sensitivity, responsivity, stability and irreversibility as multiple objectives in civil systems

Advances in Water Resources Vol. 1 No. 2 1977

Sensitivity, Responsivity, Stability and lrreversibility as Multiple Objectives in Civil Systems

YACOV Y. HAIMES

P r o f e s s o r of Sys tems Engineering & Civil Engineering, Case Institute of Technology, Case Western Rese rve Universi ty , Cleveland, Ohio

and

WARREN A. HALL

Elwood Mead P r o f e s s o r of Engineering, Depar tment of Civil Engineering, Colorado State Universi ty, Fort Collins, Colorado

Received April 1977 Revised and Accepted June 1977

This paper recommends the consideration of sensitivity, stability, risk, and irreversibility as objective functions in water resource management models within the framework of multiobjective analysis. Six major sources of uncertainties and errors in systems modeling are identified. They are associated with the following model characteristics: model structure (topology), model parameters, model scope or focus, data, optimization technique, and human subjectivity. In particular, the major objective of this paper is to set the stage for the development of an analytical and operational multiobjective framework which will provide decision-makers and planners with alternatives that consider systems' sensitivity, responsivity, stability and irreversibility along with cost and other performance indices as multiple objectives, This type of a framework should have a very. wide spectrum of applications in water and related land resources, environmental studies, energy, and others. The Surrogate Worth Trade-off method is proposed for the solution of the resulting multiobjective optimization problem.

I. INTRODUCTION

Man-made s y s t e m s are planned, designed, constructed, opera ted and modified under many r i sk s and uncon- t ro l lab le uncer ta in t ies . While in genera l the t e r m s r i sk and uncer ta inty can connote the same thing, for analyt ical purposes we shall define these as two dist inct concepts . Risk is cha rac te r i zed by a f r e - quency distr ibution of events following reasonably well-known or measu rab l e probabi l i t ies , even though the specific sequence of events is l a rge ly controlled by chance.

In contras t to r i sk , uncer ta inty is cha rac te r i zed by no known, reasonably valid probabi l i ty distr ibution of events . The t e r m r i sk is ass igned to measu rab le chance-cont ro l led fac to r s , while uncer ta inty appl ies to all o thers . In water r e s o u r c e s for example (see Haimes , Hall, and F r e e d m a n [1975] and Haimes [1977]), there a r e uncer ta in t ies a s soc ia t ed with the growth of population, industry, ag r icu l tu re and urban a r ea s , the p ro jec ted cost of labor, ma te r i a l s , and inflation, the a s s e s s m e n t of future advancement in engineering, science, and technology, and the p ro jec ted benefits a s soc ia ted with the p ro jec t s .

At p resen t most , if not all, mathemat ica l models t r ea t important sy s t em cha rac t e r i s t i c s such as r i sk , uncer ta inty, sensi t ivi ty , s tabil i ty, responsivi ty , i r - r evers ib i l i ty , etc. , e i ther by means of sys tem con- s t ra in t s or by ar t f f ica l ly imbedding them in the ove r - all index of pe r fo rmance . The s y s t e m s analyst (the

modeler) a s s u m e s both the ro le of the profess iona l analyst and the dec i s ion -make r by explicit ly or implicit ly assigning weights to these and other non- commensura t e s y s t e m c h a r a c t e r i s t i c s , t h u s commensura t ing them into the p e r f o r m a n c e index (the mathemat ica l mode l ' s object ive function). Obviously, this p roce s s is questionable even where the analyst is the dec i s ion -make r . When he is not, the resu l t will seldom be the d e c i s i o n - m a k e r ' s opt imum.

It is a rgued here that the above sys t em c h a r a c t e r i s - t ics can and should be quantified and included in the mathemat ica l models as s epa ra t e objective functions. These should then be opt imized along with the original model ' s object ive function (index of p e r - formance) , to allow the dec i s ion -maker ( s ) to se lect a p r e f e r r e d policy (solution) f r o m within the Pa re to opt imal set . Any p rocedure short of recognizing these cha r ac t e r i s t i c s as object ive functions in thei r own right essent ia l ly c o m p r o m i s e s the modeling p r o c e s s . One of the major sources of skep t ic i sm in many agencies about the utility of actually implement- ing systems optimization methodologies in water resources planning and management, for example, is the failure to recognize the above important system characteristics as objective functions, and to generate and present the associated trade-off values to the decision-makers.

The subject of sensitivity has been studied by Sobral [1968], Guardabass i [1969],Wilkie et al [1969], Tomovic [1963], Kokotivic and Rutman [1965], Ngo

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Multiple Objectives in Civil Systems Yacov Y. Haimes and Warren A. Hall

[1971], and others . Gembicki [1974], Gembicki and Haimes [1975],Haimes,Hall and Freedman [1975], Rarig [1976], and Haimes [1977] fur ther studied the sensitivity problem in the context of a multiobjective optimization problem as will be fur ther discussed in this paper .

The subject of r i sk and i r revers ib i l i ty has recently been discussed in the l i te ra ture by severa l r e - sea rche r s including Fisher , Krutilla and Cicchetti [1972], Arrow and Fisher [1974], Cummings and Norton [1974], Abras sa r t and McFar lane [1974], F isher , Krutilla and Cicchetti [1974], Krutilla and Fisher [1975], and others. In the above cited work, the classical commensurat ion of costs and benefits is used along with the expected value concept. The dynamics of the sys tem are handled through the use of the Hamiltonian and optimal control theory. Al- though the above work contributes to the understand- ing of the difficult issue~ associa ted with r isk, uncertainty, and i r revers ib i l i ty , it does not t rea t the problem f rom a multiobjective optimization point of view, which is the major departure advocated in this paper .

The Committee on Public Engineering Policy (COPEP) [1972], of the National Academy of Engineering o r - ganized during April 26-27, 1971, a colloquium on 'perspec t ives on benef i t - r isk decis ion-making. ' The colloquium was pr imar i ly addressed to r i sks to life, health, or safety. The following three major categories of decis ion-making were focused on:

(i) Individual or voluntary r i sks (e.g. sports , smoking, etc.)

(ii) Risks where the individual 's options are somewhat limited by regulations

(iii) Risks in which governmental action preempts voluntary individual decision making (e.g., air pollution, nuclear energy, and public health).

In that colloquium, the COPEP extended the benefit- cost concept to include the evaluation of all the benefits and the costs of a proposed action. The COPEP also identified what they call ' the necessa ry ingredients of a p rocess of rational analys is ' when addressing the benef i t - r isk subject. These are:

(i) The explicit recognition of uncertainty

(ii) Consistency in a s sessment of values

(iii) Distinguishing between decisions and outcomes (i.e., a good decision could, because of bad luck or unforeseeable events, lead to an undesirable outcome)

(iv) Considerat ion of t ime pre fe rences (i.e., giving proper weighting to s h o r t - t e r m and long- te rm benefits and r isks) .

The COPEP Colloquium and its findings essential ly establish the needs and importance of the approach proposed in this paper . Chauncey Starr [1969] who published his f i r s t paper on quantifying r i sk and benefits in 1969, contributed a paper in the above meeting (see COPEP [1972]) where he states:

'What we a re discussing, however, is inclusion of all societal costs , indirect as well as direct, and all measures of utility, direct as well as indirect . Clearly, existent social - technical sys - tems, over a per iod of many years , have developed an empir ical ly acceptable balance between utility and social cost . In addition, we

have examples of national decision making involving future social - technical sys tems that contain implicit predict ive t rade-offs of societal benefits versus societal cos t s . '

This paper recognizes the above importance of t rade- off analysis addressed by D r . S t a r r . Once sys tems charac te r i s t i cs such as risk, sensitivity, responsivity, i r revers ib i l i ty , and others a re quantified, t rade-offs among all benefits and costs can be generated via multiobjective optimization analyses . In his paper, Chauncey Starr concludes:

'It is evident that we need much more study of the methodology for evaluating social benefits and cos ts . The fatality measure of public r isk is perhaps more advanced than most because of decades of data collection. Nevertheless , even the use of crude measures of both benefits and costs would ass is t in the development of the insight needed for national policy purposes . We should not be discouraged by the complexity of this problem--the answers are too important, if we want a rational socie ty . '

2. SENSITIVITY, RESIK)NSIVITY, STABILITY, AND IRREVERSIB~LITY

The definition of r isk in the sense of an objective to be minimized appears deceptively simple but is in fact extremely complex. The question is, r isk of what? The answer to this question is usually a long list of undesirable outcomes and combinations of outcomes, each with a non-negligible probabili ty of occurr ing.

While in some cases a specific quantitative r i sk index can be defined, and used as the objective, more often there will be an excessive number of such indices. In such cases it is possible that certain r i sk related charac te r i s t i c s of the sys tem can be identified, quantified and used to serve as a single measure of many of those individual r isk objectives. Among these charac ter i s t ics , sensitivity, responsivi ty, stability and i r revers ib i l i ty appear to be par t icular ly important.

Although it is recognized that the current s ta te -of - the- ar t in sys tems analysis is not yet fully capable of quantitatively t reat ing all of these charac ter i s t ics , it is essential that they be considered as thoroughly as possible . They are defined as follows:

Sensitivity re la tes changes in the sys t em ' s pe r fo r - mance index (or output) to possible variat ions in the decision var iables , constraint levels and uncontrolled pa rame te r s (model coefficients).

Responsivity represen t s the ability of the sys tem to be dynamically responsive to changes (including random variat ions) in the decisions over a per iod of time.

Stability re la tes to the degree of variat ion of the mean sys tem response to fixed decisions. A stable sys tem yields an invariant mean response to the mean value of a decision set. A sys tem may be stable and still have an important random component.

Irreversibility measures the degree of difficulty involved in res tor ing previous states or conditions once the sys tem has been a l tered by a decision (including the 'decis ion ' to do nothing).

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2.1 Sensitivity

One can cons t ruc t hypothet ical s i tuat ions in which the d e t e r m i n i s t i c ma themat i ca l opt imum decis ion would be the wors t poss ib l e unless the decis ion va r i ab le could be ve ry p r e c i s e l y control led . F igu re 1 shows such a si tuation, in which it is p r e s u m e d that the decis ion va r i ab le can be cont ro l led only within l imi t s , Xc, and that x may take on any value with equal l ikel ihood within these l imi t s . The d e t e r m i n i s - t ic ma themat ica l maximum is obviously far f rom being the p r a c t i c a l opt imum decis ion. In this con- t r i ved example , x~ is c l e a r l y a ' b e t t e r ' dec is ion than x~, unless the d e c i s i o n - m a k e r is more in t e re s t ed in gambling than avoiding r i s k .

Even if the example is t r e a t e d by maximiz ing the mathemat ica l expectat ion of f(x), it does not follow that a resu l t ing 'op t imum' at x 1 is s u p e r i o r to x 2. F o r this to be t rue the app rop r i a t e object ive must indeed be to maximize or min imize the expected value of f(x). This is se ldom t rue where r i sk is a ma jo r cons idera t ion . The ' g a m b l e r ' s ru in ' p rob lem is the c l a s s i c example where this is c l ea r ly not the objec t ive .

As in many p r a c t i c a l p rob lems in water r e s o u r c e s , the dec is ion which max imizes the expected value of the re turn to the gamble r will a l so co r r e spond to a maximiza t ion of the r i s k of get t ing l i t t le or nothing. In r e a l i t y the re a r e two noncommensurab le ob jec t ives in this case , avoiding r i s k and gaining economic r e tu rn .

2.2 Responsivity

This is the capabi l i ty of the sy s t em to respond in a r easonab le t ime to a va r i ab l e (changing) decis ion. It is gene ra l ly r e l a t ed to ' f r i c t i ons ' in the sys t em and de layed r e sponse . One of the most impor tan t r e spon- s iv i ty c h a r a c t e r i s t i c s of water and other c ivi l s y s t e m s is the long lead t ime usual ly r e q u i r e d to obse rve a need, to conceive a poss ib l e means of meeting that need, to develop a p r e l i m i n a r y plan, to get bas ic po l i t i ca l approva l of the plan during a 'po l i t i ca l has s l e p e r i o d , ' to comple te the f inal design, and to cons t ruc t or o therwise c a r r y out the decis ion. This p r o c e s s often takes more than twenty y e a r s , and some t imes more than for ty or fifty y e a r s . Even for smal l , a lmos t inconsequent ia l p rob lems , it se ldom takes l e s s than two y e a r s . Since objec t ives can and do change much more rap id ly , this kind of r e spons iv i ty has become exceedingly impor tan t in water r e s o u r c e s planning.

There a r e many other fo rms of r e spons iv i ty in water r e s o u r c e s . A c l a s s i c example is t ime delay in rout ing water down an open-channel aqueduct sy s t em. Another is the r e l a t ed p rob l em of flood routing. Yet another is the abi l i ty of a 'move ' type supplementa l i r r i ga t i on sy s t em to cover the ent i re f ie ld in the face of drought. The response of hyd roe l ec t r i c s y s t e m s to r ap id f luctuat ions in demand is an economica l ly useful r e spons iv i ty of these s y s t e m s .

The r e sponse of wate r use to p r i c e is another very impor tant e lement of r e spons iv i ty in water r e s o u r c e s s y s t e m s . In many ins tances , cos t s which vary with the amount of water used a r e quite s m a l l r e l a t ive to costs which a r e insens i t ive to the amount used ( largely i r r e v e r s i b l e capital investment) . This may r e su l t in a r e sponse delay on the o r d e r of magnitude of the economic life of the inves tments involved.

A(x)

Xc

x~ I x2

Figure 1. Sens i t i v i ty Band, x c

2.3 Stabi l i ty

Stabi l i ty m e a s u r e s the r e s i s t a n c e to non-dec is ion modif icat ion of the mean r e sponse of the s y s t em. Frequent ly in water r e s o u r c e s the r e sponse of the s y s t e m will va ry apprec iab ly even for a fixed decis ion. If the effect of the va r i a t ion is to re turn the sys tem au tomat i ca l ly to the 'output ' o r objec t ive value r e p r e s e n t e d by the decis ion, the decis ion sy s t em is s tab le . If, on the o ther hand, au to -ca t a ly t i c effects cause the r e sponse to move away f rom that intended by the dec is ions , the dec is ion sy s t em is unstable . Many water r e s o u r c e s and other civi l s y s t e m s have highly unstable decis ion s y s t e m s . One obvious example is the flood control decis ion sy s t em. It has been a s s e r t e d that providing pa r t i a l flood con- t ro l , commensura te with one set of p r ed i c t ed future conditions, has r e s u l t e d in a t t r ac t ing more economic ac t iv i ty into the ' p r o t e c t e d ' a rea - -making the o r ig ina l dec is ion for pa r t i a l control quite i m p r o p e r for the new si tuat ion.

T ranspor t a t ion routing is another c l a s s i c example . In most ins tances , w a t e r - b a s e d r e c r e a t i o n has a l so responded unstably . On the other hand, many e s t i - mates of future water needs, made as long as fifty y e a r s in advance (e.g. Mulhol land 's a lmos t pe r fec t t i m e - f r a m e e s t ima te of need for 1600 cfs for Southern Cal i forn ia coas ta l a r ea ) have proved to be r e m a r k a b l y uncanny, suggest ing that highly s table , se l f - fu l f i l l ing effects may be involved in these ca se s .

2.4 Irreversibility

This is a m e a s u r e of the diff iculty in re turn ing a s y s t e m to i ts o r ig ina l s ta te once a decis ion change has been made. Suicide is an ex t r eme example of an

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i r r e v e r s i b l e dec is ion . In other cases , a decis ion can be r e v e r s e d , but only at g r e a t soc ia l or economic cos t . Humpty-Dumpty is the l i t e r a r y pe rson i f i ca t ion of this ve ry impor tan t object ive of water r e s o u r c e s and many other c ivi l s y s t e m s .

Some dec is ions a r e comple te ly i r r e v e r s i b l e but in a continuous sense . That is , the s ta te of the sys tem, s, can be changed by a r b i t r a r y sma l l modif ica t ions over t ime, t, or space in one d i rec t ion but it cannot be r e v e r s e d . Mathemat ica l ly , this fo rm can be r e p - r e sen t ed by OS/at >- 0. We can burn foss i l fuel but we cannot 'unburn ' it . Other dec i s ions are comple te ly i r r e v e r s i b l e in e i ther d i rec t ion . In some cases Jr- r e v e r s i b i l i t y is a ma t t e r of degree (i .e. soc ia l and economic cost) , e i ther continuous or discont inuous. A highway is an excel lent example of a va r i ab l e ' i r r e v e r s i b i l i t y ' s ince it can be r emoved or expanded only at cons iderab ly g r e a t e r cost than if the p r o p e r decis ion had been made o r ig ina l ly .

3. UNCERTAINTIES DUE TO ERRORS IN MODELING

Not a l l of the unce r t a in t i e s or r i s k s in water r e - sou rce s and other c ivi l s y s t e m s have to do with the actual s y s t e m i t se l f . A s ignif icant uncer ta in ty , a l l too often ignored in the quest for quanti tat ive p red ic t ive models , i s how well the models used actual ly r e p r e s e n t the r e a l s y s t e m ' s s ignif icant behavior . This uncer ta in ty can be in t roduced through the mode l ' s topology, i t s p a r a m e t e r s , and the data col lect ion and p r o c e s s i n g techniques . Model u n c e r - t a in t i es will often be in t roduced through human e r r o r s of both commiss ion and omiss ion . An ' op t imized ' decis ion set is t r u ly opt imal only if the ma themat i ca l model used to gene ra t e it c losely r e p r e s e n t s the s ignif icant behavior of the actual sy s t em over t ime and space . The fact that some soc io -economic e lements of the r ea l sy s t em can r e a c t compet i t ive ly or complemen ta r i l y to the chosen decis ion set only emphas i ze s th is shor tcoming of most ma themat i ca l models . In fact , t he re a r e ac tua l ly no civil s y s t e m s with a s ingle d e c i s i o n - m a k e r , de- spi te this cus tomary assumpt ion in opt imal decis ion modeling.

The n e c e s s a r y condition for r ea sonab le use of any decis ion set obtained through opt imizat ion is that the impor tant r e s p o n s e s of the r ea l sy s t em to those dec is ions a r e the same as those produced by the ma themat i ca l model, within a t o l e r ab l e l imi t for e r r o r . Since water r e s o u r c e s dec is ions a r e very often made only once, it may be difficult to evaluate modeling e r r o r s , let alone reduce them to quanti ta t ive p robab i l i ty m e a s u r e s . This s ignif icant source of uncer ta in ty is p robably one of the ma jo r r e a sons for the slow cautious adoption in civi l s y s t e m s of the produc ts of r e s e a r c h , p a r t i c u l a r l y s y s t e m s ana lys i s modeling.

There is an extens ive l i t e r a t u r e d i r ec t ed toward the evaluat ion of sens i t iv i ty of the opt imizing solution to va r i a t ions in the p a r a m e t e r s (Luenberger [1969], Hadley [1964], Bigelow [1974], Saaty [1959], Saaty and Webb [1960],Webb [1962], P i e r r e [1966], Gavr i lovi~ and Pe t rov id [1966], and o thers ) , most of which is s u m m a r i z e d in Haimes [1977]. In gene ra l these evaluat ions a r e based on the p r o p e r t i e s of the f i r s t pa r t i a l de r iva t ives and higher o r d e r p a r t i a l d e r i v a - t ives of the object ive function with r e s p e c t to the

cons t ra in t s or o ther model p a r a m e t e r s at the op- t i m i z e d values of the objec t ive . To the extent that point p r o p e r t i e s r e f l ec t the r i s k concerns any of these p o s s i b i l i t i e s can be u t i l i zed as object ive functions for mul t ip le objec t ive opt imiza t ion . They a re l imi ted only by the degree to which the dec i s ion- make r can unders tand the i r s ignif icance in context with his often qual i ta t ive ve r s ion of the r i s k p rob l em. Obviously one can c rea t e s i tua t ions where 'point ' p r o p e r t i e s evaluated at the opt imum a re poor in- d i ca to r s of the r i s k impac ts at o ther points r emoved even a r e l a t i ve l y sma l l d is tance f rom the ana ly t i ca l opt imum. The example p rev ious ly given under the d i scuss ion of sens i t iv i ty is one such case . Thus for some p r o b l e m s where control is i m p r e c i s e a n d / o r ind i rec t , a spac ia l ly d i s t r ibu ted index may be p r e - f e r ab l e over a point index.

L i t t l e or no work has been done with r e s p e c t to the quantif icat ion of indices for the other sys t em c h a r a c - t e r i s t i c s although some crude m e a s u r e s will be obvious in each case . One can define a c h a r a c t e r i s t i c r e sponse t ime for a s y s t e m which would be va l id for choices among s i m i l a r s y s t e m s .

There a r e s e v e r a l impor tan t p r o p e r t i e s of any index u t i l i zed to r e p r e s e n t these s y s t e m s and modeling c h a r a c t e r i s t i c s which must ex is t if they a r e to be useful in p r a c t i c a l appl ica t ion .

F i r s t , the index should m e a s u r e per t inent c ha r ac - t e r i s t i c s of the p rob lem. In p a r t i c u l a r , if a p rob lem contains a l a rge number of p a r a m e t e r s , one must decide whether or not to use an index which m e a s u r e s the sens i t iv i ty due to each individual p a r a m e t e r . F u r t h e r m o r e , only those p a r a m e t e r s with deviat ions having the g r e a t e s t effect on the opt imal solution should be cons ide red . This will avoid excess ive computation and the genera t ion of i r r e l e v a n t informat ion.

Second, informat ion conveyed by the index should be c l ea r ly unders tood. The conceptual bas i s under ly ing the sens i t iv i ty m e a s u r e must be easy to g rasp , because it may be that the d e c i s i o n - m a k e r s analyzing the p rob lem have l i t t le technica l unders tanding . This is often the case when solving l a r g e - s c a l e mul t i - object ive p r o b l e m s involving publ ic inves tment .

Third, the index should not be difficult to ca lcu la te . When making a mul t iobjec t ive ana lys i s it is often n e c e s s a r y to genera te many noninfer ior points before a p r e f e r r e d solution can be found. Should the sen- s i t iv i ty be evaluated at each noninfer ior point, this may entai l a heavy computat ional burden if the ca l - culat ions used in de te rmin ing the index a re complex. Accord ingly , it would be de s i r ab l e to have an index which u t i l i zes informat ion ca lcu la ted by the p a r t i c u l a r opt imizat ion a lgor i thm used in solving the p rob lem.

4. CHARACTERIZATION OF MODELING ERRORS

The val idi ty of the opt imal solution x* to any maxi - mizat ion or min imiza t ion p rob lem d-epends (among other things) on the accu racy with which the mathemat ica l model r e p r e s e n t s the r ea l sys t em. In p a r - t i cu la r , th is accu racy depends on the c loseness of the r ea l sys t em to the mode l ' s input-output r e l a t i on - ships . The s ou r c e s of unce r t a in t i e s and e r r o r s can be a s s o c i a t e d with s ix ma jo r model c h a r a c t e r i s t i c s : Model Topology--(~ 1), Model P a r a m e t e r s - - ( ~ 2 ) , Model

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Scope or Focus--(~a) , Data--(a4), Optimization Tech- nique--(~5) , and Human Subjec-tivity--(a__ 6). These six categories are discussed hereaf ter in some detail.

4 .1 Model Topology (a 1)

Model topology re fe r s to the order , degree and form of the equations which represent the real system. For example, a dynamic water system might be represented by a sys tem of differential equations (ordinary or partial); a static system, by sets of algebraic equations such as polynomials, etc.

Consider for example, a groundwater system of both confined and unconfined aquifers. To model the dynamic response of the aqui fer ' s hydraulic head to any future demands (withdrawals or recharges) on the groundwater system, one may use a system of differential equations. Linear, second-order , part ial differential equations may be adequate for modeling the confined aquifer, whereas nonlinear, second- order , part ial differential equations (PDE) might be needed for the unconfined aquifer. Fur the rmore , a homogeneous aquifer may be adequately modeled by a two-dimensional system, but a stratified, non- homogeneous one ought to be modeled by a three- dimensional PDE, etc. Clearly, in each case, selecting one model topology over another introduces a source of uncertaint ies and e r r o r s in the accuracy of the model.

Model topology is par t icular ly important in decision making for optimization. Almost any functional form can be used to approximate the absolute value of any cause-effect relationship. However, optimal decisions are usually not as concerned with the magnitude of these functions as with their derivatives (or incremental ratios). Thus a linear, l eas t - squares regress ion model of a basically non-l inear response, because of the charac te r i s t i cs of l inear system optimization, is very apt to select 'decis ions ' at points which in fact have the greates t e r r o r in the r ep re - sentation of the true derivative.

4.2 Model Parameters (a 2)

Once the model topology has been selected, the choice of model parameters (often called parameter identification, parameter estimation, system identification, model calibration, etc.) determines the accuracy with which the model represents the real system. Consider the groundwater system discussed earlier. Once the customary system of parabolic partial differential equations is selected, the proper values of the coefficients need to be determined (e.g. storativity and transmissivity as functions of the spatial co- ordinates). This parameter estimation process introduces a source of uncertainties and errors in the accuracy of the calculated values of the parameters and in turn in the model itself.

4.3 Model Scope (a 3)

Model scope re fe r s to the type and level of resolution used in the model for the description of the real system. Four major descript ions are identified in water r e sources sys tems (naimes and Macko [1973]).

These are: temporal description, phys ica l -hydrological description, pol i t ical-geographical description, goal or functional description. The above descriptions are discussed in some detail hereaf ter . The charac - te r i s t ic pa rame te r s of uncertaint ies and e r r o r s as -

sociated with the selection of the model scope are denoted by the set _~3-

In re fe r r ing again to the groundwater system, one may wish to study the behavior (response) of the system under planned development for short, inter- mediate and long- term planning. The groundwater system itself, which may consist of several aquifers, may be decomposed on the basis of the physical- hydrological charac te r i s t i cs or pol i t ical-geographical boundaries. Finally, if the groundwater system is to be managed as part of a la rger water r e sources system with concern for water quality, s torage, r e - Charge, and so on, then different decompositions may be more advantageous, such as goal description. Clearly, while these four descriptions have individual meri ts , each descr ibes the system f rom a narrowed point of view. The system in totality may never be well represented by any one description, and thus the selection of a model ' s scope introduces yet another source of uncertaint ies and e r r o r s into the sys t em ' s representat ion. Scope is par t icular ly important where the system is controlled by many relat ively independent dec is ion-makers , each with somewhat different objectives. Even so, such sys tems are often modeled as though a single ' ra t ional ' dec is ion-maker were at the helm, i.e., as though a single point of view could be asser ted .

4 .4 Data (U4)

Access to enough representat ive data for model construction, calibration, identification, testing, validation, and, hopefully, implementation is obviously very important in sys tems analysis . Clearly a lack of either accura te or sufficient data due to problems of collecting, acquiring, process ing, analyzing it, etc., may cause substantial e r r o r s . Consider again the above groundwater system: the value of the model pa ramete r s identified is likely to depend on the available data. An insufficient number of sampling sites, the number of samples, and sampling accuracy (within each location) may introduce significant uncertainties and e r r o r s into the system model.

4.5 Optimization Techniques (a 5)

Once the mathematical model has been constructed and its parameters identified, selecting and applying suitable optimization methodologies (solution strategies) introduces another source of uncertainties and errors into the solution of the system model. In the groundwater system discussed earlier, selecting the method of numerical integration of PDEs with the associated grid size, boundary and initial conditions, computer storage capacity and accuracy, etc., all are potential sources of uncertainties and errors in the solution. As another example, consider a nonlinear objective function with a nonlinear system of inequality constraints representing a power and water supply system. If the optimization method for solving this system is the simplex method (via linearization of the system model), then the accuracy of the solution obtained may be questionable. This is particularly true for highly nonlinear systems.

It is important to note that selecting the optimization technique generally coincides (or should coincide) with the model's construction. Consequently, any exchange between the sophistication (or simplifica- tion) of the model and the accuracy (or approximation) of the solution should be made earlier.

75

Multiple Objectives in Civil Sys tems Yacov Y. Haimes and Warren A.Hall

4 .6 Human Subjectivity (a s)

Human subjectivity strongly influences the outcome of sys tems analysis in water r e sources (as well as in other areas) . It may include: the background, training and experience of the analystis), his personal preference and se l f - in teres t , and his proficiency. Clearly, human subjectivity can influence all of the other five major categories of model charac te r i s t i c s .

A civil engineer, a hydrologist o r a sys tems engineer, for example, all addressing the problem of planning for developing the above groundwater sys tem and predict ing the water head response to withdrawals and recharges may each conceive a different approach or methodology• While human subjectivity plays a very important role in the selection of all major model charac ter i s t ics , each of which could introduce uncertaint ies and e r r o r s into the sys tem model, there is no way to analyze to what extent this could happen• Rather than t ry to quantify such cause and effect relat ionships here, the importance of each charac te r i s t i c is indicated and a f ramework for its analysis is suggested.

In analyzing the sources of uncertaint ies and e r r o r s as they affect sensitivity, stability, i r revers ibi l i ty , and ultimately, optimality, the sys t em ' s analyst may en- counter any of three conditions (a) a complete knowledge of a is available, i.e.,c~ is a determinist ic variable;-~2) alternatively, the vector ~ could be a stochastic variable but an est imate of~ts probabili ty distribution function is available; or (3t the vector c~ could also be a s tochast ic variable where no knowledge is available on the probability distribution function.

It is assumed that for any given sys tem some analytical functions can be constructed relating sensitivity, stability and i r revers ib i l i ty to c~. Fur the rmore , depending on which element of ~ is under consideration, the knowledge of its mean and var iance can vary between full knowledge and no knowledge. In any event, noncommensurable objective functions will result r egard less of the degree of knowledge of ~.

5. FORMULATION OF THE MULTI-OBJECTIVE OPTIMIZATION PROBLEM

The major task yet to be accomplished through r e - search is the quantification of the concepts of sensitivity, responsivity, stability, i r revers ib i l i ty , r isk, and uncertainty and the construct ion of the associa ted indices so that they can be considered as objective functions in a multiobjective optimization f ramework. Examples of such indices were introduced in the previous section.

In their most general form, these indices can be presented as follows:

~I~, i X , Or) . . . . , ~ j i x , ~t where

x is a vector of decision (control) variables,

is a vector of model pa rame te r s ,

,l 1(" ), • •., ~J(" ) a re functions represent ing sensitivity, responsivi ty, etc.

Given the following classical formulation of an optimization problem:

rain fl(x, at (1) X E X - - __

where × is the set of all feasible solutions. Specifically,

× -- {xl gkix-- ~) --< 0, k = 1,2 . . . . , K}

'P rob l em (1) can be modified to include one or more of the above indices ~j(x_j ~), j -- 1, 2, . . . , J, such as

min If 1 (x~ _~1 1 xsx L~lix__~)J (2)

Problem (2) is a multiobjective optimization problem.

It is possible, of course, that the original problem itself is given in a multiobjective optimization form, and with the addition of sensitivity and other indices, the new problem may have the following form:

min {l(X_~) I (3) x jx

t.~x. otj 1

%(x~_~)]

It is assumed that all functions fi(x~ ~), @i(x, ~), gk(x~ ~) a re proper ly defined and c-on-tinu~u-s_

The Surrogate Worth Trade-off method and its ex- tensions can then be used to solve problems (2) and (3) [Haimes, Hall and Freedman 1975].

6. EXAMPLE FORMULATION OF RISK OBJEC- TIVES FOR WATER RESOURCE SYSTEMS

There probably is no standard approach to the speci- fication of r isk objectives in general , given the almost infinitely possible number of combinations of sys tem- modeling charac te r i s t i cs discussed previous- ly. To a large extent, each case may have to be t rea ted de novo to a s sure that the modeling e r r o r s introduced by a s tandardized approach do not in t ro- duce more uncertainty than the r isk element being analyzed.

The basic questions is: Risk of what? It may be the r isk of a r e se rvo i r going dry. Or in the same situa- tion, it may be the r isk of failing to meet a minimum presc r ibed level of service , such as ' f i rm water ' or ' f i rm energy. ' Thirdly, it may be the r isk of diver- gence f rom the p resc r ibed or anticipated level of service . In fact, all three r i sks may exist s imul- taneously. The f i rs t two would constitute d iscre te units of the r isk objective vector; the third, an objective vector with an infinite set of components between zero and the p resc r ibed level of service .

As an example, let Fix) be the expected net economic benefit of selecting a level of water supply serv ice of x. It is desi red to set F(x) as high as possible while minimizing the r isk that the r e se rvo i r (of total capacity Qmax) and the stochast ic inflow y(t) will not be sufficient to provide minimum service at all t imes within the next n time per iods. This is perhaps the s implest form of the r isk problem, but it se rves

76

A d v a n c e s in W a t e r R e s o u r c e s Vol. 1 No. 2 1977

to i l lustrate a number of important charac ter i s t ics which must be carefully considered.

Proceeding with this simple formulation, a knowledge of the stat ist ical charac te r i s t i c s of the hydrology might allow development of a large number of 'equally likely' hydrographic sequences of n t ime periods each (see Askew, Yeh and Hall [1971] for a discussion of non-allowable conditions). The probability of failure to meet minimum service levels at least once in n t ime periods is calculated as a function of serv ice level. If this quantity is designated as Pn(x), then the vector optimization problem is

max [1-Pn(x), F(x)] X

subject to: constraints on input hydrology, constraint on r e se rvo i r capacity, and non-negative constraints on initial r e se rvo i r conditions.

Since Pn(x) and F(x) a re fundamentally different quantities, this is a vector optimization of noncommensurable functions and it can be t reated using the Surrogate Worth Trade-off method. Note that the optimum policy and acceptable r isk levels will depend on the initial s torage level chosen, hence this represen ts a family of optimizations.

There are several other representat ions of the r isk element of this problem. For example, the r isk objective can be defined as the probabili ty that the decision level x will not result in a fai lure within n t ime periods (n = 1, 2, 3, . . . ,N ) . In this case a probability distribution can be generated for

Pn(X) for each level of x considered. The resul t is a family of optimization problems of the form

m a x [ 1 - ~ n ( X ) , f ( x ) ] n = 1, 2, . . . , N

subject to constraints as before. Once again the problem can be t reated using the Surrogate Worth Trade-off method.

At this point the reader will have probably wondered why we did not simply determine the probabili ty of failure for each value of f(x) and assess an appro- pr iate economic penalty function, and proceed to maximize the mathematical expectation of the r e - suiting single economic objective. This is a very valid question and in certain c i rcumstances it would be the cor rec t model topology to follow. The validity of this approach, however, depends upon the skill and accuracy of constructing the penalty function for dropping below a delivery of x in any t ime period. In fact, this is what the Surrogate Worth Trade-off method does, except that instead of attempting to evaluate what the penalty should be (a very subjective matter in r isk cases), attention is focused on the s impler question of whether the dec is ion-maker is willing to accept a specific (computable) marginal increase in r isk in order to obtain a specific (computable) increase in his benefit. As was shown by Haimes and Hall [1974] it is not real ly necessary to know the answer to the lat ter question in an absolute quantitative sense, but ra ther only in the ordinal (rank order) or qualitative sense of one being of greater value than the other. A penalty function on the other hand must be numerical ly accura te over all possible values of x, otherwise the derivatives on which optimization usually r e s t s may be badly in

e r r o r . If the proper penalty function can be accura te - ly determined, both methods should lead to identical results , p r o v i d e d certain additional conditions are met. F i rs t , the decis ion-maker must expect to have a large number of applications for the decision, large enough so that his experience with it can reasonably be expected to be an adequate, unbiased sample of the corresponding probability distributions.

A decis ion-maker who only gets one trial with its corresponding result , is not readily consoled by the mathematical expectation. He still must consider

.separately whether the r isk is worth the gain. Anyone would be willing to make a se r ies of 1,000,000 bets of $1.00 each on the black numbers on the roulette table if he were paid even money for the two green house numbers as well as the black. However, very few would bet $1,000, 000 with one and only one bet allowable under exactly the same ci rcumstances . The mathematical expectancy is exactly the same, but the relative desirabil i ty is obviously quite different.

Second, the objective function itself must not be rendered invalid because of the occurrence of any par t icular chance event. There is no meaning to the mathematical expectation of the house winning at roulette if one ser ies of outcomes resul ted in blowing up the establishment. As a less drast ic example, f i rm power contracts in the Central Valley project of California require the level of f i rm power contracted to drop to the lowest power output actually delivered whenever actual power output falls below the ' f i rm ' contract level. This event constitutes an i r revers ib le discontinuity in the economic objective function itself, and mathematical expectation of the objective as normally defined is invalid.

Finally, it is important to a ssess whether or not the true objectives include optimization of mathematical expectation in the f i rs t place. While seemingly obvious, examples exist where this is not true.

An interesting problem in sensitivity and r isk a r i ses in the optimal construction sequencing problem (see Butcher, Haimes, and Hall [1969], Haimes and Nainis [1974] or Haimes [1977]). The methodology developed there determines the optimal order of construction for N water supply projects , each having a specific fixed capacity of Qi. Any given est imate of the future demand for service resul ts in some specific order of construction, depending on the t ime pattern of this demand. However, the latter is an uncertainty usually obtained by extrapolation p rocesses which, while somet imes valid for short periods into the future, become much more unreliable as the time span in- c reases . If it is presumed that an equally likely future demand (or requirement) function can be est imated for a short t ime period on the basis of past t rends (estimating the mean and standard deviation of the e r r o r f rom past variances) , it is possible to generate a number of possible future demand functions by methods s imilar to those used to generate simulated hydrologic sequences. In the ab- sence of any other knowledge, we can asse r t that these are the best possible es t imates of a number of 'equally l ikely' future demands.

Using such a set of equally likely future demand functions, the optimal sequence of project cons t ruc- tion for each simulated future can be computed. If they are equally likely, it is possible to determine

77

Multiple Object ives in Civil Sy s t ems Yacov Y. Haimes and Warren A. Hall

that f r equency with which any p a r t i c u l a r p ro j e c t would be c o n s t r u c t e d f i r s t . The rank o r d e r of this f r equency r e p r e s e n t s a n u m e r i c a l m e a s u r e of the sens i t iv i ty of the supp ly -use s y s t e m to the ini t ial p ro j ec t dec is ion under condi t ions of demand unce r t a in ty .

The joint 'optimality' and ' sensi t ivi ty ' problem can be written in a multiobjective framework as follows:

Min I fl(x' ~) ] x Lf2(x , ¢~)

(11)

7. E X A M P L E P R O B L E M (Haimes [1977])

Cons ide r the fol lowing m a t h e m a t i c a l model:

y(x, ~) = 2x 2 -- 2x((~--l) _~2

where

y(x, ~)

X

and

Le t

denotes the s y s t e m ' s output r e s p o n s e

denotes the m o d e l ' s dec is ion va r i ab le

denotes the m o d e l ' s p a r a m e t e r .

denote the nomina l value of (~,that may be d e t e r m i n e d via any s y s t e m s ident i f ica t ion p r o - cedure . This is the value ac tua l ly used in the op t imiza t ion p r o c e s s .

Le t the m o d e l ' s objec t ive funct ion be f l (x , ~). Fo r s impl ic i ty the objec t ive is to min imize the output (e.g., cost):

f~(x , ~) = y(x , ~)

or

f l (x , ~) = 2x 2 -- 2x(ot--1) -- ~2.

(4)

(5)

Both y(" ) and fl(" ) a r e wr i t t en as funct ions of both x and a to e m p h a s i z e this dependency not only on x alone but a l so on ~.

(6)

Let

= 2; (7)

the c o r r e s p o n d i n g nominal output r e s p o n s e is given by (8):

y(x, ~) = 2x 2 -- 2x -- 4. (8)

Define a sens i t iv i ty index, f 2 ( ' ) which m e a s u r e s the changes in the mode l ' s r e s p o n s e to changes in (~ as fol lows:

(9)

where ,

T h e r e a r e no cons t r a in t s on x. Substi tuting (6)-(10) "into (11) y ie lds :

Min [ f l ( x ' ~ ) = 2 x 2 - 2 x - 4 ] (12)

x Lf2(x, ~) 4X 2 + 16x + 1 6 J .

P r o b l e m (12) can now be so lved us ing the s a m e p r o - cedures d i s c u s s e d e a r l i e r in this chap te r . T r a n s f e r (12) into the e - c o n s t r a i n t f o rm:

Min{2x 2 -- 2x -- 4} (13) X

subject to the cons t r a in t :

4x2"+ 16x + 16 -.< e 2. (14)

F o r m the L a g r a n g i a n funct ion for (13) and (14):

L(x, ~, h12) = 2x 2 -- 2 x - - 4 + ~1214x2 + 16x + 16 -- (2]

(15)

The K u h n - T u c k e r n e c e s s a r y condi t ions fo r s ta t ion- a r i ty c o r r e s p o n d i n g to (13)-(14) yield:

~L

Ox 4 x - - 2 + ( 8 x + 1 6 ) ~ 1 2 = 0 (16)

8L = 4x 2 + 16x + 16-- ~2 "-< 0 a)~12

f2 (x, ,:/)

. 2 0

x x / - 1 0 10 20 30

/

f j (x*, (~)

1 40

(17)

~ ' ~ ' = - 2 x - 2 ~ . 8~

(10) Figure 2. Non-In fer ior Solution in the Functional Space

78

h ( ' )

/ I I

/

I

I

F i g u r e 3.

s I ~

I \ I

/ \

f,6<, a) . . . . .

f~ (x*, a)

l t t I

t I

l I t l

,, I , ~6 ~ ; ; 4 15 7

t I

I

I I 0 I

The Funct ions f l(x*, a) and f l(~, ol) vs. a

h 1 2 1 4 X 2 + 16x + 1 6 - e2] = O.

AI2 ~ 0

Solving equation (16) yields:

2 - -4x k12 -- 16+8x

(18)

(19)

(20)

A d v a n c e s in Water R e s o u r c e s Vol. 1 No. 2 1977

the noninfer ior solution in the functional space f1(" ) and f2( ' )"

Let x* and x denote the decision va r i ab les which minimize f l (x, dr) and f2(x,/~) respect ive ly :

Min fz(X, a) -- f l (x*, ~) (21) X

Min f2(x,/~) = f2(:~, &). (22) X

Both x* and ~ can be eas i ly obtained (e.g., f r om (12)) "resulting in:

x* = O. 5 (23)

= - - 2 . (24)

To d rama t i ze the t r ade -o f f s between the sensi t ivi ty objective function f2( ' ) and the optimali ty objective function fl(" ), the la t ter is evaluated at x* and ~. The resul t ing functions f l (x*, ~) and f1(£, ~) a r e plotted in F igure 3 as functions of c~. These functions a re given by (25) and (26) respect ive ly :

f l (x*, ~) = 0 . 5 - ( a - - l ) - ot 2 (25)

fl(R, ~) ----- "8 + 4(c~--1) - - ~2 (26)

Note that at the nominal value of ~, i .e., at ~ = 2, fl(x*, ~) changes rapidly with a slope equal to --5 where at the s ame point (6=2), fl(~, 8) is s table with a slope equal to zero .

F u r t h e r m o r e , F igure 4 depicts the changes that take place in f l (x*, c~) and f l(~, ~) when the nominal value

is pe r tu rbed by A ~ :- - -0 .5 . The cor responding var ia t ions a re f l(x*, ~) = - -4 .5 , f l(x*, 6--0.5) = 2.25, and If1(x* , 6 ) - - f1(x*, 6 -- O. 5) I = 2.25.

Let 7/(x*, O. 75 ~) denote the percentage of change in f1(x*, ~) with a per turbat ion of 25% in e; then ~(x*, 0.75 c~) = 50%

Similar ly , f l(~, ~) = 8, f l(~, 6--0.5) = 7.75, and If1(~t, d t ) - - f 1 ( ~ , ~ 0.5-) I = 0 . 2 5 .

Table 1 Non-Inferior Solutions and Trade-Off Values for the Example Problem 1

x f l (x, ~) f2 (x, ~) ~12

O. 50 - -4 .50 25.00 0

0 - -4 .00 16.00 0.13

--0.20 --3.52 12.96 0.19

- -0 .50 - -2 .50 9.00 0.33

- - I . 00 0 4 .00 0.75

- - 1 . 5 0 3 . 5 0 1 . 0 0 2 . 0 0

- - 1 . 6 0 4 . 3 2 O. 64 2 . 6 3

- - I . 75 5.63 0.25 4 .50

- - l . 80 6.08 0.16 5.75

- - l . 90 7.02 0.04 12.00

10

f, (x, ~) = 8

6

4.

2-

- 2 -

- 4 -

f , (x *, a) = - 4 . 5 - 6 -

-8

l ~ a = - 0 . 5

f, Ix, a ' - 0.51 = 7.75

^

f , (x*, u - 0 3 ) = - 2 . 2 5

Table 1 l i s ts s eve ra l noninfer ior solutions with the corresponding t rade-of f values . F igure 2 depicts

Figure 4. The Funct ions f t (x*, a) and f t (x*, ~ ) vs. Per turba t ion in o

79

Multiple Object ives in Civil Sys t ems Yacov Y. Haimes and Warren A. Hall

Let ~(~, 0.75 ~) denote the percentage of change in f1(~i, c~) with the same perturbation of 25°/° in d. Then, ~(~, 0.75 d) = 3%

The resul ts given in Figure 4 indicate that following a conservative policy which t rades optimality (cost objective) for a less sensitive outcome, provides a very stable solution (3% vs. 50% changes in f l ( ' ) with a deviation of 25% from the nominal value c~). Clearly, neither the solution x* nor ~ is likely to be r e c o m - mended. The use of Table 3 and the SWT method with an interaction with a dec is ion-maker should evolve the selection of a p re fe r r ed level of x where:

:K~ X~ < X*.

8. SUMMARY

In this paper a number of questions associa ted with r isk and uncertainty have been tentatively explored to stimulate fur ther analysis and r e sea rch into the quantifications of these fac tors for use in multiobjective optimization analysis. A great many problems exist in water r e sources sys tems and other civil sys tems involving r e sources in which avoidance of r isk and uncertainty is often in fact the dominating objective. If suitable quantitative measures of these objectives can be formulated, then the Surrogate Worth Trade-off method or other multiobjective optimization methodologies can determine the optimal or at least superior combinations of r isk and various fo rms of re turn .

Direct measures of r i sk to be avoided can be defined in certain situations. Examples of hydrologic r isk quantification (to be minimized) were developed as an example of t reat ing r isk due to chance-control led non-decisioned inputs or sys tem pa ramete r s .

More complex r isk and uncertainty situations develop when decisions cannot be made with precis ion control but ra ther will vary about the decision values in some random or quas i - random manner. When the number of repeti t ions of that decision is also small so that mathematical expectations may be meaningless to the decis ion-maker , an important fo rm of r i sk is introduced.

Such imprecis ion of control may introduce r i sk or uncertainty through severa l sys tems charac ter i s t ics . Four such charac te r i s t i cs were identified and de- scribed: sensitivity, responsivi ty, stability and Jr- revers ibi l i ty . In addition, several types of modeling e r r o r s were identified which can lead to imprecise control, imprec ise predict ions of the real response, or both, hence having equivalent ability to create or accentuate r isk and uncertainty.

Because they are somewhat singularly related to the specific sys tems concerned, general izat ions on r e - sponsivity, stability and i r revers ib i l i ty a re not discussed here. Sensitivity, on the other hand, would appear to be amenable to more general ized quantifications as discussed in the previous section. Each of the above measures is useful under success ively more general c i rcumstances , ranging f rom sys tems controllable within close l imits to those which can only be approximately controlled within broad limits.

This somewh~tt pre l iminary analysis and discussion indicates that quantitative measures of r i sk can be defined and utilized as objectives to be optimized in

a multi-objective control . In some instances even uncertainty (no probabili ty distribution data) can be t rea ted adequately.

An indication, however, is not an accomplished fact and much insight and analysis will be required to quantify the major r i sk fac tors involved in common water r e sources sys tems well enough to include them in multi-objective decision analysis .

The proposed planning and management f ramework might be used systemat ical ly to:

(i) Assis t planners, profess ionals and decision- makers involved in r e sources planning and management in general and in water and related land r e sources in par t icular .

(ii) Quantify and display the t rade-offs involved in reducing risk, sensitivity, i r revers ib i l i ty and other sys tems charac te r i s t i cs (viewed as systems objectives) along with reducing cost or other per formance indices, where all these objectives are kept in their noncommensurable units.

(iii) Insure comprehensive consideration of economic issues social well-being, health hazards, environmental issues, i r r evers ib le impacts , and other costs and benefits r egard less of commensur - ability through the use of the multiobjective approach.

(iv) Reduce the uncertainty surrounding r e sources planning and development in general , and water and re la ted land r e source development and management decisions in par t icular , by the development of more rea l is t ic sys tems models, and accurate ly displaying the probable conse- quences of the decisions and policies followed.

REFERENCES

Askew, A. J., W-G Yeh and W.A.Hal l , 'A Comparat ive Study of Cri t ical Drought Simulation, ' Water Resources Research~ vol. 7hi, 1971.

Bigelow, J. H. and N. Z. Shapiro, ' Implici t Function Theorems for Optimization Prob lems and for Systems of Inequali t ies , ' RAND Report R- 1036-PR, The RAND Corporation, Santa Monica, California, 1974.

Butcher, W. S., Y. Y. Haimes, and W. A. Hall, 'Dynamic P rogramming for the Optimal Sequencing of Water Supply P ro j ec t s , ' Water Resources Research, Vol. 5, No. 6, p. 1196, 1969.

Committee on Public Engineering Policy, ' P e r s p e c - tives on Benefit-Risk Decision Making,' National Academy of Engineering, Washington, D.C., 1972.

F isher A, J. Krutilla, and Cicchetti, 'The Economics of Environmental Preserva t ion: Fur ther Discussion, ' Amer i can Economic Review, Vol. LXIV, No. 5, 1974.

Gavrilovid, M., and R. Petrovid, 'On the Synthesis o f the Leas t Sensi t ivi ty Control, ' in Sensi t iv i ty Methods in Control Theory, Proc. International Symposium on Sensi t ivi ty (Dubrovnik, Yugoslavia, 1964), L. Radanovid, Ed., New York, Pergammon, 1966.

Gembicki, F., 'Vector Optimization for Control with Pe r fo rmance and P a r a m e t e r Sensitivity Indices , ' Ph.D. Dissertat ion, Department of Systems Engineering, Case Western Reserve University, Cleveland, Ohio, January 1974.

80

Gembicki, F. and Y. Y. Haimes, 'Approach to Pe r - formance and Multiobjective Sensitivity Opti- mization, the Goal Attainment Method,' IEEE Aurora atic Control, Vol. AC-70, No. 6, 1975, pp. 769- 771.

Guardabassi, G., et al, 'Structural Uncontrollability of Sensitivity Systems,' Automatica, vol. 5, 1969, pp. 297-301.

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ACKNOWLEDGMENTS

The authors would like to thank F. Gembicki and H. Rarig for their contributions to the ideas presented in this paper. The paper presents some results of the project 'Multiobjective Analysis in the Maumee River Basin: A Case Study on Level-B Planning.' The project was supported in part by the National Science Foundation, Research Applied to National Needs Program under Grant No.AEN75015820, and the Office of Water Research and Technology, U.S. Department of the Interior under Grant No. 14-34- 0001-6221.

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sensitivity, responsivity, stability and irreversibility as multiple objectives in civil systems

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