Stochastic Optimization Models for Lake Eutrophication Management

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

STOCHASTIC OPTIMIZATION MODELS FOR LAKE EUTROPHICATION

MANAGEXENT

Ldszld Somly6dy*

Roger J-B Wets**

April 1985 CP-85-16

*

R e s e a r c h Center f o r

Water

Resources Development, VITUKI, Budapest, Hungary

**

IIASA and University of California

(Supported in p a r t by a g r a n t of t h e National Science Foundation)

Cottaborative P e p e r s r e p o r t work which h a s not been performed solely

at

t h e International Institute f o r Applied Systems Analysis and which h a s received only limited review. Views

or

opinions e x p r e s s e d h e r e i n do not necessarily r e p r e s e n t those of t h e Insti- t u t e , i t s National Member Organizations,

or

o t h e r organizations supporting t h e work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

The development of s t o c h a s t i c optimization techniques, r e l a t e d software, and i t s application, w a s a major p a r t among t h e a c t i v i t i e s of t h e System and Decision S c i e n c e s P r o g r a m during t h e p a s t f e w y e a r s .

This p a p e r d e s c r i b e s r e s u l t s of t h e application of s t o c h a s t i c program- ming to water quality management. I t provides a n example of both important issues f o r investigation: a r e a l i s t i c problem with i n h e r e n t s t o c h a s t i c i t y , and a valuable

test

problem f o r t h e algorithms under development. I t a l s o gives some insights into t h e n a t u r e of solutions of c e r t a i n c l a s s e s of

sto-

c h a s t i c programming problems, and t h e justification f o r t h e consideration of randomness in decision models.

Alexander B. Kurzhanski Chairman

System and Decision S c i e n c e s P r o g r a m

ABSTRACT

W e develop a g e n e r a l framework f o r t h e study and t h e c o n t r o l of t h e eutrophication p r o c e s s of (shallow) lakes. The randomness of t h e environ- ment (variability in hydrological and meteorological conditions) i s a n i n t r i s i c c h a r a c t e r i s t i c of such systems t h a t cannot b e ignored in t h e analysis of t h e p r o c e s s or by management in t h e design of c o n t r o l measures.

The models t h a t w e suggest t a k e into account t h e s t o c h a s t i c a s p e c t s of t h e eutrophication p r o c e s s . An algorithm, designed to handle t h e resulting

sto-

c h a s t i c optimization problem, is d e s c r i b e d and i t s implementation i s out- lined. A second model, based on expectation-variance considerations, t h a t approximates t h e "full" s t o c h a s t i c model c a n b e handled by s t a n d a r d l i n e a r

o r

nonlinear programming packages. A case study i l l u s t r a t e s t h e a p p r o a c h ; w e compare t h e solutions of t h e stochastic models, and examine t h e e f f e c t t h a t randomness h a s on t h e design of good management programs.

Key Words: Lake eutrophication, eutrophication management, s t o c h a s t i c programming, r e c o u r s e model, water quality management, probabilistic con- s t r a i n t s .

CONTENTS

INTRODUCTION

1. MANAGEMENT GOALS AND OBJECTIVES 2. T H E APPROACH

3. FORMULATION O F T H E STOCHASTIC MODELS L E M P AND NLMPZ 3.1. A g g r e g a t e d L a k e E u t r o p h i c a t i o n m o d e l (LEMP)

3.2 N u t r i e n t L o a d M o d e l (NLMPZ)

4. CONTROL VARIABLES, COST FUNCTIONS AND CONSTRAINTS 4 .I. C o n t r o l V a r i a b l e s

4 . 2 C o s t F u n c t i o n s 4 . 3 C o n s t r a i n t s

5 . FORMULATION OF T H E EUTROPHICATION MANGEMENT OPTIMIZATION MODEL: EMOM

6. EMOM: AN EXPECTATION-VARIANCE MODEL 7. SOLVING EMOM: STOCHASTIC VERSION 8. APPLICATION TO L A K E BALATON

8.1. B a c k g r o u n d f o r L a k e B a l a t o n

8.1.1. D e s c r i p t i o n o f t h e l a k e , its w a t e r s h e d a n d possible c o n t r o l m e a s u r e s

8.1.2. S p e c i f i c a t i o n o f e l e m e n t s o f EMOM f o r L a k e B a l a t o n 8.2. R e s u l t s o f t h e e x p e c t a t i o n - v a r i a n c e m o d e l

8.3. R e s u l t s o f t h e Stochastic R e c o u r s e M o d e l

8.4 Comparison of the Deterministic and Stochastic Solutions

-

Sensitivity Analysis 77

9. SUMMARY 83

STOCHASTIC OPTIMIZATION MODELS FOR LAKE EUTROPHICATION MANAGEIIIENT

b?'

Ldszld Somly6dy and Roger J-B Wets

INTRODUCTION

Yan-made ( o r artificial) eutrophication h a s been considered

as

one of t h e most serious water quality problems of lakes during t h e last 10-20 years. Increasing discharges of domestic and industrial waste water and t h e intensive use of c r o p fertilizers

-

all leading t o growing nutrient loads of t h e recipients

--

can b e mentioned among t h e major causes of this undesirable phenomenon. The typical symptoms of eutrophication a r e among o t h e r s sudden algal blooms, water coloration, floating water plants and debris, excreation of toxic substances causing taste and odor problems of drinking water and fish kills. These symptoms can easily result in limita- tions of water use f o r domestic, agricultural, industrial o r recreational purposes.

One of t h e major f e a t u r e s of a r t i f i c i a l eutrophication i s t h a t although t h e consequences a p p e a r within t h e lake, t h e cause

-

t h e gradual i n c r e a s e of nutrients (various phosphorous and nitrogen compounds) reaching t h e lake

-

and most of t h e possible control measures l i e in t h e region. Conse- quently, eutrophication management r e q u i r e s analysis of complex interac- tions between t h e water body and i t s surrounding region. In t h e l a k e , dif- f e r e n t biological, chemical and hydrophysical p r o c e s s e s

--

a l l being time and s p a c e dependent, f u r t h e r m o r e non-linear

-

^are important, while in t h e region one must t a k e into account human activities generating nutrient r e s i - duals and control measures determining t h a t portion of t h e emission which r e a c h e s t h e water body.

Eutrophication management r e q u i r e s a sound understanding of a l l t h e s e p r o c e s s e s and activities which, in f a c t , belong t o quite d i v e r s e disciplines.

Additionaily, various uncertainties and stochastic f e a t u r e s of t h e problem have t o be a l s o t a k e n into account, f o r example, t h e estimation of loads from infrequent observations and t h e dependence of water quality on hydro- logic and meteorologic f a c t o r s , respectively. The f a c t t h a t we a r e dealing with a stochastic environment, i s especially important f o r shallow lakes, due primarily t o t h e a b s e n c e of thermal stratification which p r e d i c a t e s a much more definite r e s p o n s e t o randomness as would b e t h e c a s e f o r deep lakes.

Models developed with t h e aim of analyzing eutrophication can b e clas- sified into two broad groups:

(a) Dynamic simulation models (ecological models) which intend t o d e s c r i b e t h e temporal and spatial changes of various groups of s p e c i e s and nutrient f r a c t i o n s (algae, zooplankton, phosphorus, and nitrogen components, etc.). These models a r e being designed

primarily f o r r e s e a r c h purposes.

(b) Management optimization models being more applications oriented which have as objective of t h e determination of t h e b e s t o r

"optimal" combination of a l t e r n a t i v e control measures.

T h e r e i s a l a r g e l i t e r a t u r e devoted t o d e s c r i p t i v e models (see f o r example, Canale, 1976; Scavia and Robertson, 1979; Dubois, 1981; Orlob, 1983), but only relatively few management models have been proposed (Thomann, 1972; Biswas, 1981; Loucks et al. 1981; Bogdrdi et al., 1983).

Although changes in t h e ambient lake water quality should b e an important element in decision making, t o o u r knowledge, no l a k e is mentioned in t h e l i t e r a t u r e f o r which a p r o p e r description of t h e in-lake p r o c e s s e s h a s been t a k e n into account in t h e management model; i.e., t h e interaction between t h e models of t y p e (a) and (b) is missing. The r e a s o n f o r t h i s is at least threefold:

(i) Models (a) are t o o complex t o be involved directly in optimization and no well-based methodology exists in t h i s r e s p e c t .

(ii) In general, f o r management purposes, i t suffices t o use some aggregated f e a t u r e s of water quality. A s will b e s e e n , microscopic details o f f e r e d by models (a) can b e often ruled out, but i t i s not easy t o determine t h a t portion of information o f f e r e d by t h e eutrophication model (a) which should b e maintained in o r d e r t o a r r i v e at a scientifically well-based management model.

(iii) In most of t h e c a s e s , not enough time and money are available t o perform a systematic analysis including t h e joint development of both models.

Consequently, t h e r e is a gap between the application of descriptive and management models; t h e objectives of decision models a r e formulated in t e r m s of nutrient loads r a t h e r than of lake water quality, which is only acceptable f o r relatively simple situations. The present paper -

^after

con- sidering management goals in Section

¹

- offers an approach which allows t h e combined use of descriptive, simulation and management optimization models. This is discussed in Section

2.

The derivation of t h e aggregated lake and planning type nutrient load models t o be used in t h e management model is t h e subject of Section

3.

Both models a r e stochastic, special emphasis is given t o shallow lakes. Section

4

discusses f u r t h e r elements of t h e management model (cost functions, constraints, etc.). Alternative management models a r e formulated in Sections

5

and

6.

Two of them were implemented:

a

"true" stochastic model (which uses a s t h e starting point of t h e iterative solution procedure the corresponding deterministic model) and a linear programming approach capturing stochastic features of t h e problem through expectation and variance. Sections

6

and

7

give details of these models. Finally, in Section

8

t h e methods developed a r e used pri- marily f o r Lake Balaton that plays t h e role of

a

major case study.

1.

MANAGEMENT GOALS AND OBJECTIVES

A s

mentioned in t h e Introduction, artificial eutrophication leads t o

water quality changes which then r e s t r i c t s t h e use of water. The objective

of a manager when considering an eutrophication problem very much

depends on t h e features of t h e particular

system.

In most of t h e cases, how-

e v e r , t h e wish of managers can be formulated in quite general t e r m s . The

basis f o r this is t h e definition of trophic classes.

Trophic

state

classification i s t h e s u b j e c t of limnology. In p r a c t i c e f o u r major classes: oligotrophic, mesotrophic, e u t r o p h i c and h y p e r t r o p h i c c a t e g o r i e s

are

used (OECD, 1982), where oligotrophic indicates relatively

"clean" water, while h y p e r t r o p h i c r e f e r s

to

a l a k e r i c h in n u t r i e n t s and a n advanced s t a g e of eutrophication. To specify t r o p h i c c l a s s e s ,

water

quality components (concentrations of phosphorus and nitrogen f r a c t i o n s , oxygen content, algal biomass, etc., see f o r example, OECD, 1982)

are

applied

as

indicators. Based on observations and studies performed on many shallow and deep l a k e s t r o p h i c classes

are

specified quantitatively by c e r t a i n r a n g e s of t h e indicators. One of t h e

m o s t

widely used indicator i s t h e annual mean or annual peak chlorophyll-a concentration, (Chl-a), a measure of algal biomass. The chlorophyll content a f f e c t s t h e c o l o r of

water

and t h u s i t c h a r a c t e r i z e s f o r example t h e r e c r e a t i o n a l value of t h e lake. In

t e r m s

of concentration (Chl -a),,, t h e r a n g e s of classes oligotrophic

...

h y p e r t r o p h i c

are as

follows: 0-15, 10-25, 25-75 and 75- (see OECD, 1982).

A s t h e s e classes are closely r e l a t e d

to

t h e use of

water,

decision mak- ers' objective i s often

to

s h i f t a lake, say, f r o m h y p e r t r o p h i c t o a n oligo- t r o p h i c

state

and

water

quality goals

are

specified accordingly. Note t h a t t h e definition of t r o p h i c

classes

(with fixed boundaries) i s not unambiguous, which i s c e r t a i n l y not a s u r p r i s e

as

t h e t r o p h i c changes

are

caused by

com-

plex ecological processes. Still, however, decision makers r e q u i r e guide- lines easy t o understand and apply, and in t h i s s e n s e t h e use of t r o p h i c

state

classification i s inevitable in eutrophication management.

F o r many l a k e s t h e r e i s a spatial variation in t h e w a t e r quality. The t y p e of w a t e r usage c a n a l s o b e different in o n e area of t h e l a k e t h a n in o t h e r s , e.g., a g r i c u l t u r a l , r e c r e a t i o n a l o r industrial. F o r t h i s r e a s o n s t h e

objective of management can b e different f o r different segments o r basins of t h e water body. Thus spatial segmentation

is a

major component of management. Similarly, t h e decision makers should decide how important t h e random fluctuations in water quality (from y e a r t o y e a r )

are as

com- p a r e d t o expectations. Would h e f o r instance b e satisfied with d r a s t i c reduction in t h e "average" water quality without excluding t h e occurence of extreme situations o r r a t h e r would h e p r e f e r t o achieve

a

modest improve- ment in t h e mean provided t h a t h e

is

now able t o limit t h e r a n g e of possible fluctuations? Moreover, how would h e judge t h e situation

if

fluctuations also strongly vary in space?

Simultaneously with developing his judgement on t h e l a k e basins and management goals, t h e decision maker has t o perform

a

careful analysis

as

t o where (and when) h e can control nutrient loads in t h e watershed t h a t determine t h e trophic state of t h e lake? How effective are t h e control measures, what

are

t h e costs, benefits and associated constraints? How would

a

control measure taken in

a

subwatershed of

a

basin influence t h e

water

quality of

all

t h e basins (including both expectations and variances)?

Finally, how would h e select among alternative combinations of projects?

The methodology developed in t h e next sections

is

aimed at answering such questions, and thus provide technical support t o t h e decision process.

2. THE APPROACH

The approach t o eutrophication and eutrophication management

is

based on t h e idea of decomposition and aggregation (Somly6dy, 1982 and

1983a). The f i r s t s t e p

is

t o

decompose

t h e complex system into smaller,

t r a c t a b l e units forming

a

h i e r a r c h y of issues (and models), such as

biological and chemical processes in t h e lake, sediment-water interaction, water circulation and m a s s exchange, nutrient loads, watershed processes and possible control measures; influence of natural, noncontrollable meteorological f a c t o r s , e t c . One can make detailed investigations of each of these issues. This s t e p is followed by aggregation, t h e aim of which is t o p r e s e r v e and integrate only t h e issues t h a t a r e essential f o r t h e higher level of t h e analysis and hierarchy, ruling out t h e unnecessary details. In this way a sequences of corresponding detailed and aggregated (mainly descriptive type) models are developed. Only aggregated models a r e cou- pled in an on-line fashion (the approach is off-line f o r t h e detailed models) thus allowing t h e i r incorporation in a management optimization model a t t h e highest stratum of t h e hierarchy. The procedure f o r deriving t h e eutrophi- cation management optimization model (EMOM) is illustrated in Figure 1 (for f u r t h e r details, s e e Somlybdy, 1983b; and Somlyddy and van S t r a t e n , 1985) and consists of f o u r stages:

Phase 1

This i s t h e development phase of t h e dynamic, descriptive lake eutro- phication model (LEM) which has two

sets

of inputs: controllable inputs (mainly artificial nutrient loads) and noncontrollable inputs (meteorological f a c t o r s , such as temperature, s o l a r radiation, wind, precipitation). The output of t h e model i s t h e concentrations vector y of a number of water quality components as a function of time (on a daily basis) t

,

and space r :

y ( t , r ) . LEM is calibrated and validated by relying on historical data; t h e inputs of LEM are t h e r e c o r d e d observations. Because of methodological and computational difficulties on one hand, and uncertainties in knowledge

Development phase Loads

Management (optimization) phase

~ ( t , r l LEM

Checking phase Met

METG

FIGURE

1. S t r u c t u r e of t h e Analysis.

Planning mode usage

'i

-

Y i ( t )

I I

I Water quality indicator(s)

I for management.

I I I

r--- ^{I-+ Y + - I}

I I

I I

I

- - - 1 . .

--.

I ^I I

I I

I I

--

NLMP 1

k -

^{LEM -4}

and observational d a t a on t h e o t h e r hand, most of t h e models are not "con- tinuous" in s p a c e , but r a t h e r they d e s c r i b e t h e a v e r a g e water quality of l a k e segments ( b a s i n s ) having differing c h a r a c t e r i s t i c s . For details, r e a d e r s are r e f e r r e d t o t h e l i t e r a t u r e on ecological modeling (e.g, Somlyddy and van S t r a t e n , 1985).

Phase 2

In o r d e r t o apply LEM

at

a l a t e r s t a g e f o r solving t h e management problem, two important s t e p s must b e taken:

(i) A decision h a s t o b e made as t o t h e kinds of loads and meteorologi- c a l inputs t o b e used in planning scenarios. For r i v e r water qual- ity planning problems deterministic "critical" s c e n a r i o s can b e generally used, but f o r lake eutrophication problems (especially f o r shallow water bodies), t h e inputs should b e considered sto- chastic functions. In Figure 1 NLMPl and METG indicate models which g e n e r a t e t h e loads and meteorological f a c t o r s in a random fashion based on t h e analysis of historical data. The p r o p e r time s c a l e of NLMPl and METG should b e established in P h a s e 1 ; i t gen- e r a l l y suffices t o u s e a month as a basis (Somlybdy, 1983b).

(ii) The planning t y p e nutrient load model (NLMPl) i n c o r p o r a t e s aggregated c o n t r o l variables, f o r deriving t h e ' loads of individual l a k e basins ( 1 S i S N). They are used t o determine classes of a g g r e g a t e load s c e n a r i o s with d i f f e r e n t expectations and fluctua- tions f o r e a c h individual basin, but they do not e x p r e s s t h e way in which a s c e n a r i o i s actually realized in t h e watershed,

The LEM can then be

run

systematically under stochastic inputs with dif- f e r e n t control variable vectors

zi

yielding t h e stochastic vectors y i ( t )

Pw

t h a t describe t h e temporal changes in various

water

quality components f o r t h e basins ( 1 S i S N) (in contrast

to

t h e y i ( t ) of Phase 1 which a r e deter- ministic and noncontrolled). These changes are of i n t e r e s t f o r understand- ing t h e eutrophication process, but f o r decision making purposes, various indicators t h a t e x p r e s s t h e global behavior of t h e system can be used (see Section 1).

Experience h a s shown (OECD, 1982; Somlyl&y, 1983b) t h a t because of t h e cumulative n a t u r e of eutrophication, t h e indicators primarily depend on t h e annual a v e r a g e nutrient load with t h e (annual) dynamics of secondary importance,

at

least f o r lakes whose retention time is not too small. This means t h a t time can b e disregarded and

w e

can work with (vector-) indica- t o r s ,

s*,

^{t h a t a r e derived from t h e} concentration vectors (3,

( t ) ,

1 S i S N). This also implies t h a t t h e LEM can be used in an off-line

N

fashion.

There

are

various ways in which t h e LEM can be involved in t h e optimi- zation model, and implications and conclusions r e q u i r e s careful analysis.

For example, t h e

xi

indicators obtained from systematic computer experi- ments can b e s t o r e d

as a

function of t h e corresponding aggregated control vectors. These form

a

"surface" on which t h e "optimal" solution is built later. The o t h e r possibility, t h a t w e follow h e r e , i s

to

parametrize t h e

* Note that the number of indicators i s generally less than that of the water quality com- ponents, and often a single (scalar) indicator i s employed (see later).

outputs of LEM (obtained under various control v e c t o r s

&

⁾ and t o a r r i v e

at

a n analytical expression which then can be included in t h e optimization model. This i s t h e LEMP model, cf. Figure 1.

A s indicated in Figure 1, t h e success of this procedure i s not a p r i o r i obvious and depends mainly on t h e complexity and major f e a t u r e s of t h e sys-

tem.

Still, observations and model r e s u l t s have shown f o r s e v e r a l lakes t h a t a linear relationship holds between t h e indicator (e.g., ((Chl-a),,) and t h e annual a v e r a g e phosphorus load (OECD, 1982; Somlybdy, 1983b; Lam and Somlybdy, 1983). Van S t r a t e n (1983) h a s proven analytically by using a sim- plified LEM, t h a t f o r nutrient limited, turbid shallow lakes t h e maximum algal level is practically proportional t o t h e phosphorus load. Based on these findings,

w e

assume t h a t a n analytically expression, moreover a linear model, can be used

as

LEMP f o r a l a r g e variety of

water

bodies.

LEMP is a n aggregated version of t h e LEM t h a t can b e employed directly f o r planning purposes. I t describes approximately t h e indicators

5

as

a

function of t h e nutrient load, including stochastic variability. LEMP is connected t o a nutrient load model, NLMP2, which includes t h e control variables, zil ( 1 5 I 5 NLf ; 1 5 i 5 N). Compared with NLMP1, t h e NLMPZ model exhibits t h r e e significant differences:

(i) i t c o v e r s only t h a t p a r t of t h e load t h a t i s thought t o be controll- a b l e on t h e basis of available, realistic measures;

(ii) i t contains more details about t h e subwatersheds of each basin (location of pollution discharges and control measures, etc.); and (iii) i t is aggregated with r e s p e c t t o time; as f o r LEMP t h e annual

a v e r a g e load

can

generally b e used.

The coupled

N L M P 2 - L E M P m o d e l s

a r e then put through an optimization procedure, which generates (depending on t h e formulation of

E M O M ,

the Eutrophication Management- Optimization Model) f o r example, "optimal"

values f o r t h e control variables,

z i t ,

t h e associated indicators

,;'?

and- say-the total annual costs, TAC, needed f o r carrying out t h e project o r fixed by budgetary considerations.

Phase 4

In the course of this procedure various simplifications and aggrega- tions a r e made without

a

quantitative knowledge of t h e associated e r r o r s . Accordingly, t h e

last

s t e p in the analysis

is

validation. That

is,

the

L E M

can be run with t h e "optimal" load scenario

as

indicated in Figure

1

by t h e dashed line, and the 5 "accurate" and "approximate" solutions can be compared.

3.

FORMULATION OF THE STOCHASTIC MODELS LEKP AND

NUiP2

The modeling procedure

w a s

outlined in Section

2

in general terms.

Here we continue t h e discussion detailing assumptions and limitations. The reason f o r these

is

threefold:

-

^{A l l}

the models used in t h e field of eutrophication a r e system specific;

- Consequently o u r objective cannot be more than to capture some of t h e major features of eutrophication management problems;

and

-

_{W e}

wish t o avoid generalization beyond o u r experiences.

The major assumptions we make a r e

as

follows:

(i) t h e lake is shallow, t h e water quality of which is vertically

uni-

form:

(ii) t h e lake can be subdivided into basins which are sequentially con- nected,

see

Figure 2;

(iii) t h e lake is phosphorus limited (like most

water

bodies) and thus nutrients o t h e r than P (phosphorous) are not involved in t h e analysis;

(iv) a single

water

quality indicator, t h e (Chla),,, concentration is used f o r defining trophic s t a t e and t h e goal of management (see Section 1);

(v)

as

discussed before linear relationship holds f o r individual basins between (Chl-),,, and P load;

(vi) s h o r t

t e r m

(a f e w years) management is considered, t h a t i s t h e renewal processes in t h e lake and i t s sediment l a y e r following

external

load reduction, and t h e scheduling of t h e investments are out of t h e scope of t h e p r e s e n t effort;

(vii) only c e r t a i n types of P sources and associated control alterna- tives

are

taken into account.

A t t h e end of this section, w e w i l l show t h a t some of these conditions can be relaxed if needed, and in fact t h e models t o be introduced have

a

b r o a d e r r a n g e of applicability than suggested by t h e above assumptions.

3.1. Aggre~ated Lake Eutrophication model

(LE W)

Based on Section 2, t h e assumptions made and t h e insight gained from t h e study on Lake Balaton (Section 8), t h e s h o r t term response of water quality t o load reduction can be written

as

follows (Somlybdy, 1983b):

where t h e elements of t h e N-vector r e p r e s e n t t h e (to be controlled)

water

quality in t h e N basins

-

ⁱⁿ

t e r m s

of (Chl-a),,, [mg/m 3

I -

and

Y,

refers t o t h e (noncontrolled) nominal s t a t e ; E i s expectation. In t h e equa- tion, t h e N-vector

4

e x p r e s s e s t h e change in load due t o control

where t h e elements of

L ^are

t h e annual mean volumetric biologically avail- able P load, BAP [mg/m3d] in each basins

(LJ = LJa/h.

h e r e

L_P

i s t h e

"absolute" load [mg/d] and

Vi

is t h e volume [m33). The BAP load covers t h e P fractions t h a t can b e taken up directly by algae and thus determine t h e s h o r t term response of t h e

water

body. Stochastic variables and stochastic parameters are bold faced. The random N-vector lt;l r e p r e s e n t s t h e random changes of

water

quality caused by noncontrollable meteorological factors.

Finally t h e elements of t h e square NxN-matrix D and t h e vector d

are

derived from t h e analysis and simulation in Phase 2 (see Figure 1).

The elements of matrix D are the reciprocals of lumped reaction

rates.

The main diagonal comprises primarily t h e effect of biological and biochem- ical processes, while t h e o t h e r elements refer t o those of interbasin exchange due t o hydrological throughflow and mixing. These elements e x p r e s s t h a t due t o water motion a control measure taken on subwatershed

of basin i will affect t h e water quality of o t h e r basins. A s will b e seen in Section 8, t h e diagonal elements dii of t h e matrix D (the slopes of linear load response relationships) are such t h a t (Chl-a),,, does not necessarily approach z e r o ( o r a relatively s m a l l value) if

Li

goes

to

0. The reason is t h a t (Chl-a),,, i s linearly r e l a t e d t o t h e sum of e x t e r n a l and internal loads.

The internal load is t h e

P

r e l e a s e of t h e sediment (a consequence of

P

accu- mulation during preceding y e a r s and decades) which practically cannot b e controlled. Since e x t e r n a l and internal loads are coupled (a reduction in e x t e r n a l load g e n e r a t e s a time-lagged reduction in t h e internal load) t h e long-term improvement of water quality is generally l a r g e r than given by Equation (3.1). The memory effect and renewal of sediment are however poorly understood (see f o r example, Lijklema

et

a l , 1983), this is one of t h e r e a s o n s why w e c o n c e n t r a t e on short-term control. Note t h a t due

to

t h e definition of

, , L A

(3.1) also yields t h e random variations of water quality in t h e (noncontrolled) nominal

state.

The c h a r a c t e r of slopes di is similar t o t h a t of t h e diagonal elements d i i , t h e

t e r m

di zui hLI e x p r e s s e s a change, linear in hLNf, in t h e random component of t h e water quality indicator in basin i. Of course, t h e effect of t h e random fluctuations

E~

caused by meteorology d e c r e a s e s if t h e sum of e x t e r n a l and internal loads diminishes.

This also means t h a t with new control

measures

t h e water quality of a lake may approach a "new equilibrium" via m a j o r fluctuations, as observed in nature.

In view of (3.1) water quality v a r i e s randomly f o r t h r e e reasons:

(i) random changes in meteorological f a c t o r s (primarily s o l a r radia- tion and temperature) (the distributions of t h e UJ~, typically

skewed, are obtained in Phase 2 of t h e procedure (Figure 1 ) and can generally b e approached by three-parameter gamma distribu- tions);

(ii) stochastic changes and uncertainties in t h e loads (see below); and (iii) t h e combined effect of climatic and load factors.

Relation (3.1) gives t h e aggregated lake eutrophication model. The model takes into account on

a

macroscopic level t h e effect of biological and biochemical processes, interbasin

mass

exchanges, t h e sediment further- more t h e influence of stochastic factors and uncertainties.

3.2 N u t r i e n t Load Model (NTXP2)

W e consider t h r e e P sources as indicated in Figure 2:

(i) d i r e c t sewage P load, LS;

(ii) indirect sewage P load when t h e recipient i s

a

tributary of the lake, LSN (both LS and LSN can be considered biologically avail- able and deterministic); and

(iii) tributary load t o which contribute various point sources (sewage discharges) and non-point sources of t h e watershed.

The biologically available portion of t h e tributary load is

La =L&

+

b(LT -&J,), (3.3) where L& i s t h e dissolved reactive P load,

LT

total P load and b availability r a t i o of t h e particulate (not dissolved) P load (the difference of

kT

and L&), b i s a%out 0.2.

Direct sewage

1

l ndirect sewage

FIGURE

2. Development of models LEMP and NLMPZ.

Now, l e t us consider t h e basic c o n t r o l options from t h e above P loads:

(i) and (ii)P precipitation by sewage treatment plants, and

(iii) p r e - r e r s e v o i r systems established on r i v e r s b e f o r e t h e y e n t e r t h e lake. These consist of two p a r t s (Figure 2): t h e removal of parti- culate P through sedimentation in t h e f i r s t segment, and t h e remo- val of dissolved P (benthic eutrophication in reed-lakes, sorption, etc.) in t h e second p a r t . The corresponding control variables are z s , x ~ , z p , and zg as indicated in Figure 2. All of them can b e

thought as removal coefficients, with

where r

-

and r + a r e lower and u p p e r limits respectively, and if

z =

0, no action i s taken.

Now consider t h e simple situation given in Figure 2 f o r t h e i t h basin of t h e lake. The original, uncontrolled load,

Li

can be expressed as follows:

where La is t h e portion of t h e load t h a t is beyond t h e controls considered (e.g., atmospheric pollution); f o r t h e s a k e of simplicity,

w e

d r o p t h e index i from t h e r i g h t hand side of Equation (3.5). The controlled load of t h e 2-th basin is

where r t is t h e retention coefficient. The expression ( 1

-

^{r t )} r e p l a c e s a r i v e r P t m n s p o r t model and i t defines t h a t portion of P t h a t r e a c h e s t h e l a k e from an indirect sewage discharge

at a

given point on t h e t r i b u t a r y ( r t

=

⁰ means no retention). I t is a p p a r e n t from Equation (3.6) t h a t the t r i - butary load can b e controlled by

P

precipitation and/or pre-reservoirs.

The

latter

influence linearly both expectation and variance of

ki,

while

P

precipitation only influences t h e expectation, thus t h e r e is an obvious trade-off between t h e s e alternatives. With equations (3.5) and (3.6) w e can now obtain

hNi

( r e c a l l t h a t bL1

= ALL/Q):

Terms

@

^and

@

e x p r e s s reduction in t h e expectation of t h e r i v e r s dis- solved P load; term

@

r e p r e s e n t s t h e effects of t h e fluctuations of t h e

L_D

load:

term @

gives t h e modification in t h e particulate P load of t h e r i v e r , while term

@

exhibits influence of d i r e c t sewage control. If we

set

all t h e control variables x t o z e r o in Equation (3.7), w e obtain t h e fluctuations in t h e original, noncontrolled load, t h e expectation of which i s zero.

Equation (3.7) is nonlinear in t h e control variables because of t h e pro- duct term XD m

xm.

which may cause difficulties when this relation is used in an optimization scheme. Several possibilities a r e available t o overcome this nonlinearity. For example, a new variable

can be introduced, a linear function of XD and XSN, which i s then included in t h e constraint equation, see e.g.,Loucks

et

al., 1981. In such a case t h e optimization r e q u i r e s a parametric analysis involving this variable.

Another possibility i s offered by t h e s u r f a c e dependent c h a r a c t e r of t h e P removal in t h e pre-reservoir (second element). Generally, f o r a r e e d lake one cannot estimate more than t h e P removal p e r unit of surface-area, independent of t h e inflow concentration. Under this approximation, how- e v e r , ^XD can b e defined in terms of t h e original uncontrolled load of t h e

inflow (in c o n t r a s t t o t h e t h e conventional definition: a c t u a l inflow minus outflow divided by t h e a c t u a l inflow) as a variable which i s not influenced by indirect sewage control. The p r i c e f o r such a n elimination of nonlinearity i s twofold:

(i) An u p p e r limit should b e specified f o r zg which

states

t h a t no more n u t r i e n t s c a n b e removed t h a n t h o s e t h a t r e a c h t h e l a k e via t h e p a r t i c u l a r t r i b u t a r y . In

terms

of expectations t h e constraint equation c a n b e written

as

(we note t h a t in a more p r e c i s e s e n s e t h e above condition should actually b e fulfilled f o r all t h e realizations of

Ld).

(ii) A new variable z$ should b e introduced with zg 5 z$ t o t a k e into account t h e f a c t t h a t t h e impact of t h e r e s e r v o i r on t h e fluctua- tion (Equation (3.7): term

@

i s not r e s t r i c t e d by t h e (physical) c o n s t r a i n t s introduced by (3.9).

For

a

more g e n e r a l situation than illustrated in Figure 2, when t h e i t h l a k e basin i s fed by

N i

d i r e c t sewage discharges ( 1

< ⁿ

⁵ N1) and N2 tribu- t a r i e s (1

s

m S N2), e a c h with

4

indirect sewage d i s c h a r g e s ( 1 9 1 S

M,),

Equation (3.7) becomes:

Equation (3.10) i s actually t h e final NLMP2 model e x c e p t t h a t w e have not y e t discussed t h e derivation of t h e stochastic load components

kT

and La, which is in itself a difficult problem because insufficient (infrequent) o b s e r - vations, s h o r t historical d a t a , and o u r lack of understanding ( s e e e.g.

Haith, 1982; Beck, 1982). Since t h e annual means are used f o r

I,T

and La and t h e dynamics are less important, t h e recommendation is t o d e r i v e t h e loads from a r e g r e s s i o n t y p e analysis as functions of t h e major hydrologic and watershed p a r a m e t e r s ( e r r o r terms should a l s o b e included).

Observations and c a r e f u l analysis of t h e composition of t h e load (point vs non-point s o u r c e contributions) and watershed are r e q u i r e d f o r such a p r o c e d u r e . In g e n e r a l

LT

^and

^{L a}

have (positive) lower bounds and c a n b e c h a r a c t e r i z e d by strongly skewed distributions. V e r y often t h e y c a n b e e x p r e s s e d as simple functions of t h e annual mean streamflow r a t e s ,

&,

t h e s t a t i s t i c s of which are generally known from much longer r e c o r d s than those available f o r loads. This way t h e basic stochastic influence of hydro- logic regime c a n b e involved. In many c a s e s , annual means of and Q are estimated from scarce observations. The uncertainty associated c a n b e investigated from a Monte Carlo t y p e analysis o r basic s t a t i s t i c a l considera- tions (Cochran, 1962: SomlyWy and van S t r a t e n , 1985). Taking into account

all t h e s e f a c t o r s , t h e loads c a n have r a t h e r complex distributions composed of various normal, log-normal, gamma,.

. .

distributions.

The model LEMP will b e used t o g e t h e r with NLMPZ given by Equation (3.10) as major components of t h e eutrophication management-optimization model, EMOM. Let us note t h a t after introducing (3.8) o r (3.9) into NLMPZ, t h e

water

quality indicator will b e e x p r e s s e d by t h e coupled model NLMP2- LEMP as a l i n e a r function of t h e decision variables, z t l .

To conclude t h i s section, we r e t u r n t o assumptions (i)-(vii) and check t h e i r "rigour" in t h e light of t h e knowledge gained in t h i s section:

(i) shallowness i s not r e s t r i c t i v e from t h e point of view of using NLMPZ;

(ii) non-sequential connection of t h e basins would simply involve a change in t h e s t r u c t u r e of D;

(iii) P is practically t h e only element t o b e controlled even in c a s e s when e.g., nitrogen i s limiting algal growth (OECD, 1982; Herodek, 1983); and

(vi) short-term management basically determines t h e long-term behavior of t h e l a k e

water

quality, t h u s t h e r e s u l t s h e r e have s t r o n g implications f o r long-term management;

(vii) c o n t r o l s d i s r e g a r d e d h e r e (e.g., sewage diversion) have similar f e a t u r e s from a methodological viewpoint t h a n t h o s e handled in t h e p r e s e n t p a p e r .

-

²³

-

4. CONTROL VARIABLES, COST FUNCTONS

AND

CONSTRAINTS

We now consider t h e management options t h a t a r e available t o control t h e eutrophication p r o c e s s and t h e r e s t r i c t i o n s t h a t limit t h e i r choice.

4.1. C o n t r o l V a r i a b l e s

We a l r e a d y touched upon some of t h e decision variables as removal

rates

of sewage treatment plants and r e s e r v o i r s . H e r e we give

a

more detailed discussion of possible control variables, including t h e i r c h a r a c t e r and associated bounds.

S e w a g e Treatment

(i) If we consider a r t i f i c i a l eutrophication as a problem of t h e lake- watershed system and w e are not interested in t h e details of t h e engineering design of each treatment plant, zs and z~/v should b e handled as real-valued variables; otherwise all t h e elements of t h e technological p r o c e s s of all t h e plants should have been taken into account with t h e aid of f0,11 variables.

Lower bounds f o r zs and XSN often exist, since in many countries t h e effluent P concentration is fixed by s t a n d a r d s (between 0.5-2 g/m3) speci- fied by environmental agencies.

(ii) Because of historical r e a s o n s

- ^at

^least in developed countries

-

P precipitation (chemical treatment) i s going t o b e realized in existing plants designed originally f o r biological treatment. If t h e efficiency of h e biological treatment i s unsatisfactory, which i s often t h e c a s e , i t should b e upgraded p r i o r

to

introducing chemical treatment. If decision variable associated t o upgrading i s

z , ,

t h i s t y p e of treatment plant management c a n

-

²⁴

-

b e taken into account by t h e condition

sgn zu 2 sgn zs,

if z, _is

a

real-valued variable.

(iii) If t h e problem incorporates not only decision about

P

precipita- tion and t h e associated upgrading, but also t h e design of t h e sewer network, this would again lead t o a n engineering type planning task (e.g., Kovdcs et a l , 1983) t h a t differs from o u r present considerations. If, however, w e wanted t o involve sewer network design in t h e eutrophication management in a simple way, t h e network of t h e subregions could b e o r d e r e d s o as t o correspond t o treatment plants and handled similarly as explained in item

(ii), after introducing decision variables zm.

(iv) Finally, a rough p i c t u r e on t h e spatial distribution of treatment plants can b e obtained defining corresponding f 0 , l j variables (0 means again no action).

(i) A s indicated in Section 3, basic P removal processes

are

s u r f a c e dependent, thus control variables zg and zp specify t h e size of r e s e r v o i r s . Lower bounds

are

generally given by t h e smallest reasonable retention time t h a t corresponds t o a given size.

(ii) Variables ZD and zp must often b e handled as 10,1{ variables.

4.2. Cost Functions

Costs and cost functions a r e discussed in t h e same o r d e r as decision variables above.

Sewage Weatment

(i) The costs of P precipitation (including investment and operation costs being s i t e specific) grow exponentially with increasing removal

rates

and decreasing effluent concentrations (see e.g., OECD, 1982; Monteith

et

al., 1980; Schiissler, 1981). The most straightforward method t o approach this type of cost functions is piecewise linearization (see e.g., Loucks

et

al., 1981) which r e q u i r e s t h e introduction of dummy variables.

(ii)-(iv) They are fixed costs.

Pre-reservoirs

Cost functions of r e s e r v o i r systems are strongly depending on which process determines primary P removal, construction and operation condi- tions. Very often not enough knowledge i s available t o define

a

cpst func- tion o t h e r than linear and usually running costs c a n b e neglected o r can be assumed t o b e compensated by t h e benefits of t h e r e s e r v o i r s (e.g., utiliza- tion of harvested reeds). A s a summary, w e can conclude t h a t costs can b e expressed as piecewise linear functions of t h e decision variables, an impor- tant f e a t u r e from t h e viewpoint of building up t h e management model EMOM.

4.3. Constraints

A s we already discussed most of t h e physical, technological and logical constraints

-

cf. (3.4), (3.8), (3.9), (4.1) and (4.2) and r e l a t e d explanations

-

we consider h e r e only t h e budgetary constraint which is p e r h a p s t h e most important one.

In o r d e r t o select among management alternatives of different invest- ment costs (IC), and operational, maintenance and r e p a i r costs (OC), t h e total annual cost (TAC)

t e r m

is used

TAC

= C

^OC1

+ C

al ICl

where al is t h e capital recovery factor that depends on t h e discount

rate

and t h e lifetime of t h e project (see e.g.. Loucks

et

al., 1981). This f a c t o r can b e different f o r "small" and "large" projects (e.g., introduction of P precipitation in treatment plants and creation of r e s e r v o i r s of considerable size, respectively) as f o r '7arrge

"

investments governments of ten guarantee finance a t low ("pure") i n t e r e s t

rates.

For this reason,

as

pointed out by Thomann (1972), al should be considered

as

a model parameter of a certain r a n g e and i t s influence on model performance should b e tested.

In most cases t h e TAC i s limited by budgetary considerations

TAC S

fl

(4.4)

o r reexpressing this in

t e r m s

of t h e control variables

This constraint, which involves t h e piecewise linear functions cil, can b e replaced by a linear constraint by substituting x $ , x i

...

f o r t h e variable

-

27

-

ztl

, each

z&

corresponding t o

a

piece of linearity of cil

.

5. FORMULATION OF TEIE EUTROPEFTCATION HANGEBIENT OPTIMIZATION YODEL: EMOH

A s already indicated in Section 1 , t h e r e are a number of variants avail- able in t h e building of t h e management optimization model t h a t allow us t o c a p t u r e t h e stochastic f e a t u r e s of t h e

water

quality management problem.

The NLMP2-LEMP model yields t h e following description of t h e eutrophica- tion process:

z = E

f z o !

^{+ % -((D + d ~ ) UN. (3.1)}

& = E

IkoI - k ,

(3.2)

where f o r t h e s a k e of convenience, t h e water quality indicator is denoted by y , and f o r i

=

1

,..., N ,

La,i

=

^{bpi +} a ( L , ~ , i

-

^{L4,t ) + Ls,~ + LsN,~ + L x , i (3.5)}

t h e last relation being valid under some additional constraints, see (3.9) and t h e related comments. Substituting, regrouping terms and renumbering (reindexing) t h e control variables, we obtain a n affine relation f o r t h e

water quality indicators y i (1 S i S N) of t h e type

N

y

= cz

^{- h}

N N (5.1)

where

kt

incorporates all t h e noncontrollable f a c t o r s t h a t affect t h e

water

quality

zi

in basin i , and t h e random coefficients associated t o t h e

z-

variables in (3.10) determine t h e e n t r i e s of t h e random matrix through t h e transformation: ( D

+

dzu)ALN. To simplify t h e presentation of what fol-

lows, w e

string t h e decision variables in a n n - v e c t o r

(xi, . . .

^{, z,), each}

zj

corresponding t o a specific control measure affecting t h e load in some basin i

.

is thus a N X n-matrix and is a n N-vector. We also write

f o r t h e preceding equation, t h e notation y(z ,ul) is used t o

stress

t h e depen-

N

dence of t h e water quality indicators y i (1 S i S N) on t h e decision vari-

N

ables

z

and on t h e existing environmental conditions, denoted by w (con- trollable and noncontrollable) determining t h e e n t r i e s of

T

and h

.

The distribution function Gy ( z , -) of t h e random v e c t o r y ( z , *) depends on t h e choice of t h e control measures x i ,

... ,z,.

W e could view o u r objec- tive as finding

z *

t h a t satisfies t h e constraints and such t h a t f o r every o t h e r feasible

z

i.e., such t h a t f o r all z E

R~

prob. [ y ( z S ,

-1 <

^{z ]} r p r o b . [ y ( z , m)

<

^{z ]}

.

If such a n

z *

existed i t would, of course, b e t h e "absolute" optimal solution, since i t guarantees t h e b e s t water quality whatever b e t h e actual realiza- tion of t h e random environment. There always exists such a solution if

t h e r e

are

no budgetary limitations: simply build all possible projects t o t h e i r physical upper bounds ! If this were t h e case, t h e r e would b e no need f o r this analysis and i t is precisely because t h e r e are budgetary limitations t h a t w e are led t o choose

a

r e s t r i c t e d number of treatment plants and/or pre-reservoirs, and unless t h e problem is very unusual, t h e r e will be no choice of investment program t h a t w i l l dominate all o t h e r feasible programs in terms of t h e p r e f e r e n c e ordering suggested by (5.3).

W e are thus forced t o examine somewhat more carefully t h e objectives w e want t o achieve. We could, somewhat unreasonably, s e e t h e goal as bringing t h e

water

quality indicator t o

a

n e a r z e r o level (depending on t h e internal load) in all basins. This would ignore t h e individual c h a r a c t e r i s t i c s of each basin,

as

w e l l a s t h e user-oriented c r i t e r i a , a s f o r example r e c r e a - tional versus agricultural. A much more sensible approach,

as

discussed in Section 1, is t o choose t h e control measures s o as t o achieve certain desired trophic

states.

Let

be water quality goals expressed in terms of t h e selected indicator, (Chla),,,, each yi corresponding t o t h e particular use of basin i

.

^The

sensitivity of t h e solution t o t h e s e fixed levels yi would have t o b e a p a r t of t h e overall analysis of t h e system. W e a r e thus interested in t h e quantities:

C Y ~ ( ~ ,

w )

- 7 i I +

^{f o r i}

⁼

^1,

..., N ,

t h a t measure t h e deviations between t h e realized

water

quality and t h e fixed goals y i , where [@I + denotes t h e nonnegative p a r t of @:

The vector

is random with distribution function

G ( z , ⁰⁾

defined on R ~ , and t h e problem is again t o choose among all feasible control measures z

l,

. . . ,

z,

, i.e., that satisfy

all

technological and budgetary constraints, a program

z s that gen-

erates the "best" distribution

G ( z ^{s ,} e)

by which once could again mean

f o r all

2 E A s

already mentioned

earlier,

such a n z s exists only in very unusual circumstances, and thus we must find a way t o compare t h e dis- tribution functions t h a t takes into account t h e i r particular characteristics but leads t o a measure that can be expressed in t e r m s of a scalar func- tional.

5.1.

The f i r s t possibility would be t o introduce a pure

r e l i a b i l i t y cri- terion,

i.e., t o fix, in consultation with t h e decision maker, certain reliabil- ity coefficients t o guide in the choice of a n investment program. More specifically we would fix

⁰

<

^{a S 1,}

s o that among

all

feasible

z w e

should r e s t r i c t ourselves t o those satisfying

where

7

=

( 7 1 , . . . , y , ~ . O r

preferably,

if w e

take into account t h e fact t h a t each basin should be dealt

with

separably,

w e

would fix t h e reliability coef-

f

icients

a t , i

=

1

,...

^{,m Z,}

and impose t h e constraints

prob.

[ y i ( z , ⁰⁾

<

y i ] r a t , i

=

1

,...,

^mo. (5.5)

The

scalars a

o r

( a i , i

=

l , , . . , m z )

being chosen sufficiently large s o t h a t

we would observe t h e unacceptable concentration level only on r a r e

occasions. In

terms

of t h e distribution function G , these constraints become

f o r (5.4), and

G i ( z , O ) 2 at f o r i

=

1

,...,

^{N , (5.7)}

f o r (5.5) where t h e Gi ( z , *) are t h e marginal distributions of t h e random variables [ y i ( z

,

0)

-

^{y i ]}

^+.

^{I n} t h e parlance of stochastic optimization models these

are

p r o b a b i l i s t i c (or chance) c o n s t r a i n t s ; one refers t o (5.4) as

a

j o i n t probabilistic constraint. To find

a

measure f o r comparing t h e distri- butions f G ! z , 0). z feasible j w e have a simple a c c e p t / r e j e c t criterion, namely if a t 0 G ( z , 0) is e i t h e r l a r g e r than o r equal t o a , o r f o r each i

=

1 ,

...,

N, Gi ( z , 0) i s l a r g e r than o r equal t o a t , t h e investment program z is acceptable and otherwise i t is rejected. This means t h a t w e "compare"

t h e possible distributions f G ( z , a), z feasible j

at

1 point, but w e do not con- s i d e r any systematic ranking. Assuming w e o p t f o r t h e more natural separ- able version of t h e probabilistic constraints (5.5), w e would r e l y on t h e fol- lowing model f o r t h e policy analysis:

find z E Rn such t h a t

r j

- s

z j

s

ff+, ^j =I

,...

, n ,

CT=l

a i j z j bi * i =I,

...

^,m

,

I I

p r o b

[zj'=l

t i j (W )zj

-

^{hi (W )}

<

7 i

1

^{2 a t , 1 51,}

... ^{, N ,}

and z

= z.j?=l

^{cj ( z j )} is minimized

where

as

before t h e v e c t o r r

-

and r + are upper and lower bounds on z , t h e inequalities

EGl

^{aij z j S bi} describes t h e technological constraints,

including (3.9), and f o r every j

is t h e cost function associated t o project j , see (4.5). The overall objec- tive would thus b e t o find t h e smallest possible budget t h a t would guarantee meeting t h e present goals yi

at

least a portion at of t h e time.

W e did not pursue this approach because i t did not a l l o w

u s

t o distin- guish between situations where w e almost

m e t

t h e p r e s e t goals yi and those t h a t generate "catastrophic" situations, i.e., when some of t h e values of t h e (pi ( x ,w ), i

=

1

,...

,N) would exceed by f a r (yi, i

=

1 ,

. . .

, N) and f o r pur- poses of analysis of this eutrophication model this i s a serious shortcoming.

Let us also point out t h a t t h e r e are also major technical difficulties t h a t would have t o b e overcome. Probabilistic constraints involving &fine func- tions with random coefficients

are

difficult t o manage. W e have only very limited knowledge about such constraints, and then only if t h e random coef- ficients ((ti, hi ^(a)) are jointly normally distributed, cf. Section 1 of Wets, 1983a f o r a survey of t h e available r e s u l t s and t h e relevant refer- ences. Since in environmental problems t h e coefficients

are

generally not normally distributed random variables w e could not even use t h e f e w results t h a t

are

available, except possibly by replacing t h e probabilistic con- s t r a i n t s by approximates ones using Chebyshev's inequality,

as

suggested by Sinha, cf. Proposition 1.26 in Wets, 1983a.

5.2. A second possibility i s t o recognize t h e f a c t t h a t one should dis- tinguish between situations t h a t barely violate t h e desired water quality o r levels (yi, i

=

1 ,

...,

N) and those t h a t deviate substa.ntially from these norms. This suggests a formulation of o u r objective in terms of a

penalization t h a t would t a k e into account t h e observed values of [vi (z ,w )

-

^{yi ] + f o r i}

=

,

. - .

,N. We expect such a function

q : R~ 4 R

t o have t h e following properties:

(i)

4'

is nonnegative,

(ii) q ( z )

=

0 if zi 5 0 i

=

l ? . . . . N ,

(iii)

4'

is separable, i.e., q ( z )

= zrZl ^qi

^(zi).

This last p r o p e r t y comes from t h e f a c t t h a t t h e objectives f o r each basin

are

o r may be different and t h e r e are essentially no "joint rewards" t o b e accrued from having given concentration levels in neighboring basins, t h e interconnections between t h e basins being already modeled through t h e Equation (3.1). A more sophisticated model, would still work with s e p a r a t e penalty functions q l ( z

I . .

-

.

'PN(zN)] but instead of simply summing these penalties, would

treat

them as multiple objectives. A solution t o such a problem would eventually assign specific weights t o each basin, making i t equivalent t o a n optimization problem with single objective function. W e shall assume t h a t t h e s e weighting f a c t o r s have been made available t o o r have been discovered by t h e model builder, and have been incorporated in t h e functions

qi

themselves; note however t h a t t h e methodology developed h e r e would apply equally w e l l t o a multiple objective version of t h e model.

In addition, t o (i)-(iii) w e would expect t h e following properties: f o r i = l , - - . ,N,

(iv)

qi

is differentiable, with derivative

qi.

(v)

+;

i s monotone increasing, i.e.,

+*

i s convex, (vi)

+;

^{( z t )}

>

⁰ whenever zt

>

^0,

-

relatively small if z t i s "close" t o 0 ,

-

leveling off when zt i s much "larger" t h a n 0.

A couple of possibilities, both with Ot ( z * )

=

0 if z f 5 0 , are + * ( z * )

= p*

2: If Z* P O ,

with

pi >

^0,

+ * ( z t )

=

pt(ez' - z t - 1 ) if z i 2 0 , a l s o with

pi >

0.

A s w e s e e , t h e r e i s a wide v a r i e t y of functions t h a t h a v e t h e d e s i r e d p r o p e r t i e s , what i s at s t a k e h e r e is t h e c r e a t i o n of a (negative) utility func- tion t h a t measures t h e socio-economic consequences of t h e d e t e r i o r a t i o n of t h e environment. W e found t h a t t h e following c l a s s of functions provided a flexible tool f o r t h e analysis of t h e s e f a c t o r s . Let 8 : R ^-,

R +

b e defined by

This i s a piecewise linear-quadratic-linear function. The functions (+*, i

=

1 ,

...

, N ) are defined through:

O r ( Z r )

=

^Q* ei 8 ( e c i z t ) f o r

=

1 ,

...,

N , (5.10) where qt and et are positive quantities t h a t allow us t o s c a l e e a c h function

+*

ⁱⁿ

t e r m s

of slopes and t h e r a n g e of i t s quadratic component. By varying t h e p a r a m e t e r s et and qt w e are a b l e t o model a wide r a n g e of p r e f e r e n c e relationships and study t h e stability of t h e solution under p e r t u r b a t i o n of t h e s e scaling p a r a m e t e r s .

FIGURE 3. Criteria functions.

The objective i s thus to find a program that in the average minimizes the penalties associated with exceeding the desired concentration levels.

This leads us to the following formulation of the water quality management problem: