) more amenable to solution is to perform adiscretization over the parameter space in orderto approximate the expected cost by a weightedcost function (Grossmann and Sargent, 1978),which reduces (12) to

minimize / w' C(d,z',0')

(13)s.t. Uti.z',0') £ 0, i ' 1.2_..n

where the weights w1 correspond to discreteprobabilities for the selected finite number ofparameter points 0'£T, i=1,2,...n. These weightscould be derived from the joint probabilitydistribution function of the parameters, or theycould be selected to reflect subjectiveprobabilities assigned by the designer. Note thatin the case where a joint distribution function isavailable, by suitable selection of the parameterbounds one can also define a minimum level ofprobability of parameter realization for whichfeasible operation of the chemical plant isguaranteed.

It is important to note that from a practicalpoint of view the simplification for the expectedcost as given in (13) does not represent a majorlimitation. The reason is that by optimizing thedesign for several parameter values one willobtain designs that in general are not toosensitive in the objective function to changes inparameter values. This has been confirmed withnumerical examples reported by Grossmann andSargent (1978) and by Halemane (1982).

With the simplification in (13) the number ofdecision variables is finite, since optimization isperformed over the vector d of design variablesand the finite number of vectors z1, Z2,...,zn ofcontrol variables. The control variables z1 areselected to satisfy the corresponding constraintsf(d.z'.0') £ 0, so as to achieve optimal feasibleoperation at the point 0' of the parameter space.Since the number of decision variables in (13) isfinite, but the number of constraints is infinitebecause the feasibility constraint is stillimposed, problem (13) corresponds to a semi-infinite programming problem (see Hettich. 1978;Polak. 1981).

" It is interesting to note that if the feasibilityconstraint is excluded in (13), the resultingstructure of the problem is equivalent to that ofthe deterministic multi-period problem given by

(2). This problem could then be interpreted asone where the plant operates in each periodwith the parameter value d\ and with the lengthof each period being proportional to w1. Sincethis problem can be solved very efficiently with.the projection-restriction strategy, a veryimportant question that arises is whether a finitenumber of points in 0-space can be selected, sothat by ensuring feasibility of the design forthose points, one " can guarantee that thefeasibility constraint in (13) will be satisfied.

As shown by Halemane and Grossmann (1981a,c),the answer to this problem is given by provingfirstly that the feasibility constraint in (13) ismathematically equivalent to the subproblem

max mmz

max f (d,z,0)j (14)

They show that if the constraint functionsf.(d,z.0) are jointly convex in z and 6, the globaland local solutions to the subproblem in (14)that lead to critical points 6C must Ire atvertices of the polyhedron T in (8) that definesthe parameter space. This then implies that ifthe design can be guaranteed to be feasible atthe vertices of T, it can also be guaranteed tobe feasible for all other 6£T.

Halemane and Grossmann (1981c) also show thatan interesting interpretation of the constraint in(14) is given if one defines for a fixed d and 6the function

f(d,0) min {uz

f (d.z.0)j (15)

Note that this function \id.9) provides ameasure of feasibility iy<0) or infeasibility (y>0)for the chosen design d at the parameter value6. Geometrically, y can also be interpreted asthe "depth" of the feasible region since itmeasures the maximum deviation of theconstraint functions with respect to the zerobound in (14). Furthermore, since the solution of(14) is given when the function f{6.0) attains themaximum over the set T, there can in general bea finite number of critical parameter values 6C

for which the degree of feasibility is thesmallest. In the convex case, Halemane andGrossmann (1981c) show that these critical

-z

z

-z

* 0

- 26

• 60

1 .

• 2 -

- 9d

£ 0 £

d

2

£ 0

* 0

£ 0

points would correspond to some of thevertices of the polyhedron T. These ideas canbe illustrated more clearly with the followingset of linear constraints

(16)

The feasible region for this set of constraints isshown in Figure 2a for d - 1, and thecorresponding function f is shown in Figure 2b.Note that f is nondifferentiable at 6 = 9/5. andthat it exhibits two local maxima at 6 = 1 and 6« 2. It is clear from Figure 2a that the size ofthe feasible region decreases at both extremepoints, 6 » 1 and 6 = 2, and gets enlargedtowards the interior point 6 = 9/5. The functionf plotted in Figure 2b reflects precisely thisinformation, since y is strictly negative for 1 <6 < 2, and zero at the two extreme points. Thus,there are in this case two critical points thatwould have to be considered for design, whichare in fact the two extreme points of theparameter 6.

0

0 25

-0 3

\ \ \

jz

I

Fig. 2. Feasible region and o(d,0) forconstraints (16) with d • 1.

Solution Procedure

As was shown in the last section, the two-stageprogramming formulation with the feasibilityconstraint is given by

w'C(d.z',0')

s.t. f(d.z',6H < 0 i * 1.2....n

max min max f (d.z.tf) £ 0z j€J J

(17)

A direct solution procedure for this optimizationproblem poses great difficulty since it involvesthe max-min-max constraint, which as has beenshown in the example of Figure 2 involves anon-differentiable global optimization problem(see Danskin, 1967; Demyanov and Malozemov,1974). Therefore, in order to derive a reasonablesolution procedure it is best to take advantageof the fact that in the convex case feasibility ofthis constraint can be guaranteed if theconstraint functions are forced to be feasible atthe vertices of the polyhedron T. Although thisprocedure would be strictly valid only for theconvex case, it may also be valid in someinstances when nonconvex constraint functionsare involved.

functions.

Step 2 - Determine the design vector dk bysolving the problem

k

minimize y w C(d.z'.0')

s.t. f(d.z'.0') < 0. i = 1.2....Nk

(18)

using the projection-restriction strategyfor multiperiod-design problems.

Step 3 - Determine the critical parameter values0c#k by solving for every vertex 01 inT, the problem

Given that the objective is to ensure feasibilityfor all the vertices of the set T, one approachto solve problem (17) would be to reformulate itas a multiperiod problem in which the 2P

parameter vertices are selected as the n pointsfor the design. However, this procedure couldclearly become very expensive computationallyif the number of parameters p is rather large(e.g. p ^ 4). To circumvent this difficulty.Halemane and Grossmann (1981c) have proposedan iterative multi-period design algorithm that isgiven by the following steps:

Step 1 - Set k = 0. Choose an initial set Tconsisting of N vertices where N <?2P . ° °

min {u | u £ f (dk.z.0'). j € J}

The vertex that gives rise to themaximum value of y is thendetermined and is denoted by #c>. Ifp(dk,0c>) ^ 0, stop. Otherwise,proceed to Step 4.

Step 4 - Incorporate the new critical point inthe design by defining

i ' ^ , - I V , I - <20>

set k * k • 1 and return to Step 2.

This can be achieved with small computing

requirements using the procedure suggested by

Grossmann and Sargent (1978), in which each

constraint is maximized individually by assuming

monotonicity. The gradients of >oQ of each of

the individual constraint functions f , j=1.2,...m,

with respect to the parameters 6 , k=1,2,...p, are

computed at Initial values of d and z, and the

signs of these gradients are analyzed. If for

each individual constraint function f , the

gradient df.ldO^ > 0, the upper bound du^ is

selected for the parameter 6 . whereas if

9f 730 < 0 the lower bound d^ is selected.

Clearly, for zero gradients either choice of the

bounds is possible. Since each constraint may

lead to a different vertex, the set of vertices

obtained for all constraints is finally merged

into the smaller set of vertices T by using ao

set covering formulation (see Garfinkel andNemhauser, 1972). It should be noted that if theconstraint functions f., are monotonic in theparameters $ , these vertices will correspond tothe maximization of individual constraint

•Note that at the termination of this algorithmthe design will necessarily be feasible for allvalues of parameters, because it will be feasiblefor the critical parameter values. Also, thealgorithm has to terminate in a finite number ofiterations since there can only be a finitenumber of critical parameter points. The initialvertices predicted in Step 1 by the method ofGrossmann and Sargent (1978) will often yieldvery good guesses for which only one globaliteration in the algorithm may be required. It isalso important to note that the minimizations in(19) may not have to be performed untilcompletion for all vertices, as they can bestopped when f reaches a negative value inwhich case the existence of a non-emptyfeasible region is detected. Thus, by the aboveconsiderations this algorithm should provide ingeneral a more efficient method of solution thanthe case when all the vertices are included inproblem (18). Halemane and Grossmann (1981c)have applied this algorithm to two exampleproblems, each one involving five uncertain

parameters: a heat exchanger network and areactor with a cooler. The computationalrequirements were modest since no more thantwo global iterations of the algorithm wererequired to obtain the optimal and feasiblesolutions.

The efficiency of the above algorithm could beenhanced further by making use of the followingprovisions. Firstly, the number of parameterpoints that must be considered in Step 2 couldbe kept relatively small at each iteration of thealgorithm if some of the vertices are eliminatedwhen new ones are added in Step 3. Theobvious criterion would be to discard thosevertices that have the smallest negative value off. since they are the ones that are most likelyto remain feasible for small changes in thedesign vector d. However, there is clearly noguarantee that these vertices would becomeinfeasible in the next iteration in which casethey would have to be included in Step 2 again.

The second provision would be related to theproblem of having to solve problem (19) foreach one of the vertices of the set T, which canclearly become a major burden in thecomputations if the number of parameters p islarge. For instance, for p = 10, 1024 verticesneed to be analyzed, whereas for p * 20 thenumber of vertices is 1,048,576. In order toovercome this problem, if one assumesconvexity in the constraints, a lower bound on pfor each vertex i can be computed veryefficiently by solving (19) with the constraintfunctions linearized at the nominal point (zN, 0N).That is, the lower bound y' at vertex i would begiven by

f\ * min uz

u a f (dk.zN,j

df

dz

8f(21)

where clearly f* will yield a rigorous lowerbound since the convex functions f (dk,z,#) willbe greater or equal than the correspondinglinearized functions in (21). Note that thecomputation of these lower bounds involves thesolution of a parametric linear programmingproblem in which only the right hand side ismodified at the different vertices. Hence, thesolution of the linear program would requirevery few simplex iterations at the successive

- vertices (Hillier and Lieberman, 1980). Preliminarynumerical results (Swaney, 1982) indicate that

the quality of these bounds is very good. Thesebounds could be used either as a heuristic toavoid solving (19) in Step 3, or otherwise theycould be used within a rigorous boundingprocedure since an upper bound y't can becomputed by simply evaluating the constraintfunctions at the control variables z predicted by(21). Unfortunately, numerical results haveindicated that the quality of these upper boundsis not very good. Further investigation wouldbe required to test the effectiveness of thisprocedure.

Discussion on locating critical parameter points

Clearly one of the major difficulties involved inthe problem of design under uncertainty is theselection of a finite number of critical pointswhose feasibility will ensure feasibility for thewhole set of parameters T. Ideally, one wouldlike a procedure by which the critical parameterscould be predicted a priori. At the simplestlevel one could think of using intuition orengineering judgment to do that, for instance byselecting what would appear to be the "worst"parameter values (e.g. low transfer coefficients,low efficiencies, high flowrates, etc.). However,as has been shown with the heat exchangernetwork example by Grossmann and Sargent(1978). this selection is not always trivial sincein their problem the feasible design is notobtained by selecting the lower bounds for theheat transfer coefficients which would be

1 normally regarded as the "worst" values.

A further complication, as was illustrated in theexample of Fig. 2. is that there may be severalcritical points that may have to be consideredfor the design. A procedure that can predictseveral points a priori is the one suggested byGrossmann and Sargent (1978). This procedurepredicts the critical points by analyzing the signof gradients of the constraints, as was outlinedin Step 1 of the algorithm. However, althoughthis procedure is very often successful, it mayin some cases fail to predict the right set ofcritical parameter values. In order to gain someinsight as to why this may happen, and also totry to understand under which conditions asingle critical point exists. consider thefollowing set of three linear constraints thatinvolve two control variables, two parametersand one design variable:

(22)

2. i * 1.2.

Since these constraint functions are linear, theirgradients are independent of the initial pointchosen for such calculations; and since they aremonotonic in 6, the maximization of each ofthese functions can be performed by analyzingthese gradients. It is clear that the threeconstraint functions get maximized at the threedifferent vertices dy ' [2.1], d2 - [1,2] and 03

* [1,1] respectively. In order to show that thesevertices can lead to designs that may beinfeasible, consider problem (15) for establishingthe feasibility at a given design d, andparameter 6. which yields

f(d.d) * min uz

s.t. f * -z i • 26 y - 6 < u

(23)

tf - -i/j r.>.X_

+ *l - t» .«:

Fig. 3. Feasible region for theconstraints in (22) withd - 3.

Figure 3 gives a plot of the feasible region forthe set of constraints in (22). with a design d =3, wherein the values of y obtained from (23)are also given for each of the four vertexpoints of the parameter-space. Since theconstraint functions are linear and hence convex,the critical parameter points must lie at avertex. Clearly, from the values of f in Fig. 3.6* » [2,2] is the critical point, since y attainsits maximum at this vertex. Also note in Fig. 3

1 2 3

Fig. 4. Feasible region for theconstraints in (22) withd - 4.

that the design d = 3 is found to be feasible forthe three vertices predicted by maximization ofconstraints. 9y = [2,1], S2 = [1,2] and 03=[1.1],whereas it is infeasible for the criticalparameter point - namely the vertex 6* = [2,2].To make the design feasible for the criticalpoint as well, consider that the design variableis increased to d = 4. The feasible region andthe value of f for the four vertices are againshown in Fig. 4 for this value of d = 4, whereas it can be seen the design is just feasible forthe critical point, whereas a finite region offeasibility exists for all other points in theparameter space. Thus, by ensuring the

feasibility of the design for the criticalparameter values as predicted by the max-min-

' max constraint, it is possible to guarantee theoverall feasibility of the design for everyparameter value within the specified range.

In order to gain some further insight as to whythe maximization of individual constraintfunctions will not always lead to correct criticalpoints, assume that for all #£T the samecommon (single) set of values z is selected forthe control variables. It then follows that themax-min-max constraint reduces to

max max f (d,z.0)J

(24)

which is equivalent to

max f iJ

(25)

Thus, if for the design d it is possible to selecta control 2. feasible and common for all 06T.then some of the parameter points predicted bymaximization of individual constraints willcorrespond to those (critical points) predicted bythe max-min-max constraint. However, it isclear that in general different controls z mayhave to be selected for different realizations of6 to maintain feasibility and, therefore, bymaximization of the individual constraintfunctions one may not always predict thecorrect critical points in a design.

Another interesting question about locating

critical points is to determine the conditions

under which only a single critical point will need

to be considered for the design. To analyze

this case assume that for a given design d, the

set of constraint functions is linear and given

by:

a B

k=1

b z • cjk k j

0,

(26)

j - 1,2,...mJ

The critical point 6C is given by the solution tothe problem:

max min u6€T z

s t -

(27)

k*1

bj k \

Assume that for any SET the problem

f{6,6) * min uz

s.t.

(28)

u. j = 1.2....mk=1

has the same set of active constraints (e.g. thefirst r. r £ m).

The Kuhn-Tucker conditions for the aboveminimization problem then yield

(a) 1 0, j

(b) 0

(c)

' k

b zjk k

(29)

c . j - 1,2....r| • 1

Since the value of u at the minimum determines), it follows that

r • 7 X ( Z % *k

(30)

-. - . , )

From this expression it is clear that y(d.d) islinear and hence monotonic in 8. for the chosendesign d. Therefore the maximization of f in(30) will lead to a single critical point. Note thatin (29) the active constraints are determined bythe values of the multipliers X , which are inturn obtained from the subset (a).(b), and since r£ m. a necessary condition for having the sameset of constraints to be active for every 0£Twould be: m ^ n • 1. In general, a necessaryand sufficient condition would be to have thevalues of all the multipliers X (j=1,2,...m) uniquelydetermined by the system (29). Furthermore, iffor every k, signOy/90^) = sign(a ) for someconstraints j, then these constraints are

.maximized at the point defined by max f(d,d),and under these conditions the maximization ofindividual constraints with respect to 0 wouldlead to the critical points.

To illustrate these ideas consider the set ofconstraints given by (22). The solution to (29 a,b)yields X s X - X - *1/3. indicating that the threeconstraints are active for all 9. Furthermore,from (30) the value for y is obtained as fi6,d) -1/3(^1 • 6 - d), indicating that p(d,0) is indeedmonotonic in 6, and that maximization of fresults in the single critical point given by 6A =[2,2]. Note that the signs of the gradients3f/B$} and dflBQ are both positive, whereasthere is no constraint in (22) that satisfies thiscondition. Therefore, in this case since themonotonicity of f cannot be related with themonotonicity of the individual constraintfunctions f , the maximization of individual

iconstraint functions does not predict the rightcritical parameter value. From the analysispresented above, it would be most interesting toinvestigate whether for nonlinear constraints

with special structure the property ofmonotonicity of y also holds when the sameset of constraints remains active for all thevertices in T.

Future directions. As has been shown above, itis the aspect of feasibility that greatlycomplicates the two-stage programmingformulation for the design problem underuncertainty. In the case when feasibility can beensured by considering only the vertices in T,

the proposed algorithm provides a reasonableway of tackling the problem. However, there isno question that there is still great incentive toenhance the efficiency of this algorithm andsome of the provisions suggested above shouldbe explored further. The greatest challenge,however, would be to devise a procedure thatcould also handle the nonconvex case for whichthe critical point may not correspond to avertex. This would require the solution of thenondifferentiable global optimization problemthat is involved in the max-min-max constraint,which at the present time appears to be anextremely difficult problem to tackle.

DESIGN WITH OPTIMAL DEGREE OF FLEXIBILITY

In the first part of the paper procedures wereoutlined which treat the case of design for afixed degree of flexibility. In that case therequired flexibility is pre-specified. either by adiscrete set of required operating conditions orby requiring feasibility of operation when a setof uncertain parameters can vary between fixedbounds. The more general problem is todetermine the design which possesses theoptimal degree of flexibility. Solution of thisproblem requires a quantitative characterizationof the property of flexibility.

For the flexibility of a design to be "optimal"requires that the economic advantages offlexibility be balanced in relation to its cost.As stated before, flexibility as a design attributerepresents the ability of a design toaccommodate variations: with a higher degreeof flexibility, the range of tolerable variations isgreater. The uncertain parameters which

describe the variations may be considered asrandom variables, and conceptually theirrealizations may be described in terms of theirjoint probability distribution. It follows then

- that a design featuring a higher degree offlexibility will have a lower probability ofencountering infeasible operation. Since there

will be an economic penalty incurred wheninfeasibilities prevent successful operation, thereis strong motivation to provide a design with anadequate degree of flexibility.

Conceptually one could construct a nonlinearprogram with an objective function whichinvolves the expectation of the compositeeconomic cost. including penalties forinfeasibility. In theory the solution to thatstochastic program would determine the optimaldegree of flexibility. The practical problems ofsuch an approach are two-fold. First, thecombined occurrence of the feasibility constraintand the expectation operator make the aboveprogram one of great mathematical difficulty.Second, it is doubtful that either the probabilitydistributions of the uncertain parameters or theeconomic penalties for infeasibility will ever beknown very accurately in a practical plant designsituation. For these reasons the complexstochastic program as outlined can hardly bejustified, and pragmatic simplifications are inorder.

Fig. 5. Trade-off curve for flexibilityand cost.

A key step towards simplification is theseparation of the composite objective function*into two components: minimizing capital andoperating costs on the. one hand, and maximizingflexibility on the other. The resultingformulation then takes the form of amulticriterion optimization problem, withannualized cost and degree of flexibility as twosimultaneous objective functions. The standardprocedure would then be to construct a trade-offcurve relating flexibility to cost as shown in Fig.5 (see Clark and Westerberg, 1982). This couldbe done for instance by using the * -constrainedmethod where one objective is optimized whilethe other objective is set to the limit < whichis varied parametncally (Haimes. Hall and

Freedman. 1975). Examination of the curvewould allow the assessment of an appropriatetrade-off, thereby establishing an "optimal"degree of flexibility. The principal requirement insuch a procedure is that a quantitative measurefor the degree of flexibility be available. Thisneed of a metric for flexibility is the motivationfor the flexibility index described below.

An Index of Flexibility

The problem at hand is to construct a scalarmetric whose value for any fixed designcharacterizes the size of the region of feasibleoperation in the space of uncertain parameters.Since for each realization of the parameterscontrol variables will be adjusted to attainfeasibility of operation (if possible), the feasibleregion in 0-space may be defined as

(3D

'where the vector of inequalities f(d,z.#) £ 0define feasibility in (z, 0)-space. In general theactual shape of this region could be rathercomplex, being defined by a boundary whosepoints are determined implicitly by the equationf{6.&) = 0 (see Fig. 6), while the function p(d.tf)is itself the result of the nonlinear programshown in equation (15). A particular example ofthis region which corresponds to the set ofconstraints in (22) with d = 4, is shown in Fig.7.

Fig. 7. Plot of region R for the setof constraints in (22) withd » 4.

Since in general the geometry of the feasiDieregion as given in (31) is difficult to treat in ameaningful way, the following approach isproposed. It may be assumed that the uncertainparameters will vary independently of eachother.2 It makes sense then to analyze thefeasible region R in terms of the maximumranges over which the parameters may varyindependently of each other while still remaininginside the feasible region. Geometrically thisapproach corresponds to inscribing a hyper-rectangle within the feasible region as shown inFig. 6. The size of the feasible region is thencharacterized by the lengths of the sides of therectangle. The remaining difficulty is that therectangle is not uniquely determined; trade-offscan result by increasing the range of someparameters while decreasing the range of others.

F••tibia

tX.

-f<4.») - 0

Fig. 6. Hyper-rectangles containedin the feasible region R.

The solution is to supply a set of scalingfactors which in effect determine theproportions of the rectangle. With these factorsin hand, positive and negative variations in theuncertain parameters may be expressed asscaled deviations from a given nominal value:

*N.a (32)

The point 0N specifies some nominal operationthat will be feasible; the scaling factors A ~ #and L~d, that represent expected positive and

1

negative deviations may well differ from eachother, since the nominal values dN may not lie atthe centers of their parameter ranges. One may

If the set of parameters tn **• O''?--a! P'os'em forma avonart depenoent. men onnciDai component a-»a'v$t$ mav oeemployed to obtain an maepenoent set.

then consider the feasible region expressed inthe space of the scaled parameters as shown inFig. 8. In the scaled space the rectangleappears as a hypercube, centered at the nominalpoint (located at the origin). The dimension ofthe largest hypercube which may be inscribedwithin the feasible region may then be adoptedas the desired measure of the size of theregion. The index of flexibility, F, is thereforedefined as one-half the length of a side of thathypercube. Note that this hypercube has theproperty that for any of the parameter pointscontained in it. the existence of controlvariables which meet the design specificationsand constraints is guaranteed.

,t f i

Fig. 8. Maximum hypercube contained inthe feasible region R.

The choice of appropriate scaling factors in (32)requires some comment. Arbitrary choice of thescales will of course give unsatisfactory results.For instance, one might consider scalingparameter deviations in proportion to theirnominal values, in effect giving equal weight toequal percentage changes for all parameters. Togive the same weight, for example, to apercentage change in temperature as to apercentage change in flowrate would in mostcases be inappropriate. Referring again to Fig.6, the flexibility index implies a rectangularregion in the space of uncertain parametersinside of which feasible operation is guaranteedfor all combinations of parameter realizations.By considering the probability distributions ofthe parameters one could in theory compute thetotal probability that parameter realizations willTie "within the rectangle by integrating the jointprobability density function over the rectangular

region. Since it is this probability of feasibilitywhich is the underlying objective of flexibility, itwould make sense to define the rectangle in away which tends to maximize that probabilityfor a given cost. Usually only approximateknowledge of the individual probabilitydistributions will be available, so that a rigorousmaximization will not be possible. However, itis reasonable to expect that some estimate ofrange or variance measure will be available foreach parameter, to be specified by the designengineer based on experience, statistical data, orrule-of-thumb target values. An appropriatechoice of scaling factors would be to use thesevariances or range estimates- directly; this choicehas the following heuristic support.

Consider the problem of defining one corner ofa rectangle in a space of two parameters asshown in Fig. 9. Given probability distributions,contours may be constructed representing thelocus of corners for rectangles which enclosethe same total probability. Since the individualprobability distributions will usually be unimodal,these contours will be concave; examples fornormal and uniform distributions are shown inthe Fig. 9. By invoking the traditional convexcost argument. another contour may beenvisioned which represents the locus of cornersfor rectangles corresponding to designs ofconstant cost. Since this cost contour willusually be convex, the rectangle whichmaximizes the probability of feasibility will haveits corner located near the "knee" of aconstant-probability contour. By scaling inproportion to the square-root of the variance ofthe individual distributions, the rectangle corneris positioned along a ray which passes throughthe "knees". By virtue of this the direct use ofthe variance estimates as scaling factors isdeemed reasonable.

Thus, the index of flexibility F represents thesize of a scaled hypercube region of guaranteedfeasibility, with that size being an approximaterepresentation of the total probability thatparameter realizations will be feasible.

In order to evaluate the index of flexibility F, anappropriate mathematical formulation isnecessary. Assuming that the state variables xare eliminated as in equation (9). thespecifications and constraints for a fixed designd are given by the vector of inequalities Hd.z.i/J£ 0, where the control vector z is to beadjusted for different realizations of the vector

a) Normal Distribution

b) Uniform Distribution

Fig. 9. Scaling of deviationsthrough distribution functions

of uncertain parameters 6 to achieve feasible

operation. It is not necessary to introduce the

variables h+ . «T . i * i....p explicitly, since a

single scalar variable <5 may be used to

characterize the hypercube of feasible deviations

T « {d\dN - OL'S

(33)

The flexibility index, F, for a given design, d. isthen given by the semi-infinite programmingproblem

F « max Z

s.t.

where the first constraint imposes the feasibilitycondition for all 6 values that lie within the

- hypercube T. Using the equivalent formulationfor the feasibility condition in (14), this problemmay also be formulated as

F « max 3

s.t. max min max f (d.z.0)0€T 2 jGJ '

(35)

f 1

As shown by Swaney and Grossmann (1982), theproblem in (35) exhibits two importantproperties. The first is that if the constraintsare jointly convex in z and 6, the maximum ofthe feasibility constraint lies at one or severalof the vertices of the hypercube T given by (33).The second one is that the function y given byequation (15) (which provides a measure offeasibility) is zero at these vertices. Theseproperties are illustrated in Fig. 8, where thehypercube touches the boundary y{6.6) - 0 atone of its vertices. Note that this vertex maybe interpreted as a critical parameter valuewhich identifies a worst-case condition for thedesign.

An Efficient Vertex Enumeration Procedure

The formulation in (35) provides the definition ofthe flexibility index for a chemical plant offixed design. As in the case of design underuncertainty, the max-min-max constraint is themajor source of difficulty when seeking anefficient solution method for (35). However, byassuming that the critical parameter values must•occur at vertices of the hypercube, the problemis simplified considerably. Swaney andGrossmann (1982) have recently developed anefficient search method to find the smallest ofthe maximum tolerable parameter deviations forall vertex directions.

Basically they consider the followingsubproblems

max 5

s.t. M.i.8) i 0

0 • 6" *

k € V (36)

where (A#)k is the vector of deviations that leadto vertex k and V is the set of vertices of thehypercube T. They show that these subproblemsdefine points on the boundary of the feasibleregion y{6.6) £ 0. The value for F is then takenas o * min J .

In order to avoid solving each subproblem in(36) explicitly, the approach suggested bySwaney and Grossmann (1982) takes advantageof the fact that the o^ value need only bedetermined for the critical, or worst-case,vertices. Since o > o for those vertices whichdo not belong to the worst case set. two pointsmay be noted for non-worst-case vertices: 1)The actual value <3 is unimportant, and 2) Thepoint 6 ' 0N • oiA&)k lies within the feasibleregion. Therefore, the worst-case set may beidentified by testing vertices for feasibility at 3- 0N • J(A#)k. A procedure may then beconstructed wherein the suboroblems in (36) foreach vertex can usually be replaced by a mucheasier feasibility test. Basically, the verticesare searched in a sequence; feasibility of eachvertex is checked at the point 6 - 0N • o(A#)k

where <S is the current upper bound on o (basedon o for the vertices in the current estimate of

k

the worst-case set). If the vertex is infeasibleat <5, then the value of o is obtained bysolving (36), the vertex enters the worst-caseset. and the bound 1 is updated. If the vertexis feasible at o, the search proceeds to the nextvertex.

A good initial estimate for the worst-casevertex direction may be obtained by determiningat the nominal parameter 6N the steepest-ascentdirection of the feasibility measure f(6.6). Thiswould involve the following steps:

Step 1. Determine at 6H the control zN suchthat

mm uz

(37)s.t. u £ f (d.0N,z) j 6 J

Step 2. For small positive perturbations A # .i=1.2,...p. along each i'th coordinate in theparameter space determine

f « mm uz

s.t. u * f (d.zN,0N) •J

df Bf

(38)

J€J

Step 3. Select the vertex direction A # * suchthat (Ar)T A0* > 0. where Af1 = fx - fN, i=1,2...p.

It should be noted that if y(d.6) is monotonic in"97 the initial estimate will lead to the correctcritical vertex for defining the flexibility index

F. However, since the property of monotonicitycannot be expected to hold in general, feasibilitymust be checked for all the other vertices.

To establish feasibility at a given point drequires only thai some z be found for whichf(d.z.O) ^ 0. The z value which solves equation(15) would be a sufficient choice, and a solutionprocedure applied to (15) will serve as aneffective feasible point procedure. However, aswas discussed in the algorithm for design underuncertainty it is in general not necessary to findan optimal solution for (15); any point z forwhich u ^ 0 will establish feasibility. Theprocedure may thus be halted as soon as thecondition u < 0 is obtained, with the result thatin many cases feasibility will be establishedwith less work than would be required to solve(15) completely. Clearly, for infeasible vertices,termination with u > 0 will result.

Significant economy in performing the feasibilitytests may be achieved by taking advantage ofthe fact that the same set of active constraintswill apply to many vertices in the feasibilitysubproblems. By carefully ordering the sequenceof vertex examination, the computational workrequired to locate a feasible z can be minimized.A heuristic procedure for vertex sequencingwhich provides an initial sequence, as well as anevolutionary re-ordering method has beendeveloped and is described in Swaney andGrossmann (1982) who apply it to an example•problem. It is also interesting to mention thatthis vertex enumeration procedure could beapplied in Step 3 of the algorithm that waspresented for design under uncertainty.

A Bounding Procedure

The rigorous evaluation of F requires that thelocation of the pointis) where the inscribedhypercube touches the boundary of the feasibleregion be determined. In the above procedure itwas assumed that the critical points will lie atvertices of the hypercube. and that they couldtherefore be identified by searching among theset of all vertices. Unfortunately, if the numberof parameters p becomes large the method canbecome expensive, since the number of verticesto be analyzed increases as 2D.

The important question that arises, then, iswhether a procedure can be derived forevaluating the flexibility index F which does notnecessarily require analyzing all of the vertices

in the parameter space. Provided one is willingto assume convexity in the feasible region, it ispossible to derive a bounding procedure that canaccomplish this objective. This bounding

procedure is based on two observations. First,a valid upper bound on F is given by ahypercube centered at the origin which containson at least one of its faces any point belongingto the boundary of the feasible region. Second.if convexity in the region is assumed, a lowerbound may be obtained by determining thelargest hypercube that can be inscribed in apolytope that is contained within the feasibleregion, and whose vertices lie on the boundary.This suggests the following procedure whosefirst three steps are depicted in Fig. 10:

min (40)

1

y * / J"7

YtL1

trrI

/f

4

Fig. 10. Lower and upper bounds formaximum hypercube

Step 1. For a fixed value of d, the maximumpositive and negative deviations for eachparameter are obtained by solving the following2p subproblems:

max 3*i

s.t.

Step 3. A lower bound F is obtained bydetermining the largest hypercube that iscontained in the polytope defined by thedeviations uT «T i=1....p obtained in step 1. Thishypercube may be determined as follows:

a) For each parameter i calculate

1

s min{ o *,6 "}

where s (41)

which will define a vertex k taken in thedirections s from the nominal point.

b) The equation aT 5 - 1 then describes thehyperplane containing that face of the polytopewhich is closest to the nominal point (origin).

c) Solve for the lower bound F using a15 = 1by setting 3. = signU )F.

Step 4. a) An improved upper bound F isobtained by solving (36) along the vertexdirection which corresponds to the face of thecurrent polytope that is closest to the origin(e.g. in the first iteration the direction is definedby (41)).

b) If F = F, or the bounds are within a specifiedtolerance, stop. Otherwise go to step 5.

Step 5. a) The polytope contained within thefeasible region is expanded by incorporating theadditional boundary point found in step 4.

b) The lower bound F is updated by inscribingthe largest hypercube in the expanded polytope.Return to step 4.

max 5" (39)z

s.t. f(d.z.0N,...,0N-i i

Step 2. A valid upper bound F for theflexibility index is given by the smallestdeviation obtained in step 1.

It should be noted that this bounding procedurerequires solving at least 2p * 1 optimizationsubproblems for determining the boundarypoints. Therefore, it is clear that potentialcomputational gains can only be achieved if thenumber of parameters p. is strictly greater thantwo. Also, since there is no guarantee that forsome cases (e.g. symmetric regions) all of thevertices will not have to be analyzed, theefficiency of this bounding procedure is

unpredictable. Nevertheless, in a number ofinstances the procedure would require analyzingonly a small number of vertices, and if theexact determination of F is not required it couldstill be a useful tool. Finally, since theassumption of convexity is crucial inestablishing the validity of the lower bound, thevertex enumeration procedure presented in theprevious section is of more general applicability.

Future directions. It is clear that the flexibilityindex defined above is only one possible choice.Although this index has the advantage of ameaningful physical interpretation, it might be*worthwhile to explore other options which, forinstance, do not require the definition of anominal point. As for the solution procedures, \more computational experience is required totest their effectiveness. Also, it would beparticularly important to develop an efficientnumerical procedure for solving the bicriterionoptimization problem of minimizing cost andmaximizing flexibility.

GENERAL REMARKS

This paper has attempted to present a unifiedapproach for the problem of design of flexiblechemical plants. As has been shown, this areaoffers a number of very interesting possibilitiesat both theoretical and practical levels.

On the theoretical side, the problems inflexibility give rise to optimization problemsthat involve large numbers of decision variablesand/or an infinite number of constraints. Thedistinct feature of these optimization problemsis that the major difficulty lies in the feasibilityconstraints, as opposed to the case wheredesigns are optimized for a single nominalparameter value. It is clear that there is stillmuch work required to derive efficientoptimization strategies for solving thechallenging problems that arise in flexibledesign. More research is required to improvethe computational efficiency of the methods thathave been described in the paper for multiperioddesign, design under uncertainty and theflexibility index. Also more work is required toextend these methods, or develop new ones, forhandling the nonconvex case. An avenue that isalso worthwhile exploring for the developmentof strategies is the use of computer graphicswhich can yield interesting insights into ihefeasibility and movement of constraints (seeArkun and Stephanopoulos. 1979; Etzkorn and

Arkun, 1982). It is interesting to note that inthe field of electrical engineering verysubstantial progress has been made in problemsrelated to flexibility, such as in the problem ofdesign centering, tolerancing and tuning ofelectronic circuits (see Bandler. 1974; Bandler,Lui and Tromp. 1976; Brayton, Director, Hachteland Vidigal, 1979; Director and Hachtel. 1977;Madsen and Schjaer-Jacobsen, 1978; Polak andSangiovanni. 1979; Mayne, Polak and Voreadis,1982). However, it should be noted that thenature of these problems is somewhat differentfrom the ones encountered in chemical processdesign. The emphasis in the work in electricalengineering has been on manufacturing where theuncertainties arise in the design variables.Another crucial difference in these problems isthat in most cases no control variables asconsidered in this paper are included.

On the practical side, the area of flexibility addsa new important dimension to the field ofoptimization of chemical processes. The reasonis that since in this area the main concern isthe feasibility of operation of the plant, there isthe possibility of obtaining designs that are notonly economically attractive, but perhaps moreimportantly, sufficiently robust to satisfy designspecifications despite uncertainties in theparameter values. Since this would seem to bethe overriding concern of design engineers (see

Blau, 1981), it is expected that systematicprocedures for flexible design could have an

'important impact in practice. Furthermore, itshould be clear that the aim of these proceduresis not only to determine rational overdesigns,but also to provide tools that can be used tohelp close the gap that currently exists betweenthe activities of design and control of chemicalplants. It is clear ihat before performing adetailed analysis on the dynamics and control ofa process, it is first of all necessary todetermine whether in fact feasible operationover a finite range of parameters can beattained with a proposed design. In principlethese tools for flexible design could also beused to synthesize systematically flexibleprocess flowsheets, although there is clearlymuch more research work required to accomplishthis goal.

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ACKNOWLEDGEMENTS

We are very grateful to Darla Pitlock for herpatience in typing this manuscript. The helpprovided by Peter Clark in the use of theSCRIBE system is greatly appreciated. Finally,the authors would like to acknowledge thefinancial support provided by the NationalScience Foundation under Grant CPE 79-26398.