Fuzzy Sets and Water Quality

CopHig ht

©

IF A.C 9 th Tric:nni.d World Congress

BuddpeSl. Hungdn . 19/'{4

FUZZY SETS AND WATER QUALITY P. W. Jowitt Departm",t of CivIl Engmemng, Impenal Col/ege, London SW7. Cl{

Abstract. The paper examines ways in which water quality and river basin objectives can be expressed linguistically, without the need for over-precise criteria based on insubstantial evidence. The achievement of these objectives is measured through a fuzzy set, generated by a sequence of linguistic arguments stemming from pollutant concentrations, water uses to the objective criterion. Some simple examples are given. Keywords.

Water pollution; fuzzy sets; objectives; river basin management.

INTRODUCTION The planning and (real-time) management of a water resource system such as a river basin is beset by two major problems. Firstly, the system is temporally and spatially varying and depends on the interaction of a large number of variables, only some of which are measurable and/or controllable. Secondly, the system behaviour affects a multitude of different parties, each in a variety of ways.

postulate, sensitivity analyses are more difficult to interpret and the very act of mathematisation lends an unwarranted and misleading air of precision. In addition, the transformation of an intuitively simple objective into a precise set of mathematical/statistical forms, often leads to criteria that are difficult to monitor in practice and are, in any event, far removed from the original intent. In this respect, there is a measure of irony: in most practical studies, a broad statement of the objective is given at the outset in the form of literal text, yet little effort is made to develop these sentiments in a formal semantic way and so develop an expert system for objective assessment. The tendency is instead to define a 'precise' mathematical measure of performance, in which the real imprecision is largely ignored.

These two difficulties are heightened by the presence of uncertainty which both hinders system identification and the specification of system objectives against which to compare the system's response. If both the behaviour and objective of the system were simple (say, a single variable, time varying system, with an objective in one dimension affecting a sole user), then intuition and common sense would often be sufficient to cope with planning and operational decisions. Even the intrusion of some uncertainty and doubt into such a system might not preclude common sense appraisal and an intuitive assessment of sensitivity. Furthermore, with such a system, common sense could be used heuristically and on the basis of bad and good judgement in the past, an expert system would develop. However for more complex systems (behaviourally and in terms of objectives), common sense can become confused and it becomes necessary to formalise the understanding of behaviour and specification of goals. The natural tendency is to mathematise both aspects. Thus, the behaviour of the system is modelled, for example, through a set of coupled differential equations and the objective is defined as a multivariate function of incommensurable attributes. The sensitivity of the model and the decision function can be assessed, though usually a posteriori. If the system and its objectives are complex, yet otherwise well-defined and not clouded by too much phenomological and semantic uncertainty, then such a mathematical resolution of the problem can work very well. The extrapolation of this technique to badly-defined systems with imprecise objectives is more difficult to countenance and, at worst, misleading: the mathematical model of the process is but a

The application of fuzzy logic is advocated as an adjunct to the modelling and decision-making process, with the advantage that model and objective imprecision is incorporated at the outset so that sensitivity analysis proceeds in parallel. Moreover, common sense appreciations of behaviour and goals are not merely reflected by the analysis, but can be used to synthesise it. In this paper, these ideas are illustrated with reference to a problem of water quality and the setting of discharge consent standards to a lowland river used as a means of effluent disposal and water abstraction. For the purposes of illustration, attention is confined to two discharges (industrial and municipal effluent) and two abstractions (for irrigation and water supply). In-stream quality per se is not considered. The upstream polluters and downstream abstractors may represent different interest groups, though it is often the case that a regional authority is empowered to act in a superior role and make the trade-offs between differing parties. In the example problem, the conflict and mutuality of interest between the four river users is not clear cut.

3145

P. W. Jowitt

3146

One particular advantage of using fuzzy logic in this type of problem is the way in which a distinction is made between "hard" and "soft" constraints or objectives. For example, maintaining the river stage to below the flood bank level is a "hard" objective, whereas, given the usual uncertainties, designing against the 100 year event is rather a "soft" objective for the simple reason that the flood-frequency relation is imprecisely known. Similarly, in the present illustration concerning water quality management, there are some "soft" constraints/objectives to be dealt with; concern has been expressed, for example, over the levels of nitrate in water supply. High concentrations are thought to be harmful, but the relationship between concentration and effect is continuously graded; consequently any "standard" is somewhat arbitrary. Thus it makes little sense to regard the range of nitrate concentrations as divisible into wholly good and bad regions. Even so, it is current practice to do just that; nitrate standards are often couched in terms of a specified non-exceedance percentile, giving recognition to the temporal variability of concentration, but imagining a rigid demarcation between good and bad. It might be argued that the specified percentile value is merely pragmatism designed to give an operating target. Nonetheless, such percentiles rapidly assume an aura of scientific fact rather than their intended notion of a workable expression of scientific opinion. Fuzzy logic as used in this paper is directed towards correcting this tendency.

The link between water quality and the objective is represented by the generalised fuzzy relation. S

S

fuzzy set giving the measure of support for success and failure .

Q

{QI'~Jj fuzzy sets describing water quality for irrigation and water

where

fuzzy relation linking Q , ~ to S. R is based on the lin~ist~c iHrormation embodied in (1),(2),(3).

RQS

LOCAL ASSESSMENTS OF WATER QUALITY The relationship between "pollutant" (nitrate and chloride) concentrations and Q and I Q themselves is similarly based upon such l!nguistic statements as (1), (2) and (3) , and not on rigid percentile standards based on dubious "dose-response" data. For example, suppose the irrigation and water supply quality are related to chloride and nitrate concentrations in the following terms:

(4)

QUALITY FOR IRRIGATION, Q I ~ite

Q I

is bad if chloride

high/high

Q I

is good if chloride low

Q 1

is quite good/good i f chloride quite low and high.

n~trate

SUCCESS AND FAILURE TO MEET WATER QUALITY OBJECTIVES

QUALITY FOR WATER SUPPLY,

Setting aside for the present the conflict between polluters and abstractors, consider first the problem of determining the relationship between water quality states at the points of abstraction and the overall objective downstream. The two abstractions, irrigation water and water for supply, have different requirements and it is sensible to couch water quality objectives which reflect their different requirements and the relative importance of them. The overall objective might be stated as: OBJECTIVE

"to provide for a rural area with very important agricultural demands and where the river is major source of water supply ..-.---(1)

~

is bad if chloride high

or if nitrate

SUCCESS: "If water quality is good for irrigation and quite good/good for water supply.

FAILURE

(2)

"If water quality is bad for irrigation or bad for water suppy-Qr quite bad for irrigation and ~ bad for water supply". (3)

Note that the precise reasons for badness or goodness for a particular user are not detailed in these statements; they are not particularly relevant at this stage. The objective and its measures of success/failure reflect the balance between different users and the general level of importance attached to the users as a whole.

high

or if chloride quite high/high and nitrate quite high ~

is good if chloride low/quite low

and nitrate

low

Expressions (4) and (5) are represented symbolically through the relations XoR XOR where

The measures of success and failure to meet this objective might be expressed as follows:

(5)

~

X =

XQI xQw

{xc'\JI

= the fuzzy set describing

the concentrations X ,X of chloride and nitr~te~ fuzzy relations linking concentrations to quality for irrigation and water supply.

In summary, there are three generalised relations linking pollutant concentrations, X = {X ,X j, through the qualities for particfila~ users, Q = {Q ,~I to a measure of objective performance S.I

]

X

Q

3 147

Fu z zy Se ts and Wat e r Qu a lity GLOBAL ASSESSMENT OF WATER QUALITY OBJECTIVES

whose axes are the fuzzy supports for success and failure (see Figure 1).

Given such a set of relations, it is possible to consider the performance of the system, as gauged by S, in response to different levels of nitrate and chloride. These levels might be specific concentrations, but more usefully, they might represent more general expressions of pollutant concentrations. Indeed, it is also possible to examine the variation in the measure of success and failure that result from (statistical) distributions of pollutant concentrations. Note that within fuzzy sets such as S (failure, success) and ~ (bad, quite bad, ••• , good), failure and success or bad and good are not necessarily mutually exclusive; in this sense they reflect the reality of an imprecise world. The relation S = QoR is a symbolic representation of li~§UistiC statements (1), (2) and (3), and is developed from a collection of fuzzy compositions (0), fuzzy unions (v) and fuzzy intersections (.). (These operations are defined in appendix 1). Specifically

FAILURE-SUCCESS FUZZY SUPPORT SPECTRA The different regions of such a plot are annotated in Figure 1. OVerall success and failure may not always result: the a priori statements used to define the relationships may contain little information for some XC' X and N so lead to ambiguity. On the other hand, contradictions may be embedded in the definitions and these are revealed by the region of contradiction. For example if

v

In this illustration, Q and ~ are described by the five labels: bad, ~ite bad, indifferent, quite good and~. Set S is characterised by just two labels, failure and success. Table 2 gives the numerical forms of RB to R

XoR

X,Qw

X N

[0,

Q

[0, 0, .1, .B, 1]

~

[1,

S

[1, .1]

...

... ]

0, 0, 1]

.6, 0, 0, 0]

(7)

I = XCORlv{XCOR2.XNoR3}

Of course, in-stream pollutant concentrations are rarely constant, so it is necessary to ask of the variation in S if C and N have some distribution with given percentiles. The populations of C and N are sampled to produce a random pair of singletons X , X. These in turn produce a random point in S: R~peated sampling leads to the "Failure-Success Fuzzy SUpport spectrum" shown in Figure 2 .

•

ll

Similarly, the link between pollutant concentrations and water quality for each user is given by Q

[0, 1, 0,

as shown in Figure 2.

and RB,R9,Rl0,Rll are fuzzy sub-relations defined through elements of (1),(2) and (3). (See Table 1).

XORX QI

Xc

I

fuzzy composition fuzzy intersection fuzzy union

0

40 mg/ l, 21 mg/ l so that

then

and where

C N

~ ={XCOR4.XNoRs}VXCOR6vXNoR7 (B)

with Rl to ~ defined linguistically and numerically 1n Tables 3 and 4, based on statements (4) and (5). For the pollutants, the labels within the sets Xc' X are discretized levels of concentration; for c~loride (C), 10 values in the range 0 to 250 mg/ l and for nitrate (N) 11 values in the range 0 to 22 mg/ l.

The "expert system" encoded in the symbolic fuzzy composition (9)

S

thus enables the objective to be assessed when the system is subject to pollutant loadings X. These may consist of fuzzy singletons or even a statistical distribution of singletons. It is also possible to use linguistic descriptors, such as "Nitrate is high" etc. to drive (9). Furthermore, if the duration of pollution episodes is important then this too can be embedded within the fuzzy logic composition, and the behaviour of the system analysed via time series simulation to produce a time history of the "failure success fuzzy support spectrum". OVERALL RIVER BASIN GOALS

Fuzzy equations (6), (7) and (B) thus define the relationship between concentration and the river basin objective. Thus, if Xc and X are N set as "singletons": e.g.

••• 0

representing a chloride level C X = N

{al·,

01·,11 xnmg/ l, 01

x

c

So far attention has been confined to goals

I .}

mg/ l .

••• 0

I .}

representing a nitrate level N = xn mg/l. then S is determined from (6), (7) and (B), producing a point in S-space. In this example, S is labelled only by measures of "success" and "failure", so that any value of S space can be represented as a point on a two dimensional plot

couched solely in terms of the downstream objectives of abstractors, such that by trial and error or a more formal search procedure, inputs XC' X can be found which give an N adequate measure of success and a tolerable level of failure. (Some aspects related to the direct synthesis of suitable X ,X are covered elsewhere; Jowitt (19B4). In ~e ~xample used for illustration herein there are two sources of pollution, one for nitrogeneous matter and the other for chloride. A balance must be struck between the efforts expended in reducing discharges and the deleterious effects such discharges have on downstream abstractors. To this end a global objective must be specified which is superior to (1), and which might read:

P. W. Jowitt

3148

GOAL:

"to give a good measure of success in terms of the downstream objective, bearing in mind treatment costs generally and the important of industry to the area as a whole". ('0)

The goal indicates that treatment costs should not be excessive and that industrial waste control might be financially limited. In other situations, it may well be that downstream interests are paramount. Furthermore, it might be necessary to state objectives which change over a number of years, initially having concern for the economic consequences of pollution control, but reducing this emphasis in favour of more stringent control as time progresses. This global objective ('0) might be delineated as follows: SUCCESS: "Downstream objective successful and wastewater treatment costs at worst quite expensive and industrial waste treatment costs at worst moderately expensive". FAILURE: "Downstream objective not successful or waste water treatment costs very expensive or industrial treatment costs expensive". The level of expense for th~ upstream polluters would be related to the discharged concentrations of nitrogeneous material and chloride. For example, for the industrial discharge; "Expensive if effluent chloride is low, low "Not expensive if effluent chloride is high, high"

the practical difficulties in defining mathematically such an objective function so precisely. The method proposed here admits this at the outset such that any imprecision in the system is visible at all stages of the decision process. CLOSURE The foregoing has sought to demonstrate the application of fuzzy logic to the problem of assessing the impact of patterns of water quality on overall river basin objectives. An important aim of the paper is to illustrate that imprecision in these stated objectives (and in the causal factors that affect them) need not preclude such an asessment. The technique is based directly on linquistic statements of such objectives and effects; as such it does not require the formulation of artificially and misleading measures of performance. REFERENCE Jowitt, P.W. ('984). Unpublished Lecture: "Fuzzy Sets in Water Resources Management". (Presented at a Workshop on Imprecision in Civil Engineering, Imperial College, '983, and to be Submitted for publication in "Civil Engineering Systems").

TABLE'

Linguistic Relations between Pollutant Concentrations and Quality for Irrigation and Water Supply (together with Associated Fuzzy Relations ~Q)

~

~

S

FAILURE/SUCCESS MEASURE,

etc. IRRIGATION QUALITY Once again, these statements are developed into suitable fuzzy relations and denoted symbolically by a fuzzy composition. Ultimately, a series of relations result which allow descriptions of chloride and nitrogeneous material concentrations to be assessed in terms of the financial burden on polluters and the effects on downstream users, and thereby into the consequences on the global objective which states the desired balance between different groups.

WATER SUPPLY QUALITY NITRATE CONCENTRATION CHLORIDE CONCENTRATION "'

S

SUCCESS i f {QI GOOD and

~

~

GOOD/QUITE GOOD

~

Ra FAILURE i f

This consequence will be expressed as a set of fuzzy supports for success and failure to meet the global objective, and may be represented once again as a failure success fuzzy support spectrum. As such, any particular outcome, even in response to precise pollutant input levels (i.e. fuzzy singletons) will not always lead to a definitive failure or success (or, as is perhaps more conventional, even a single numerical value of a mathematical objective function); some interpretation will be required by the decision maker. It would be wrong to regard this as a material weakness of the method. It is argued that any imprecision in the conclusion realistically and properly reflects imprecision in the stated objective. It is indeed unlikely that large scale decisions will ever be made solely on the basis of a numerical value of some function of incommensurate attributes, not because such a notion is theoretically unrealisable (at least for a single decision-maker) but because of

FUZZY SETS

{Q!

R9

QUITE BAD and

or

Q I

BAD

R,O

or

~

BAD

R"

F,

QIORa

F2

QWOR9

F4

QIOR,O

FS

QN OR ,

Le.

S

F3

' ' '}

S

"--"

~

QUITE

F3 v F4 v FS

{QIOR8·~OR9 }vQIOR,O vQWOR "

where Q I

XcOR,V{XcOR2·XNoR3}

~

{XCOR4·XNoRS}VXcOR6vXNoR7

BA~}

3149

Fuzzy Sets and Water Quality Numerical Forms of

TABLE 2

TABLE 4

Numerical Forms of RQS for R, to

Rl

RIO

Re

1.0 1.0 0.6 0.1 0.0

0.0 0.0 0.2 0.' 0.8 1.0 1.0 0.7 0.' 0.1

0.0 0.1 0.' 0.7 1.0 1.0 0.7 0.' 0.1 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.0

0.0 0 .0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.0

0.0 0.0 0.0 0.0 0. 0 0.0 0. 0 0.0 0. 0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.6 0.8 1.0 1.0 0.8 0.' 0.2 0.0 0.0 0.0

1.0 1.0 0•• 0.7 O.S O.l 0.1 0.0 0.0 0. 0

0.0 0.0 0.0 0.2 0.' 0.8 1.0 1.0 1.0 1.0

0.0 0.0 0.2 0.' 0.8 1.0 1.0 0. 7 0. ' 0.1

0.0 0.1 0.' 0.7 1.0 1.0 0. 7 0.' 0.1 0. 0

0.0 0.1 0.' 0.7 1.0 1.0 0.7 0.' 0.1 0.0

0.6 0•• 1.0 1.0 O.B 0. ' 0.2 0. 0 0.0 0. 0

0.0 0.0 0.0 0.0 0.2 0.' O.B 1.0 1.0 1.0 1.0

0.0 0.0 0.0 0.2 0.' O.B 1.0 1.0 0.7 0.' 0.1

0.0 0.1 0.' 0.7 1.0 1.0 1.0 0.7 0.' 0.1 0. 0

0.0 0.0 0.0 0.0 0.1 O.l O.S 0.7 0 •• 1.0 1.0

0.0 0.0 0. 0 0.0 0. 1 O. l O. S 0.7 0 •• 1.0 1.0

0.0 0.0 0.0 0.1 O.l O.S 0.7 0 •• 1.0 1.0

0.0 0.0 0.0 0. 2 0 •• O. B 1.0 1.0 0.8 0.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Rl

TABLE 3

Q

I

BAD

if

Xc

HIGH/Q.HIGH

Q

GOOD i f

Xc

LOW

GOOD i f

{Xc

I

Q I

\.

and

Q LOW

-....

1

J

XCOR,

Q 2

X OR C 2

Q

X OR N 3

3

J Q4

i.e.

Q

w

BAD

~

= Q2· Q3

Q I

}

Q I

Q 6

X OR C 4

Q7

XNOR S

Qg

XC OR6

R,O

~

0.0 0.0 0.0 0.0 0.2 0 •• 0.8 1.0 1.0 0.8 0.6

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

o

K

I

I.

~'I.-<.""

I ~l

.~...

~

-

f'ISUCCES s

./,-=

X oR v {X OR .X OR } c I C 2 N 3

if

Xc HIGH

R6

or

X HIGH N

R7

or

{Xc Q.HIGH

and

X Q.HIGH} N '-"-w

-

~

R2

i f {Xc LOW/Q .LOW

'~i

I-

"

..,'"

I

...

~

"'''1-

~ (,

.0(

~

and

X LOW } N

s ~ 11If~i.4.re ~ -1/ "","ssJ $

;; £1,'1J

1

Q s

Q,.Q,

~ ~

Q vQ vQ S 9 10

X OR N 7

i.e.

0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.0

R,

Q, v Q4

~

GOOD

0.0 0.0 0. 0 0.0 0. 0 0.0 0.0 0.0 0. 0 0.0

O.~

0.1 0.0 0.0 0.0 0.0

XN HIGH}

R,

w

1.0 1.0 0 •• 0.7 O. S

R7 0.0 0.0 0.0 0.0 0.1 O.l O.S 0.7 0•• 1.0 1.0

R3

~

Q

0.6 O. B 1.0 1.0 O. B 0.' 0.2 0.0 0.0 0.0 0.0

~

R2 Q,

1.0 1.0 1.0 0 •• 0. 7 O.S O.l 0.1 0.0 0.0

R6

0.0 0.0 0.0 0.0 0.0 0 .0 0.0 0.0 0.0 0.0 0.0

Linguistic Relations between Water Quality for Irrigation and Water SUpply and OVerall River Basin Objective (together with associated fuzzy relations R ) QS

Q I

0.6 0.8 1.0 1.0 O.B 0.' 0.2 0.0 0.0 0.0

RS

R2

0.0 0.0 0.0 0.0 0.0

1.0 0.8 0.15 0.1 0.0

0.0 0.1 0.6 1.0 1.0

R'

0.0 0.0 0.0 0.2 0.' 0.8 1.0 1.0 1.0 1.0

Rll

R9

1.0 1.0 0.6 0.1 0.0

0.0 0.0 0.0 0.0 0.0

1.0 0.8 0.6 0.1 0.0

0.0 0.1 0.4 0.7 1.0

~

Fig.,.

{X OR ·X OR } v X OR VX OR C 4 N S C 6 N 7

Graphical Representation of Fuzzy Set S (Failure, Success) Showing Regions of "Failure", "Success", "Ambiguity" and "Contradiction".

P. W. Jowitt

315 0

FAILURE/SUCCESS FUZZY supPORT

50 Xi}. CHLORIDE

40.

95 Xi}_ CHLORIDE

120.

50 Xi}. NITRATE

3.

95 Xi'. NITRATE

9.

SeECTR~

0.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

0.0

o

o

o o

o o

o o

o

o

o o

o

o

o o

o

.1

o o

5

9

26

.2

o

o

o

o

o

o

o

o

4

o

o

.3

o

o

o o o

o

o o o

26

o o

o o

o o o

o

o

o o

o

o

o o o

o o o

o

o

o

14

o o o

o

o

o

o

o

o o

o o

o

o

7

o

5

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

.5

o

o o o

.6

o

o

.7

o o o o

o

.8

1.0

Fig. 2.

o

o o

o

1.0

o

FIBURES IN SPECT_ REFER TO X OCCURRENCE OF BET B

o

"Failure/Success Fuzzy SUpport Spectrum" for Log Normally Distributed Chloride and Nitrate concentrations of specified percentiles.

APPENDIX ,

for example:

.51

2,

.81

.71

2,

.2

Given the fuzzy sets X

L lIi I xi

~

11,

I x"

1121

X

2

I

•••

lJ i

I xi

•••

)In

I xn

I 2 X vX

I

11

LnilYi

,I " o I',

2

.7

Y2, ••• nil Yi •.• nml Ym -

In"

n2 ••• nj

I nm)

~

and the relation

then the Union of xl and x2 is defined

I 2,

.51 2,

and if

n,1 Y,' n21

,

I 3,

,

I4 I4

la, .5, .8,,) 10 ,.7,.2" )

then

X .X

y

3,

I 3, , I 4 .21 3, , I 4 .8

' l

.. :I .7

., .2

.8

1".7, .8,,) 1".5, .2,' )

-

J

2

.7

.7

.5

3

.2

0

.9

4

'0

20

30

and X

=

0I

"

.51 2, .81 3, ,

I4

la,

.5,

.8,

then and the Intersection of xl and x2 is defined

la,

y

.5,

.8,

' )0 ' .8

.7

.'] .2

.7

.7

.5

.2

0

.9

[

with the Composition of X with

~y

defined as

L max Imin ( 11.1 , PJ1. . ) ) I yJ. . 1

II,P

1.7, .7,

.9 )

.71'0, .7120, .9130

, )