Multicriterion modeling of wastewatermanagement : a comparison of techniques

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Authors Tecle, Aregai,1948-

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MULTICRITERION MODELING OF WASTEWATER MANAGEMENT:

A COMPARISON OF TECHNIQUES

by

Aregai Tecle

A Thesis Submitted to the Faculty of the

DEPARTMENT OF HYDROLOGY AND WATER RESOURCES

In Partial Fulfillment of the RequirementsFor the Degree of

MASTER OF SCIENCEWITH A MAJOR IN HYDROLOGY

In the Graduate College

THE UNIVERSITY OF ARIZONA

1986

01) 3)(Lc- Ji3Date

LV-c14V- EAA.,

LUCIEN DUCKSTEINProfessor of Hydrologyand Water Resources

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of

requirements for an advanced degree at The University of Arizona and is

deposited in the University Library to be made available to borrowersunder rules of the Library.

Brief quotations from this thesis are allowed without specialpermission, provided that accurate acknowledgment of source is made.

Requests for permission for extended quotation from or reproduction ofthis manuscript in whole or in part may be granted by the head of the

major department or the Dean of the Graduate College when in his orher judgment the proposed use of the material is in the interests of

scholarship. In all other instances, however, permission must be

obtained from the author.

SIGNED:

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

ACKNOWLEDGEMENTS

I wish to extend my most sincere thanks to my mentors Dr. Lucien

Duckstein and Dr. Martin M. Fogel for their invaluable impact on my

education. Special thanks are in order to both of them: to my thesis

director, Dr. Lucien Duckstein whose encouragement and untiring guidance

made the completion of this thesis possible, and whose profound

influence on my conception of the theory of multiobjective decision

making process has been greater than I realize yet; and to Dr. Martin M.

Fogel whose continuous support and guidance made my overall education at

this institution possible. A particular note of thanks also goes to my

committee member, Dr. Soroosh Sorooshian for his careful evaluation of

this work and his helpful comments.

I wish to thank Dr Nathan Buras and the Department of Hydrology

and Water Resources for the help provided while completing this program

of study.

Special thanks go to my friend Rachid R. Labgaa for his help and

comments during the writing of this thesis. A note of appreciation also

goes to Dr. Jerry Harwood for his friendly moral support and help in

using the word processor.

Finally, I would like to thank my wife, Negisti and our

daughter, Selam whose inspirational support was always a source of

strength and courage.

TABLE OF CONTENTS

Page

LIST OF TABLES vii

LIST OF FIGURES ix

ABSTRACT xi

1. INTRODUCTION 1

Preliminary Considerations 1Purpose and Organization 4

Purpose 4Organization 5

2. CASE STUDY PROBLEM 7

Physical Description of the Case Study 7Geography 8Existing Wastewater Treatment System 13

Institutional and Economic Consideration 21Administration of Wastewater Treatment Plant 21Economic Considerations of theWastewater Treatment plant 22

3. MULTICRITERION PROBLEM FORMULATION 24

Objectives 25Specifications 26

Prevent Groundwater Pollution 27Satisfy Required Effluent Quality 28Promote Treated Wastewater Reuse 28Ensure System Dependability 29Optimize Resource Utilization 31

Criteria 33Criterion Scores 37Generation of Alternatives 39

Pure Action Alternatives 40Supplementary Alternative Activities 41Action Mix Alternatives 45

Evaluation Matrix 45

iv

TABLE OF CONTENTS -- Continued

4. MULTICRITERION DECISION-MAKING TECHNIQUES

Outranking Types ELECTRE I ELECTRE II

Page

50

505054

Multicriterion Q-analysis I 60Multicriterion Q-analysis II 67

Distance-Based Techniques 68Compromise Programming 69Cooperative Game Theory 71

5. APPLICATION OF MODELS AND ANALYSIS OF RESULTS 74

Application of Models, Solutions andSensitivity Analyses 74

ELECTRE I 75ELECTRE II 83Multicriterion Q-analysis I 89Multicriterion Q-analysis II 97Compromise Programming 99Cooperative Game Theory 104

Comparative Evaluation of Techniquesand their Application Results 105

Comparative Discussion of Model Performances 106Comparative Discussion of MostPreferred Alternatives 109

6. SUMMARY AND CONCLUSIONS 112

Summary 112Conclusions 115

APPENDIX A: DESIGN PARAMETERS OF THE EXISTING NOGALESINTERNATIONAL WASTEWATER TREATMENT PLANT • • • • 119

APPENDIX B: ESTIMATED COST CRITERIA FOR EACHCONSIDERED ALTERNATIVE WASTEWATERTREATMENT SCHEME (IN 1988 DOLLARS) 122

APPENDIX C: OPTIMAL (P,Q) VALUE DETERMINATIONAND SENSITIVITY OF ELECTRE I WITHRESPECT TO CHANGES IN (P,Q) VALUES 123

TABLE OF CONTENTS -- Continued

vi

PageAPPENDIX D: ILLUSTRATION OF THE STEP BY STEP

RANKING PROCEDURE IN ELECTRE II 124

APPENDIX E: SENSITIVITY ANALYSIS OF MCQAWITH RESPECT TO CHANGES OFSCALAR SLICING PARAMETER S(K) 128

APPENDIX F: RESULTS OF MCQA I AND MCQA II IN TERMSOF DISTANCE FROM THEIR RESPECTIVEREFERENCE POINTS OF (1,1) AND (1,1,1)

129

APPENDIX G: ECONOMIC RANKING OF ALTERNATIVES 131

REFERENCES 132

LIST OF TABLES

Table Page

1. Annual Average Wastewater Influent and EffluentCharacteristics in the NIWWTP and EPA Acceptable Criteria . 19

2. Types of Measurement Scales 38

3. Objectives, Specifications, Criteria and Criterion . . . . 39

4. Considered Alternatives 46

5. Evaluation Matrix 47

6. System Parameters 52

7. Point Value Used in Discordance Matrix Determination . . . 53

8. Type of Outranking as a Function of Concordanceand Discordance Levels 58

9. Payoff Matrix 61

10. Concordance Matrix 76

11. Discordance Matrix 77

12. ELECTRE I Results 79

13. Characteristic Makeup of the Seven Cases forSensitivity Analysis of ELECTRE II Application 84

14. Ranking of Alternatives Using ELECTRE II forthe Seven Different Cases 87

15. Binary Incidence Matrix Obtained Using SlicingParameter Value of 0.9 90

16. Results of Q-analysis 93

17. Ranking of Alternatives Using MCQA-I Techniques 95

viii

LIST OF TABLES -- Continued

Table Page

18. Ranking of Alternatives Using MCQA-II Techniques 98

19. Alternative Ranking Using Compromise Programmingfor the 4 Sets of Weights and p=1,2 and 101

20. First Through Fourth Ranked Alternatives inEach of the Six Models 109

21. Effects of Model Sensitivity Analysis with Respect toParameters and Model-Wise Preferred Alternatives 115

LIST OF FIGURES

Figure Page

1. Location of Wastewater Treatment Plantin the Upper Santa Cruz River Basin 8

2. Nogales International Wastewater Treatment System 15

3. Average Daily Wastewater Flow (1972-1985) 16

4. Comparison of Wastewater Treatment PlantInfluent and Effluent Rate 17

5. Relationship Between Objectives, Specifications,Criteria and Alternative Schemes 34

6. Graphical Illustration of the Concept ofDistance Based Techniques: CompromiseProgramming and Cooperative Game Theory 70

7. Composite Graph of ELECTRE I Used toObtain the Kernel (Nondominated Alternatives)for (p,q) Values of (0.7,0.2) 78

8. Occurrence Frequency of Alternatives in a Kernelout of Total of 31 Trials Using ELECTRE I 80

9. Number of Alternatives Selected (in a Kernel) withRespect to Different Pairs of Combinations ofthe Thresholds p and q Values 81

10. Reduced Graph of the Strong Relationshipof ELECTRE II Application 85

11. Reduced Graph of the Weak Relationshipof ELECTRE II Application 86

12. Connectivity Structure of the Simplicial Complex Kx(Y;L). 91

13. Connectivity Structure of the Conjugate Simplicial Complexthat is the Inverse of Figure 12 92

14. Illustration of the Number of Selected AlternativesVerses Slicing Parameters-the Lower the SlicingParameter the Less the Selectivity of Alternatives . . . . 94

ix

X

LIST OF FIGURES -- Continued

Page

15. Graphical Illustration of the Most and Least PreferredAlternatives Using Compromise Programming 102

16. Ranking Specificity of Cooperative Game Theory 105

ABSTRACT

Multicriterion modeling of wastewater management problem is

presented in order to select the most preferred wastewater scheme. The

Nogales International Wastewater Treatment Plant which serves the

binational cities of Nogales, Arizona and Nogales, Sonora is used as

case study in the modeling process. The process includes identifying of

objectives, specifying of treatment alternatives and defining criteria

to relate the objective satisfactum level to the alternative schemes.

Six different multicriterion decision making techniques are

applied to analyze and obtain preference ordering among the alternative

treatment schemes. Analyses on the individual techniques and comparison

among them are performed to arrive at the following conclusions: (1)

all the techniques except one can be confidently used to obtain complete

ordering of alternatives, (2) there is inter-model consistency in the

ordering process, (3) in performing this function, the techniques are

fairly robust with respect to parameter changes, and (4) only two

treatment alternatives of fifteen considered are consistently ranked

higher than the rest.

x i

CHAPTER 1

INTRODUCTION

1.1. Preliminary Considerations.

This study is concerned with the application of multicriterion

decision making techniques to select an appropriate wastewater

management scheme. The need for this kind of investigation has

manifested itself in different ways. Phenomenal population growth and

urbani zati on i nrecentyearshave 1 ed to i ncreased production of

municipal and industrial wastes which must be properly treated and

disposed of. Not long ago, some surveys by the U.S. Environmental

Protection Agency showed that more than 60% of the wastewater treatment

plants in the United States were not operating as well as they should

(Council on Environmental Quality, 1979). Inadequately treated sewage

is being discharged into steams, rivers, and lakes which in many

instances may be due to improper operation and maintenance of wastewater

treatment plants. This has been a matter of great concern in the United

States for sometime now.

To cope with the problem, stricter regulations on wastewater

treatment, and disposal mechanisms have been issued at many levels.

Congress passed the Federal Water Pollution Control Act of 1977 (PL 92-

500) and, subsequently, the Clean Water Act of 1977 (PL 95-217) which

require stringent water quality management practices by municipalities,

industries and other dischargers by 1984. Section 208 of PL 92-500

1

2

requires regional facility planning through phasing out and integration

of existing treatment plant facilities.

The Nogales International Wastewater Treatment Plant (NIWWTP) is

such a regional facility serving the twin cities of Nogales, Arizona

(U.S.A.) and Nogales, Sonor (Mexico) and is projected to serve other

nearby growing communities such as Kino Springs and Rio Rio. By

treating wastes from Mexico, the NIWWTP serves not only within-country

regional level as dictated by section 208 of PL 92-500 but international

communities, the growth of which in the last few years seem to approach

a critical stage. The opening of border industries through the Mexican

Border Industrialization Program (Dominguez, 1980), the worsening of the

Mexican economy coupled with the continuing drastic devaluation of the

peso, and the desire of many Mexicans to cross the border and find

employment in the United States are combined to lead to an exploding

population growth on the Mexican side of the border. Thus wastewater is

being produced well above the projected level making the existing

treatment plant unable to achieve its mandated performance level,

instead resulting in the release into the Santa Cruz River of effluent

that does not meet EPA and Arizona Department of Health Services

standards. In addition sewage from broken sewer lines and unanswered

systems have been entering and polluting the Nogales wash (Montano,

1981; Vega, 1983). The situation has affected businesses and proved to

be health hazard at many instances in Nogales, Arizona (Alegria, 1980;

Greenberg, 1982).

3

The wastewater pollution problem has caught the attention of

officials at all levels of government (Dandoy, 1978; Friedkin, 1978;

Lindeman, 1978; Condes and Alegria, 1979; Vega, 1983; Arizona Department

of Health Services, 1985) resulting at times in authorization of studies

to determine the most appropriate treatment schemes to solve the problem

(John Carollo Engineers, 1979; Arthur Beard Engineers, Inc., 1982,

1984). These studies used a traditional cost-effectiveness approach

which expresses all aspects of the problem in monetary terms to

recommend certain treatment alternatives. Although usually useful, such

methods are sometimes grossly inadequate and/or inappropriate because of

the inherent multiobjective nature of the problem of wastewater

management planning (Major, 1977; Nakamura and Riley, 1981; Hiessl et

al., 1985; Tecle and Fogel, 1986). One major weakness of the economic

oriented single objective wastewater management screening method is the

difficulty to handle non-commensurable conflicting objectives such as

cost and water quality. Another problem with the traditional method is

its inability to handle non-numerical objectives such as aesthetic

values of wastewater treatment projects. Consequently, the need for

research on multicriterion wastewater management system cannot be

understated.

If properly managed wastewater may serve as an important

resource. The nutrients and other chemicals in it may enhance

aquacultural and agricultural productivity. Treatment plant sites may

become important recreational facilities and haven for wildlife. Most

importantly, the treated wastewater can become an essential water

4

resource of an area. This is particularly true in areas of scarce water

resource such as the region in which the wastewater treatment plant

under study is located. Thus, the recognition of wastewater both as a

waste product in one hand, and a useful resource on the other makes it a

convenient subject for application of multicriterion evaluation methods

and strengthens further the need for research on its multicriterion

management aspect as stated above.

Even though the need for multicriterion planning in water

resources in general (Maass et al., 1962; Marglin, 1967; U.S. Water

Resources Council, 1973; Major, 1977; Duckstein and Opricovic, 1980;

Gershon et al., 1982), and wastewater management in particular (Lohani

and Abulbhau, 1979; Nakamura and Riley, 1981; Hiessl et al., 1985; Tecle

and Fogel, 1986) has long been recognized, the practical application of

MCDM techniques are not widespread (United States General Accounting

Office, 1978). This study, therefore, attempts to prove the

applicability and promote the wide use of multicriterion decision making

techniques in wastewater management.

1.2. Purpose and Organization.

1.2.1. Purpose.

This study is conducted with two purposes in mind. The first

one is concerned with multicriterion formulation of a wastewater

management problem which is suitable for analysis using six different

multicriterion decision making techniques. The second one, on the other

hand, is focused at evaluating the comparative performances of the six

multicriterion decision making techniques in selecting the most

5

'satisficing' wastewater management scheme. The satisficing condition

is viewed with respect to 12 non-commensurable criteria while 15

different wastewater treatment alternatives are presented to compete for

selection.

1.2.2. Organization.

The thesis is organized to present a step by step development of

multicriterion modelling of the problem under study. Chapter 2 provides

a descriptive overview of the case study. At first, the area's

geographical features considered to be relevant to this investigation

are discussed. Then the physical status of the existing wastewater

treatment plant including its design and present treatment capacities,

future trends with respect to wastewater flow and major quality

parameters followed by a brief review of the institutional and budgetary

arrangements for the wastewater treatment plant are presented.

Chapter 3 develops multicriterion formulation of the problem to

make it suitable for evaluation using MCDM techniques. The formulation

consists of six separate but not necessarily independent steps that

include identification of project objectives, specifications, criteria

and criterion scales, generation of alternative treatment schemes, and

constructing an evaluation matrix. In the process 5 objectives, 12

criteria and 15 treatment alternatives are presented and explained.

In chapter 4 the theory and mathematical procedures behind each

of the six MCDM techniques applied in this study are described. These

procedures are by no means exhaustive but they do adequately reflect the

6

algorithms of each technique. The six MCDM techniques utilized are

ELECTRE I, ELECTRE II, MCQA I, MCQA II, compromise programming and

cooperative game theory. The first four belong to the class of

outranking types while the last two are of the distance based group of

MCDM techniques. All of them, however, are conveniently used to

evaluate a complex wastewater management problem with discrete and non-

commensurable objectives.

Applications of the six MCDM techniques to the case study are

discussed in chapter 5. In addition application results and model

sensitivity analyses with respect to criterion weights and other

parameter changes in each technique are made. Then a comparative

evaluation of techniques and their application results are provided

toward the last part of this chapter. At the end a comparison between

the two top-ranked alternatives is made to gain more insight of their

relative attributes.

The last chapter, chapter 6, contains a summary of the lessons

gained in this study and certain conclusions drawn from it.

CHAPTER 2

CASE STUDY PROBLEM

An essential requirement to resolving a problem is a complete

understanding of its background. This chapter, therefore, focuses on

the status of the existing wastewater treatment plant and its immediate

and relevant environment. This chapter is made up of two sections. The

first one consists of a physical description of the case study in which

the historical status of the existing wastewater treatment plant along

with the area's physiography and population trends are treated. The

second section, on the other hand, reviews the economic and

institutional aspects of wastewater management in the study area.

2.1. Physical Description of the Case Study.

Numerous field trips and extensive literature review have been

made by the author in the course of three years to study the water

quality in the upper Santa Cruz River Basin (Figure 1) in general, and

the problems of wastewater management in the Ambos Nogales area in

particular (Tecle et al., 1985; Tecle and Fogel, 1986, Fogel and Tecle,

1986). The descriptive features of the case study in this

investigation, the Nogales International Wastewater Treatment Plant

henceforth abbreviated as N1WWTP is, therefore part of the product of

the above efforts.

7

ARIZONA

MEXICO

0 6 12 16

Seale In Kilometers

8

2.1.1. Geography.

Geographical features can play important roles in an area's

waste assimilative capacity. As a result, a complete knowledge and

understanding of these features is necessary in the selection and design

of biological wastewater treatment systems suitable for the particular

area (Gloyna, 1971). Factors such as climate and soil type may

influence the rate of wastewater treatment while the population size and

regional activities condition wastewater production and concentration

level. For these reasons, a brief synopsis of the study site's

geographical conditions is provided.

Figure 1. Location of wastewater treatment plant in the Upper SantaCruz river Basin.

9

a. Location. The Nogales International Wastewater Treatment

Plant is located 14.5 kilometers north of the U.S.-Mexico international

boundary line at the confluence of the Nogales Wash and Santa Cruz River

(Figure 1). The Santa Cruz River is an ephemeral stream that comes from

the southeast side of the Treatment Plant and flows northward while, the

Nogales Wash is a polluted perennial stream that comes through the

centers of both Nogales, Arizona (U.S.A.) and Nogales, Sonora (Mexico)

and joins the Santa Cruz River near the Wastewater Treatment Plant.

Geographically, the Wastewater Treatment Plant is respectively located

at 31°251 north latitude and 110057, west longitude. It has an

elevation of 3560 feet above sea level (IBWC U.S.A. Mexico, 1983).

b. Climate. The climate of the area which is typical of the

southwestern United States is characterized by warm summers and mild

winters. This makes it quite conducive for wastewater treatment

operations of the biological treatment type or land application schemes.

On the basis of a 30-years record (1953-1983) the mean annuals for

temperature, rainfall, wind speed and evaporation at the wastewater

treatment plant weather station are 60.30

F, 17.08 inches, 1.8 miles per

hour and 93.38 inches respectively. The corresponding minimum and

0 0maximum measurements for 1983 are respectively, 19F and 106 F; on a

monthly basis 0.18 inches (April) and 4.70 inches (July); a monthly

average of 1.1 miles per hour (August) and 2.5 miles per hour (March),

and 3.32 inches (Feb.) and 13.53 inches (June) on a total monthly

basis. The instrumentation at this weather station consists of maximum

and minimum thermometer, standard 8-in rain gage, 48-in evaporation pan

10

with stillwell and hook gage, anemometer (registers miles), to

respectively measure the above, and psychrometer, hand turbine type to

measure relative humidity (IBWC U.S. and Mexico, 1983).

The above climatic factors are important in the wastewater

treatment process. Temperature affects the rate of biochemical

degradation. The average temperature, daily fluctuations, and yearly

variations all influence the biological, physical and chemical processes

in lagoons. Wind is usually the principal source of energy for mixing

the water in the facultative lagoons. Mixing, in turn is an important

physical parameter affecting the growth of algae, as most algae are non-

mobile and mixing, therefore is necessary to bring them into the zone of

effective light penetration. Duration and intensity of rainfall affect

the rates of infiltration and direct storm inflow for which the sewer

system and treatment facilities must be designed, while humidity affects

evaporation from the wastewater lagoons (Gloyna, 1971). These

conditions show the need to carefully evaluate climate factors before

selecting the type of a wastewater treatment system in a particular

area.

C. Soil and Geology. The soil and the rock materials in the

underlying geological structure may serve as physical, chemical and

biological filters for wastewater (Thomas, 1973; Ellis, 1973; Miller,

1973). The pore size distribution and the nature of wastewater movement

through the channels in the soil enables the removal of suspended solids

making the soil serve as a physical filter. Many organic and

inorganic reactions take place between the soil and the wastewater as

11

the latter passes through the soil profile making the soil also serve as

a chemical filter (Fuller and Warrick, 1985). In addition, there are

various groups of organisms in the soil such as bacteria, fungi,

protozoa, other soil micro-and macro-organisms and higher plants that

help the soil act as a biological filter. Higher plants can help

renovate wastewater through their absorption and transpiration

activities while micro-and macro-organisms in the soil and the

geological rock strata help in the degradation of organic compounds

contained in the wastewater. For these reasons a careful assessment of

the soil and geologic materials at the Wastewater Treatment site is

important (Goldstein and Moberg, 1973).

The soil type in and around the Nogales Wastewater Treatment

Plant is of the Comoro-Pima association type with deep sandy loans and

clay loans. These soils consist of well-drained soils that are 60

inches or more in depth and 0 to 3 percent slopes. The soils are formed

in recent alluvium weathered from mixed rock and are found on flood

plains and alluvial fans (Richardson et al., 1979). Geologically, the

Wastewater Treatment Plant site and the flood plains around it are

underlain by Younger Alluvium which is a Quaternary age depositional

product (Halpenny, 1964a; Simons, 1974). This material is made up

mostly of unconsolidated sands and gravels with occasional lenses of

silt or clay. This deposition ranges from 40 to 100 ft in thickness has

a coefficient of storage in the order of 17% of total volume in the

saturated zone and a high permeability which make it an excellent water-

bearing, unconfined aquifer (Harshbarger, 1978; Montano, 1981; Halpenny,

12

1964b, 1983). The aquifer is usually in the range of 0 to 10 feet from

the surface making it prone to pollution by percolating wastewater.

d. Population Records and Projections. A clear understanding

and knowledge of the population dynamics and their economic activities

are important in order to determine the amount and type of wastewater

generated in the future. Such information in turn helps in selecting

the size and type of wastewater treatment scheme needed in the area.

From 1965, when the Border Industrialization Program (BIP) was

established until 1985, the population growth in the twin cities of

Nogales, Arizona and Nogales, Sonora has respectively increased by 240%

and 500%. At this rate, the population in Nogales, Sonora may grow to

more than half a million while that of Nogales, Arizona may reach about

40,000 by the year 2000. At the same time, if BIP continues to attract

maquiladoras or 'twin-plant' industries (Halpenny, 1966; Dominguez,

1980) then the number and size of industries in the area may also grow

by leaps and bounds. The consequences is that more wastewater will be

generated to affect the selected wastewater treatment plant.

e. Water Quality. Increasing production of wastewater without

an equivalent attention to its management becomes a threat to water

quality and public health in the community. Such a threat has been

eminent in the Ambos Nogales area for sometime now (Alegria, 1978;

Swanson, 1979, 1980; Scot, 1980; Montano, 1981; Greenberg et al., 1982;

Arizona Department of Health Services, 1985). Analyses of samples from

Nogales Wash in 1986 have shown total coliform and fecal coliform counts

of up to 16,000,000 per 100 ml and 9,000,000 per 100 ml respectively.

13

This is in spite of the specific standard for any one sample analyzed in

any one month not to exceed a total coliform group density of more than

2,000 per 100 ml, or a fecal coliform count of more than 4,000 per

100 ml (Mische and Sevilla, 1977; Swanson, 1979). Such a stream

pollution has affected groundwater quality in the area as evidenced by

the fact that some water supply producing wells located close to the

Wash have been found to be contaminated and closed. In addition,

periodic analyses of groundwater samples by the U.S. Geological Survey

from other locations in the area have indicated a continuing rise in

chemical concentrations such as phosphates and nitrates (SEAGO Nitrates

Task Force, 1982). Furthermore, the existing wastewater treatment plant

as discussed below is not capable of treating the wastewater to an

acceptable level (Vega, 1983; Arthur Beard Engineers, Inc. 1984). As a

result, there is a need to improve the existing wastewater transport,

treatment and disposal systems in order to solve the water quality

problem.

2.1.2. Existing Wastewater Treatment System.

In an effort to select an appropriate wastewater treatment

scheme, the existing system is reviewed. This will help to analyze the

weaknesses and strengths of the operating system and design alternatives

accordingly. The review is conveniently presented as follow.

a. Wastewater Treatment Plant Design. The existing wastewater

treatment was designed and constructed in accordance with an agreement

reached by the International Boundary and Water Commission (1967) Minute

No. 227, and approved by both the U.S.A. and Mexican government. The

14

agreement stipulated construction of a treatment facility that enables

complete treatment of 8.2 mgd of wastewater from the twin cities of

Nogales, Arizona and Nogales , Sonora (McNealy et al., 1973). According

to that agreement the Treatment Plant capacity allocation to the cities

was 4.95 mgd to Nogales, Sonora, and the remaining 3.25 mgd to Nogales,

, Arizona.

The plant was so designed by the IBWC with the technical advice

and assistance of John Carollo Engineers of Phoenix, Arizona and

constructed by John Murphy Construction Company of Spring Valley,

California. It consisted of a 5-cell aeration lagoon-stabilization-

ponds-type plant (Figure 2), with each aerated lagoon having four

electric motor driven aerators. A total area of lagoon-system of 83.6

acres consisting of two aerated lagoons each having an area of 6.7

acres, two first stage stabilization ponds with an area of 23.3 acres

each, and one second stage stabilization pond having an area of 23.6

acres was constructed. Chlorination facilities are provided that

includes a chlorine contact basin, and a chlorination building with one

active and a standby chlorinator. Two Parshall measuring flumes, one at

each end of the plant were also installed. A detail design parameter

are provided in Appendix A.

b. Wastewater Flow Rates The Parshall flume at the entrance to

the aerated lagoons measures the total wastewater influent from both

Nogales, Arizona and Nogales, Sonora, while the one located at the end

of the lagoons measures the treated wastewater effluent. A third

Parshall flume located at the International Boundary measures the flow

ARIZONA

INTERNATIONAL BOUNDARY

SONORA

UNITED STATESMEXICO

15

Figure 2. Nogales International Wastewater Treatment System.

16

from Nogales, Sonora. Flow rates for Nogales, Arizona are obtained by

the numerical difference between the total and the Nogales, Sonora

measure. Average daily flows during the period 1973-1985 for the two

sources and the total from the two are drawn in Figure 3 for purposes of

Figure 3. Average daily wastewater flow (1972-1985).

comparison and forecasting future flows. Figure 4 is drawn to do the

same for the plant effluent rate. The straight lines overlying each

curve in both figures are the corresponding linear regression lines for

their respective flow rates. Even though the curves show a pronounced

fluctuation on a monthly basis, they do show a gradual increase on an

annual basis with a correlation coefficient greater than 0.6 for each

Total Effluent

14. ...

IL, t A tilit i I a s I a a t i I g Ilia allit. 1.111 I 1 I I I it

1 17. VOL 443. 115. SI. 27. 112. 122. 145.

TIME IN MONTHS C11172 —. 1025)

Figure 4. Comparison of wastewater treatment plant influent andeffluent rate.

curve. The average increase can be approximated from the slopes of the

linear regression lines. The average total daily flow rate for 1985 was

8.9 mgd, about 9% higher than the design average flow rate of 8.2 mgd.

To forecast future rate of wastewater production one of three

methods can be followed. They are: 1. by regressing historical rate of

flow, that is, extending the linear regression lines in Figure 3 into

the future, 2. by multiplying the expected per capita rate of

wastewater production in each city by the respective projected number of

population in each of the two cities, and 3. by simulating future flows

using parameters derived from historical flow data through maximum

17

18

likelihood or least squares method. Since the second one involves

uncertainties in both the per capita wastewater production and in

estimating the future number of population in the absence of adequate

data particularly in the case of Nogales, Sonora, and the third one

involves lengthy computational procedures, the first method is used in

this study to estimate future wastewater inflow rates. It is also used

to estimate future Treatment Plant effluent rate by extending the

regression lines in Figure 4.

c. Wastewater Influent Composition. Another important

wastewater characteristic is its composition (Everett, 1980).

Wastewater components which are of significance in the selection and

design of wastewater treatment facilities are gross solids, settleable

solids, suspended solids, dissolved oxygen, and oxygen demanding

substances. The concentration of the various chemical constituents are

also required for purposes such as water reclamation or for estimating

effects on certain downstream water uses. The process characteristics

along with the annual performances for the years 1981 and 1983 are shown

in Table 1. The differences of the results from year to year may be due

to the large amounts of infiltration entering the collection system in

recent years (Arthur Beard Engineers, Inc., 1984).

d. Wastewater Treatment Plant Performance and Effluent

Deposition. Performance records presented in Table 1 indicate that

secondary treatment standards established by EPA have not been met. The

annual BOD5 concentration is 34 mg/1 compared to EPA standard criteria

of 30 mg/l. The overall annual average suspended solids concentration

19

Table 1. Annual Average Wastewater Influent and EffluentCharacteristics in the NIWWTP and EPA AcceptableCriteria.

ParameterInfluenta Effluenta EPAb

AcceptableCriteria

1984 1985 1984 1985

Total Influent (mgd) 8.64 8.92 8.2aTotal Effluent (mgd) 9.28 9.07Temperature ( F) 70 68 67 66pH 6.91 6.89 7.83 7.84 6.5 to 9.5Dissolved Oxygen (mg/1) 4.8 5.0 9.3 10.0BOD5 (mg/1) 142 135 34 34 30Suspended Solids (mg/1) 196 167 78 59 30Settleable Solids (m1/1) 6 5 0.3 0.11 0.1Fecal Coliform Countfor Contact (/100m1)

5591 764 200

Total Dissolved Solids

(m9/ 1 )497 532

Chlorine Residue 0.4 0.5 0.530 minutes (mg/1)

Sources: a- Vega (1972-1985) Monthly Reports on NogalesInternational Wastewater Treatment Plant.

h- Arizona State Department of Health Services (1972).

is 65 mg/1 compared to EPA concentration criteria of 30 mg/l. Since the

suspended solids concentration tests includes algae, however, specifying

the actual suspended solid concentration in a lagoon such as the case

under study is difficult.

Table 1 also indicates an annual average pH of 8.9 which falls

well above the median level of the established pH criteria of 6.5 to

9.5. Other effluent quality levels that show low treatment plant

performance are also provided in the Table. Such substandard

performance levels can he partly attributed to the decreased detention

20

time of the wastewater in the lagoons due to the over capacity

wastewater inflow and/or the accumulation of sludge in the lagoons.

Based on design criteria found in Appendix A, the required lagoon

detention time in the Treatment Plant is 13 days during winter and 7

days during summer. This apparently cannot be met when the average flow

rate is above the design level of 8.2 mgd or when the effective design

volume of the Treatment Plant has decreased due to the accumulation of

sludge and grit deposits in the bottom of the aerated lagoons. Such

accumulated sludge deposit occupied approximately 25% of the available

space of the Plant until most of it was removed at the beginning of 1986

(Arthur Beard Engineers, Inc., 1984).

In addition to the hydraulic capacities of the Nogales

Wastewater Treatment Plant being exceeded, more stringent effluent

limitations are being required from the treatment plant. In the mid-

1970's the state of Arizona classified the reach of the Santa Cruz River

immediately downstream of the Treatment Plant as "effluent limited"

which in effect dictates secondary treatment and establishes nutrient

limitations of effluent prior to discharge to the Santa Cruz River in

accordance with 40 CFR Part 120.104, FR, Vol. 41, No. 121, 6/22/76 (John

Carollo Engineers, 1979; Arthur Beard Engineers, Inc., 1984).

To date this reclassification of the Santa Cruz River is not

complied. The Treatment Plant effluent is being dumped into the River

without meeting State and EPA standards creating a threat to downstream

users. Some studies have been made to determine the options for

improving the situation. Lack of commitment and coordination among the

21

decision making bodies, however, see to have hindered any development in

the wastewater management. A review of this problem is the main theme

of the following section.

2.2 Institutional and Economic Considerations.

In this section the institutional structure and economic

conditions behind the wastewater management under study are reviewed in

order to provide further background on the problem and help in selecting

appropriate alternative wastewater treatment schemes. In this respect

the administrative and jurisdictional management and the resources

needed to operate the existing Wastewater Treatment Plant are discussed.

2.2.1. Administration of the Wastewater Treatment Plant.

The construction and operation of the binational wastewater

treatment plant in Nogales, Arizona is considered by some to be an

outstanding example of cooperation along the border between the U.S.A.

and Mexico (McNealy et al., 1973; Bradley and DeCook, 1978; Jamail and

Ullery, 1979). Through a bilateral agreement between the two countries,

arrangement has been made for the wastewater treatment plant to be

staffed and operated by the City of Nogales, Arizona. At present the

staff consists of one operator, one assistant operator, one laborer and

a Plant Manager. Overall administration of plant operation and

maintenance is the responsibility of the Plant Manager. The

International Boundary and Water Commission provides general guidance

22

and supervision as necessary while the Arizona State Department of

Health Services oversees quality control (McNealy et al., 1973).

A major problem in running the treatment plant is that no one

has overall jurisdictional control on it. This can be a serious matter

if toxic materials are released into the plant. Locally, Nogales,

Arizona, has adopted an industrial pre-treatment ordinance, with EPA

approval, which will provide the City with enforcement authority over

Arizona dischargers, including levying substantial penalties for

noncompliance. This can help to ensure non-discharging of toxic wastes

from Arizona users of wastewater treatment system. Nogales, Arizona,

however, does not have any control over its twin city, Nogales, Sonora,

and the possibility of a toxic spill into the sewer system and

eventually into the treatment plant does exist. There has been an

unconfirmed report that the Mexican Federal Government prohibits

discharge of toxic wastes into the public sewers. However, there

appears to be no agency in Mexico to provide enforcement of sanctions

against this problem. Therefore, in order to cope with such problem in

the future, the selected alternative need to have the ability to handle

occasional toxic spills.

2.2.2. Economic Considerations of the Wastewater Treatment Plant.

Construction of the treatment plant was financed jointly by the

International Boundary and Water Commission (U.S.A. and Mexico), the

City of Nogales, Arizona, and the United States Environmental Protection

Agency. Mexico shares the international maintenance and operational

costs, based on the volume of sewage crossing the border from Nogales,

23

Sonora. Most of the 0 and M costs go to energy consumption, labor costs

and cost of chlorine. Monthly costs for these items are provided in the

Nogales International Wastewater Treatment Plant monthly report sheet

prepared by the Treatment Plant manager.

Other resources needs include the area of land upon which the

treatment plant is situated. A total of 143 acres, 83.6 acres of which

are taken by the treatment plant lagoons and ponds, are currently

occupied by the treatment plant. If land application is considered to

be a component of future wastewater treatment schemes, however, the

required size of land will be significant. About 2000-2500 acres of

land will be needed to accomodate irrigation of wastewater in the future

(Arthur Beard Engineers, Inc., 1982, 1984). At present there are only

1,200 acres of grandfathered land which can be irrigated at any time

according to the 1980 Arizona groundwater law. The remaining land, by

necessity, will have to come from conversion of nongrandfathered land

for which legislation may be required.

CHAPTER 3

MULTICRITERION PROBLEM FORMULATION

In order for a problem to be evaluated using a multiobjective-

multicriterion procedure, the problem must be presented in a format

suitable for analysis using a multicriterion decision-making (MCDM)

technique. Accordingly, to formulate the problem of wastewater

management in a multicriterion context the following six steps appear to

be useful (David and Duckstein, 1979; Duckstein and Opricovic, 1980;

Tecle and Fogel, 1986). The steps are:

1. Defining the desired objectives that the system is to fulfill.

2. Identifying the mission requirements or desired specifications from

such objectives.

3. Selecting evaluation criteria that relate system capabilities to

specifications and hence to objectives.

4. Determining measurement scales to describe the range of possible

values (quantitative) or relative position (qualitative) an

alternative system can attain in terms of a particular criterion.

5. Generating alternative schemes for attaining the desired objectives.

6. Formulating an evaluation matrix, an element of which represents a

particular value or relative position of an alternative in terms of

one criterion.

24

25

3.1. Objectives.

An objective can be defined as the direction of change of state

of a system desired by a decision maker(s). In this study, the

objectives indicate the major purposes the wastewater management system

is desired to fulfill. Now, given the environmental conditions, as

described in chapter 2, of the study site in general and the Nogales

International Wastewater Treatment Plant in particular, five objectives

directed at pollution abatement, wastewater treatment, treated

wastewater reuse, system dependability and optimal resource use are

presented. The objectives are verbally described as follows:

1. Prevent Groundwater Pollution: Hundred percent of the water supply

in the Upper Santa Cruz Basin in which the current study site is

located, is groundwater pumped from the underlying aquifer. Thus,

it is necessary to prevent wastewater from intruding into the

aquifer to pollute the areas water resource supply.

2. Meet Required Effluent Quality: In order to maintain a healthy

environment and prevent unnecessary fines, the adopted wastewater

management scheme should comply with local, state and federal

regulations to meet the needed level of effluent quality

requirements.

3. Treated Wastewater Reuse: Attempts should be made to put the treated

wastewater to the best possible uses available. This particularly

is important in an area with limited water resource supply such as

the Upper Santa Cruz River Basin.

26

4. Ensure System Dependability: To be dependable a wastewater

management system should have the capacity to continue to produce

the required effluent quality as much as possible under different

adverse conditions. Dependability in this study is also taken to

include system adjustability to meet new needs and regulations, and

system recoverability from failure.

5. Promote Optimum Resource Utilization: Minimal and efficient use of

physical and socioeconomic resources should be maintainedin

managing the wastewater system.

3.2. Specifications.

For a better interpretation and comprehension by the decision

maker, the wastewater management objectives must be quantified to the

fullest extent possible. A method to achieve this quantification is to

express each objective in terms of a set of specifications.

Specification in this case can be defined as the clarification of the

individual objective by redefining it in terms of detailed measurable

characteristics including applicable constraints such as process

(physical, biological, chemical), resource, legal, institutional, etc

that are inherent in the wastewater management problem. The fulfillment

of the specifications is for the most part essential but not necessary

sufficient for the attainment of the desired objectives of the

wastewater management under consideration. The specifications for each

particular objective in this study are described as follows.

27

3.2.1. Prevent Groundwater Pollution.

In defining the specifications with respect to groundwater

pollution abatement from wastewater intrusion, a distinction is made

between wastewater movement and the effect the intruding wastewater will

have on the quality of available water supply in the aquifer. Each one

of these specification categories is discussed as follows:

a. Pollutant Movement. This category (subobjective) deals with

the protection of the groundwater resource supply from intruding

wastewater-borne contaminants. In this regard the specifications to

prevent groundwater pollution may be seen in terms of wastewater

collection, storage, and disposal of both treated and raw wastewater.

Such specifications include capacity of the sewer system; amount of

leakage from sewer lines, and bottom and sides of the treatment ponds;

treatment pond overflows; method of effluent disposal, and the

characteristics of the soil matrix and underlying geologic formation

through which the wastewater flows in either horizontal or vertical or

both directions.

b. Water Quality. This refers to the quality of the

groundwater supply susceptible to wastewater pollution. This

susceptibility may be a function of the groundwater supply's proximity

to the source of the wastewater and the concentration of pollutants in

it. The main sources of wastewater in the study area are domestic,

commercial and industrial activities, and there is evidence that

wastewater originating in Mexico from such sources is polluting both

28

surface water and groundwater in some parts of Nogales, Arizona

(Swanson, 1979, 1980; Scot, 1980; Montano, 1981; Vega, 1983).

The basic water quality specifications consist of determination

of concentrations of BOD, COD, DO and other known potential

contaminants, pH, and temperature of the wastewater. Other

specification of this category may include wastewater movement and soil

aquifer characteristics.

3.2.2. Satisfy Required Effluent Quality.

The treated wastewater standard should comply with all federal,

state and local effluent quality requirements and receiving water

objectives (U.S. EPA, 1976; Freeman, 1978). The specifications for this

objective includes Federal standards and regulations (PL. 92-500, 33 USC

1251 et seq., 40 CFR; PL 92-500 section 42, 40 CFR Part 122), Arizona

State regulations (ACRR Title 9, chapter 20 and chapter 21; ACRR Title

12, chapter 15); and local regulations and ordinances. Some basic water

quality specifications are listed in the last column of Table 1.

3.2.3. Promote Treated Wastewater Reuse.

In light of the area's limited water resource there is a need to

conserve water by effluent reuse or by effluent reclamation to produce

water suitable for agriculture, industry, recreation or other uses in

accordance with U.S. EPA and Arizona Department of Health Services,

Bureau of Water Quality Control (1978) regulations. By utilizing

treated wastewater for the above uses, water supply previously used for

these purposes would be devoted to other uses or stored for future use.

29

Some of the specifications of this objective include utilization

of treated wastewater for agriculture, recreation, industrial, and other

purposes. Other specifications may include effluent quality standards

for various uses, and the economic and other benefits accrued from such

reuses.

3.2.4. Ensure System Dependability.

Webster's Third New International Dictionary defines dependable

as trustworthy, a thing that can be relied on as in a need or emergency

and often connotes steadiness. Dependability in this study is,

therefore taken to include the level of holistic (i.e., all-embracing)

trust one can have on a particular wastewater management scheme's

capacity to perform to his/her full satisfaction.

In defining the specifications with respect to this objective, a

distinction is made between reliability and resilience. These two

specification categories represent different aspects of system

dependability.

a. Reliability. Reliability can be defined either as the freedom

from failure of a component or system equipment while maintaining a

specific performance (structural reliability) or as a measure of

dependability or trustworthiness of a system in accomplishing a certain

mission for a particular period of time (target reliability) (Frankel,

1984; Duckstein et al.,1987). To define it quantitatively for the case

30

under study, let F represent a failure incident of all or part of the

wastewater treatment system such that

4 1 if the system failsF=

0 otherwise

then reliability can be described as an estimate of the relative

frequency in which the wastewater treatment system is not in mode F

during an operational period of T for t=0,1,2,...,T. To be precise, the

reliability of the wastewater treatment system with respect to the

failure mode F during the operational period T is defined (Plate , 1984;

Moy et al., 1986; Duckstein et al., 1987) as

R(T,F) = (T+1- E F)/(T+1)

(1 )

t=1

In this case, the specifications for wastewater management with

respect to reliability can include structural design, system operation,

maintenance, manpower availability, operational cost, wastewater flow

rate, influent type and pollutant concentration, and climatic

conditions. Such components should be carefully considered in order to

fully evaluate the reliability of the wastewater management schemes under

study.

b. Resilience. In addition to functional dependability on a

system to perform as desired, one would also wish for a system to be

dependable on its ability to bounce back after mishaps. This phenomenon

is resilience, and it describes how quickly a system is likely to

recover to normalcy from failure once failure has occurred (Hashimoto et

al., 1982). Resilience can also be taken to mean the condition by which

31

a system adapts to disturbances by moving to an equilibrium point

different from its initial point (Salis and Duckstein, 1983). In

wastewater treatment operation such a phenomena may take place when

operation returns to normalcy after mishaps such as power interruptions,

accidental chemical spills, equipment breakdown, and inclement weather.

Thus, the importance of resilience in the system is clear. Another

characteristic which can be very important in wastewater management

operation is system flexibility. This enables system amenability to

revision of existing operating conditions in order to accomodate

changing patterns of urban development without undue financial loss or

compromise of performance. This may be like system capacity to benefit

from a developing technology.

Some of the specifications with respect to resilience may

include system flexibility, operation and maintenance cost, design

criteria such as lagoon dimensions and sewer capacity, wastewater

volumes and pollutant concentrations, etc. All these specific

characteristics and some others are important to enhance system

resilience. When a system is flexible, for example, an increase in

wastewater loading from rapid urbanization may have only a minimal

effect on its operational capacity. Such a system can adjust to the new

condition thereby reaching an equilibrium situation different from the

original one and stabilize without much problem.

3.2.5. Optimize Resource Utilization.

Optimal utilization of resources should be an important aspect

of any project planning in order to avoid waste and unnecessary

32

expenditures. This is more so in the case of wastewater management in

which the incentive of direct profit to individual is practically

minimum if not nonexistent. In any case, optimal utilization of the

resources in any project management requires complete understanding of

the individual resource's absolute and relative contributions to the

project. Towards this end, the objective under discussion is

categorized into monetary and non-monetary groups of specifications.

a. Monetary Resource Requirements. This includes the

specifications of the project costs, and operation and maintenance

costs. The first case consists of construction cost and other first

costs such as engineering, inspection, legal and administrative,

contingency, and land and easements. The 0 and M costs, on the other

hand, are the estimated annual operation and maintenance costs prepared

using standard guidelines (Gumerman et al., 1979), and past 0 and M cost

records, along with some inherent assumptions and limitations. In this

respect, the system not only needs to be operationally effective but

must also incur reasonable monetary cost.

Other specifications that should be considered in this category

are the hidden costs. These consist of both internal and external

diseconomies which are usually inherent in any wastewater management

system. Internal diseconomies can be thought of in terms of benefits

foregone by choosing one management scheme over another, while external

diseconomies are the adverse effects of the management on groups outside

the area of concern. These are groups that have nothing to do with the

original problem or its solution. However, since these kinds of costs

33

also constitute a part of the overall costs to society, consideration

must be made to avoid or at least minimize them.

b. Non-monetary Resource Needs. The specifications in this

category include resources such as land, energy, water and manpower.

These resources are usually essential components of wastewater

management systems and their scarcity in many instances is an indication

for the need to conserve and use them efficiently. This is particularly

important in an international wastewater management system such as the

case study under consideration in which resources utilized are supposed

to be shared between the two countries.

3.3. Criteria.

An essential component of any multicriterion evaluation concerns

the criteria by which an evaluation is performed. The notion of

criterion can be defined as a measurable aspect of judgment by which a

dimension of the various choice possibilities under consideration can be

characterized (Voogd, 1983; Duckstein et al., 1986). For the Nogales

International Wastewater management plan a number of criteria are

deductively generated to measure the degree to which each of the

alternative systems meet the specifications. One or more criteria are

provided for each specification. Conversely, a common criterion, such

as cost, may be related to several specifications (Figure 5). These

criteria, in many cases, are non-commensurable as they cannot be

expressed in common units. The criteria used in this study are:

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34

35

1. Vulnerability to pollution: This refers to the degree of likelihood

the area's water resources system would get polluted under the different

wastewater management scenarios. The criterion values are subjectively

determined ordinal points.

2. Water quality: This criterion describes the impact each management

alternative would have upon existing water quality level. It relates to

actual level of pollution rather than to its possible future happening

as in criterion one above. The quality of water, in this case, is

measured against standards set (State of Arizona, Water Quality

Standards) for various uses. Since quantitative data are not available,

criterion measures are again assigned ordinal values.

3. Level of treatment: The final level of treated wastewater effluent

is evaluated by averaging various quality parameters such as BUD, DO, SS

and pH. As a result, the criterion values are provided as standardized

cardinal points for each alternative.

4. Effluent use: This refers to the total amount of treated effluent

used for various purposes such as irrigation of crops and golf courses,

aquaculture, industrial cooling or process water for mineral and mining

operations, recreation and land beautification. There is no data

available for the amounts actually usable for various purposes, as a

result ordinal points are provided for the future relative use.

5. Reliability: Reliability is both verbally and quantitatively described

in section 3.2.4.a. To summarize those definitions, the reliability of

a wastewater treatment system, R, in equation (1) can be said to

represent the frequency or probability that the wastewater treatment

36

system remains in a satisfactory condition. To evaluate the treatment

alternatives for reliability the following factors will be considered:

equipment failure, hydraulic and organic shock loadings, power

interruptions, inclement weather and accidental-chemical spills (Arthur

Beard Engineers, Inc., 1984).

6. Compatibility: The alternative treatment schemes are evaluated with

respect to their compatibility with local and regional plans for

wastewater management systems, including wastewater reclamation.

Ordinal points (scales) are assigned to each alternative scheme based on

its degree of ability to get along with other related regional or local

plans.

7. Resilience: This describes a system's ability to recover after

failure. High resilience value implies higher system capability to

bounce back to normalcy or reach a new equilibrium point after failure.

The relative criterion value in this case is expressed on an ordinal

scale.

8. Flexibility: This measures the ability of the alternative treatment

systems to accomodate to new management plans with minimal financial

loss or compromise of performance. Its value is expressed in ordinal

terms.

9. Capital cost: Project costs for each alternative treatment system

are evaluated to compare their relative impact on selecting the most

preferred alternative(s).

37

10. Operation and maintenance cost: The same is done as in criterion 9

with the annual 0 and M cost. The criterion values for both costs are

expressed in dollars.

11. Land: The area near the present site of the treatment plant has

been developed for various uses. The plant's location in a narrow

valley by the confluence of the Santa Cruz River and Potrero Creek

(Figure 1) has a limiting effect on available land. In light of this

situation, alternative systems will be evaluated according to their

individual land occupancy. Since there are no precise measurements

available of the amount of land needed by every alternative scheme, the

criterion is assigned an ordinal scale based on its expected relative

land requirement.

12. Manpower: Alternatives will be assessed according to the number of

personnel and level of expertise required to operate them. The less

complex an operating system is the lower demand for manpower it has.

Lack of precise information, however, makes it more convenient to assign

an ordinal value to this criterion.

3.4. Criterion Scores.

In the next section, a number of wastewater treatment techniques

are presented as alternative action plans from which the most

satisficing alternative plan is to be chosen using a number of

multicriterion evaluation techniques. These techniques employ criterion

scores that reflect the degree to which an alternative plan meets a

certain criterion. Determination of scores is made using a number of

indicators or indices which are based on the best available information.

38

By indicator is meant the measuring rod by which the effect of

alternative treatment scheme on a desired objective with respect to a

particular criterion is determined. In this thesis, the results of

previous studies on the problem will be used as indicators to determine

the criterion scores that constitute the individual elements of the

evaluation matrix in section 3.6.

In determining the criterion scores both quantitative and

qualitative measurement scales are considered. Different types of such

measurement scales are illustrated in Table 2.

Table 2. Types of Measurement Scales (after Voogd, 1983, p.75)

MeasurementUnit

Origin Order(rank)

1T.: >)

= ro

Ratio ScaleInterval Scale

KnownKnown

KnownUnknown

KnownKnown

• w-- >= m'74-)

Ordinal ScaleBinary Scale

Nominal Scale

UnknownUnknown

Unknown

UnknownUnknown

Unknown

KnownCan be partly

knownUnknown

When the measurement unit is known such as in 0 and M cost, a

quantitative scale, cardinal or ratio scale is used, but when the

measurement unit is not known, only an ordinal scale can be used.

Table 3 which represents a summary of the objectives, specifications,

criteria and criterion scales discussed in this chapter shows in its

last column the types of measurement units used in each criterion.

39

Table 3. Objectives, Specifications, Criteria and Criterion Scales.

Objectives Specifications Criteria Criterion Scales

GroundwaterPollutant Movement Vulnerability

to PollutionOrdinal (A to G)

Protection Water Quality Water Quality Ordinal (A to G)

Effluent Effluent Level of Treatment Ratio [0,1]Quality Quality Level Achieved

Was Effluent Used Amount of Effluent Ordinal (A to G)Reuse Used

Reliability Reliability Ratio [0,1]System Compatibility Ordinal (A to G)

Dependability Resilience Resilience Ordinal (A to G)Flexibility Ordinal (A to G)

Monetary Cost Capital Cost $/1000 gal/dayResources 0 and M Cost $/1000 gal/dayNeeded Non-Monetary Land Ordinal (A to G)

Resource Need Manpower Ordinal (A to G)

Note: Ordinal Implies a 7-point qualitative scale: A=best and G=worst.

These measurement units are assessed in a 7-point scale to allow a wide

latitude of preference space or range to the DM (Osgood, Suci and

Tannenboum, 1957; Davidson and Farquhar, 1976; Saaty, 1977; Voogd,

1983).

3.5. Generation of Alternatives.

The set of objectives of wastewater management can be approached

using several alternative wastewater treatment options. Based on the

amount and type of wastewater generated, the level of treatment

required, and monetary and other resources availability, a number of

40

wastewater treatment alternative schemes can be generated. The

alternative schemes selected in this study have, for the most part, been

found to be appropriate for the particular problem under consideration

(Arthur Beard Engineers, Inc., 1982, 1984). The alternatives take into

consideration the optimum use of the existing facilities and site. The

generated alternatives are composed of three groups of alternative

schemes: 1. pure actions, 2. supplementary actions, and 3. action

mixes.

3.5.1. Pure Action Alternatives.

This group includes 3 wastewater treatment alternative

techniques each of which is capable of treating 44,000 m 3/day (11.5 gpd)

of wastewater with an effluent BOD 5 of 30 mg/l. However, none of them

has nutrient removal capabilities. These alternatives are:

a. Facultative lagoons. In this technique, wastewater is

stabilized from artificially accelerated transfer of oxygen from the air

to the wastewater using aerators. The contents in the aerated lagoons

are not completely mixed. Most of the solids settle to the bottom of

the lagoon, a portion of which will undergo anaerobic decomposition.

The facultative lagoons are followed by a polishing pond where the

remaining solids are allowed to settle out and decompose further under

anaerobic condition. Thus the effluent from this type of treatment is

highly stabilized. The facultative lagoon alternative is essentially

the method of treatment currently in use at the existing wastewater

treatment plant (Arthur Beard Engineers, Inc, 1984).

41

b. Aerobic Lagoons. This method is designed as a completely

mixed process, with all solids maintained in suspension. The entire

lagoon is kept in an aerobic condition with the help of mechanical

aerators. The aerators must supply both sufficient oxygen for bio-

oxidation and sufficient power to mix the lagoon contents. As

facultative lagoons, aerobic lagoons are followed by polishing ponds

where the solids are separated from liquids and allowed to settle out

and decompose further anaerobically.

c. Oxidation Ditches. This is a modification of the

conventional activated sludge process (Brass, 1962). It is considered

as an alternative system because it would enable upgrading the plant to

higher flows in the future and provide considerably higher quality

effluent than alternatives 1 or 2 above. This system is flexible enough

to accomodate any future changes in ADHS/EPA requirements (Arthur Beard

Engineers, Inc., 1984). The oxidation ditch treatment plant combines

physical and biological processes to stabilize wastewater. In this

system only a small portion of the organic matter undergoes chemical

oxidation while the bulk of the organic matter is stabilized by the

biochemical activities of the micro-organisms.

3.5.2. Supplementary Alternative Activities.

Both facultative lagoons and aerobic lagoons described above

produce algae as a basic part of the processes. This makes the two

alternatives to be incapable of reliably producing an effluent suitable

for discharge into the Santa Cruz River (Arthur Beard Engineers, Inc.,

1984). In addition, all three alternatives need additional nutrient

42

removal facilities in order to have tertiary level of treatment

capacity. As a result, both algae removal and nutrient removal

facilities are referred to as supplementary processes that may be

included with any one of the pure alternative actions to make the

function of the latter complete. Four methods of supplementary

alternative activities are considered in this study. They are: 1.

chemical addition with sedimentation, 2. rapid sand filtration; 3.

nutrient removal through chemical addition, and 4. land application.

a. Algae removal 121 chemical addition with sedimentation. In

this technique, the lagoon effluent is processed first by coagulation

and flocculation with alum [Al 2(SO4)3.16H20] or other chemicals (Pavoni

et al., 1977) followed by sedimentation. Coagulation and flocculation

are mechanisms by which the dispersed particles are made to agglomerate

into larger particles. Sedimentation, on the other hand, is a solid-

liquid separation process. The separated solids are eventually

deposited in the lagoon bottom. The clarified effluent, on the other

hand, is discharged into the chlorine contact tank, where chlorination

takes place before it is released out of the treatment plant.

b. Rapid Sand Infiltration. In this method, algae are removed

from the lagoon effluent by filtering through a sand filter. A sand

filter is basically a basin of graded and selected sands through which

algae filtration takes place. In this process algae are trapped on the

top of the sand as the effluent is applied to the filter, thereby

removing them from the liquid stream. In order to reduce filter

clogging problems by the algae, the filter is backwashed using filtered

43

effluent, air scouring being used to assist the separation of sludge

from sand (Mara, 1976; Pavoni et al., 1977, Arthur Beard Engineers, Inc.

1984).

c. Nutrient Removal. The presence of nitrogen and phosphorus

compounds in an effluent causes pollution in a receiving waterway.

Phosphates and nitrates stimulate eutrophication. Nitrates and nitrites

may cause health hazard. In addition ammonia has high oxygen demand,

interferes with chlorination and can also be toxic to aquatic life.

Furthermore, phosphates may also interfere with coagulation processes

used in water treatment (Winkler et al., 1981).

Due to the aerobic conditions and the relatively long detention

times associated with the aerated lagoon process, only partial

nitrification can be expected in the effluent. Phosphorous removal with

the aerated lagoon process, however, is insignificant (Arthur Beard

Engineers, Inc., 1984). As a result, the lagoon type systems will

require nutrient removal facilities. Various types of nutrient removal

methods are known. Chemical treatment at the discharge point of the

aerated lagoons and polishing ponds are commonly done to remove

nutrients. Alum as Al 2(SO4)3, ferric salts and lime as Ca(OH) 2 are the

most common chemical compounds used for the purpose (Winkler et al.,

1981; Arthuer Beard Engineers, Inc., 1984).

c. Land Application of Wastewater. Land treatment of wastewater

implies that the land or soil is used as a medium to treat the

wastewater. Roughly defined, land application is any technique which

utilizes the interactions between natural soil and vegetation and

44

wastewater to upgrade the quality of the wastewater (Pavoni et al.,

1977; Fuller and Warrick, 1985). There are three main groups of land

application methods: 1. irrigation, 2. overland flow, and 3.

infiltration-percolation.

Irrigation may be defined as the application of wastewater on

land to sustain plant growth. During irrigation, wastewater is recycled

completely by the land through either evaporation, transpiration, or

incorporation into plants, or percolation into the subsoil. In the

overland flow method, wastewater is sprayed into vegetation on gently

sloping ground. Biological treatment occurs as the wastewater contacts

the biota in the ground cover vegetation and the soil. After

transmission losses the wastewater is collected in ditches. In

contrast, the infiltration-percolation technique is based on wastewater

application to very permeable soils, with the purpose of getting the

water percolate through the soil and enter the groundwater. In this

study only the first group of techniques, i.e. irrigation, will be

considered.

With agricultural irrigation, nitrogen is removed by crop up-

take and harvest, and by denitrification. Even though, nitrogen removal

by crop up-take is dependent on the type of crop, it is in the order of

65% while denitrification through the soil averages about 20% making a

total nitrogen removal of 85% to 90%. Phosphate removal with

irrigation, on the other hand, is primarily through the chemical

absorption process of the soil, with plants removing less than 20%. The

total amount of phosphate removal is about 95% (Arthur Beard Engineers,

45

Inc., 1982). In addition, the removal of viruses and pathogenic

bacteria has bean found to be 99% in similar projects in which effluents

are not disinfected prior to land application. Therefore, with effluent

disinfection prior to land application, comparable or better results

than without disinfection can be expected.

3.5.3. Action Mix Alternatives.

As pointed out in section 3.5.2. above, more preferred

wastewater treatment techniques can be obtained by combining a pure

alternative treatment scheme with one or two of the supplementary

treatment activities. Twelve such action mixes are selected to make the

total number of treatment alternatives considered in this study to be

fifteen as shown in Table 4 and Figure 5. Using these alternative

schemes and the twelve criteria described previously in section 3.4, an

evaluation matrix is constructed as the last step of formulating the

wastewater management problem in a MCDM format.

3.6. Evaluation Matrix.

The last step of formulating a wastewater management planning as

a multicriterion decision making problem consists of constructing an

evaluation matrix. This matrix is made up of the alternative systems

versus criteria array of Table 5. The elements of the evaluation matrix

represent criterion values for the wastewater management alternatives.

The different possible types of the criterion values and how they can be

determined were presented in section 3.4, while the measurement units in

each criterion were stated in section 3.3.

Table 4. Considered Alternatives

Code Description

Al facultative lagoonsA2 aerobic lagoonsA3 oxidation ditchesA4 Al+chemical algae removalA5 A2+chemical algae removalA6 Al+filtration algae removalA7 A2+filtration algae removalA8 A4+nutrient removalA9 A5+nutrient removalA10 A6+nutrient removalAll A7+nutrient removalAl2 A3+nutrient removalAl3 Al+land applicationA14 A2+1and applicationA15 A3+1and application

At this stage, the source of the actual criterion score each

alternative wastewater management scheme received is considered. The

major sources of data are a couple of previous studies on the Nogales

Wastewater treatment plant (Arthur Beard Engineers, Inc., 1982, 1984).

The annualized capital, and operation and maintenance costs for each

treatment alternatives are taken from these studies. Level of

treatment, reliability, compatibility, resilience, flexibility, land

manpower, and effluent use values, for the main part, are also

approximated from the above studies. For illustrative purposes, the

estimated capital, and annual operation and maintenance costs of each

alternative treatment system in 1988 dollars are provided in the Table

of Appendix B. The last two columns are the capital, and 0 and M

46

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48

costs entered in Table 5. They are respectively calculated from the

second and third columns of the Table in the Appendix using the equation

C = (1000xACx(1+r) 4 )/(365x11.5x10 6 ) (2)

in which C stands for the treatment plant improvement or construction

capital cost or annual 0 and M cost in 1988 dollars, AC is a 1984

capital cost or annual 0 and M cost in 1984 dollars as estimated by

Arthur Beard Engineers, Inc. (1984), while r is the average annual

inflation rate taken to be 4%. The numbers 4, 1000, 365, and 11.5x10 6

are respectively the number of years between 1984 and 1988, gallons,

number of days in a year, and the capacity of the future treatment

plant.

The scales or values for the other criteria can also be

similarly illustrated. Arthur Beard Engineers, Inc. (1984), in

particular, discusses the effect of every wastewater treatment

alternative considered on these criteria. Additional pieces of

information on these criteria were also obtained from other sources

(Sullivan et al., 1973; Thomas and Harlin Jr., 1969; Giravolo et al.,

1977; Mische and Sevilla, 1977; Fuller and Warrick, 1985) which show

criterion values for some of the alternatives in different places at

different times. The remaining two criteria, vulnerability to pollution

and water quality indices were not specifically discussed in either of

the above publications. Their ordinal values are, therefore, assessed

subjectively from area field works, experiences on similar projects

elsewhere (Bard and Krutchoff, 1974; Giravolo et al., 1977; Reynolds et

49

al., 1979; Magette et al., 1983), the study of the water quality

parameters and the major types of wastewater generated in the area

(Gloyna, 1971; Goldstein and Moberg Jr., 1973; Mara, 1976; Arceivala,

1981; Montano, 1981; Winkler, 1981; Vega, 1983).

Having determined the rating for every alternative with respect

to every criterion, the evaluation matrix given in Table 5 is

constructed in order to be able to apply multicriterion decision-making

techniques.

CHAPTER 4

MULTICRITERION DECISION-MAKING TECHNIQUES

It was stated above that one of the major objectives of this

research is comparative application of multicriterion decision-making

(MCDM) techniques to several wastewater management alternatives. For

this purpose, six MCDM techniques are provided. These techniques are

ELECTRE I, ELECTRE II, MCQA I, MCQA II, compromise programming and

cooperative game theory. These techniques can be categorized into two

groups: 1. outranking type and 2. distance-based techniques.

4.1. Outranking Types.

ELECTRE I, ELECTRE II, MCQA I and MCQA II belong to the

outranking type of techniques. These techniques can be used to rank a

discrete set of alternatives with respect to a set of criteria (Gershon,

1981). This is accomplished through pairwise (ELECTRE) or global (MCQA)

comaprisons among-members of the set of alternatives. The end products

in these methods are either a partial or complete ordering of the

nondominated alternatives. The partial or complete ordering of the

systems depends on the particular technique used.

4.1.1. ELECTRE I.

ELECTRE I which stands for elimination and (et) choice

translating reality, was initially developed by Benayoun, Roy and

Sussmnan (1966) and improved by Roy (1971). A fundamental feature of

50

51

ELECTRE I is the use of pairwise comparisons among members of a set of

discrete alternative systems in order to eliminate a subset of less

desirable alternatives while choosing those systems which are preferred

for most of the criteria without causing an unacceptable level of

discontent for any one criterion (Nijkamp and Delft, 1977; Gershon et

al., 1982). Therefore, both elimination and choice are essential

ingredients of ELECTRE I. The methodology involves three important

concepts: concordance, discordance, and threshold values.

The concordance between any two alternative actions m and n is a

weighted measure of the number of criteria for which action in is weakly

preferred to action n (mPn or mEn, that is, action in is preferred to or

equivalent to action n). Now, if we let wi, represent the

weight given a priori to criterion i by the decision maker for use in a

specific algorithm (Table 6), then the concordance (or concord) index

between actions in and n can be determined as follows:

C(m,n) = z wi / ( z wi) (3)icS(m,n) i=1

where S(m,n) [iimPnUmEn], that is the set of all criteria for which in

is preferred to n or equal to n. The weights, w i which are elicited

from the decision maker, reflect his/her preference structure. The

concordance matrix, C(m,n) can be thought of as representing the

weighted percentage of all criteria for which one action is weakly

preferred to another. By definition, 0 < C(m,n) < 1 (Gershon et al.,

1982).

52

Table 6. System Parameters.

CriteriaBest

ValuesWorstValues

Weight Sets

1 2 3 4

1 150.00 75.00 5 7 2 12 140.00 80.00 6 5 4 13 210.00 60.00 7 7 3 14 119.00 34.00 6 4 4 15 150.00 75.00 7 6 1 16 84.00 36.00 4 3 4 17 120.00 40.00 5 6 2 18 105.00 30.00 3 2 1 19 175.00 25.00 5 7 7 1

10 100.00 40.00 4 6 5 111 90.00 30.00 3 2 3 112 84.00 24.00 2 6 2 1

Discordance is complementary to the concordance concept.

Accordingly, discordance represents the maximum discomfort one

experiences when confronted with criteria for which alternative m is not

preferred to alternative n. To compute the discordance matrix in this

study, each criterion is assigned a different range of scale, the upper

value of which ranging from 84 to 210. In other cases, an interval

scale common to each criterion but with each criterion having different

levels can be defined (Szidarovszky et al., 1986). In any case, the

interval scale is subjectively determined to represent the degree of

dissatisfaction the DM may experience in moving from one scale point to

the next less desirable scale point in one criterion compared with a

similar operation on another criterion. A seven-point scale as shown in

Table 7 is selected for each criterion in order to enable the widest

Table 7. Point Value Used in Discordance Matrix Determination.

Criteria Levels Value Criteria Levels Value

1 G 25 7 G 20F 50 F 40E 75 E 60D 100 D 80C 125 C 100B 150 B 120A 175 A 140

2 G 20 8 G 15F 40 F 30E 60 E 45D 80 D 60C 100 C 75B 120 B 90A 140 A 105

3 0.3-0.4 30 9 .50-.75 1750.4-0.5 60 .75-1.00 1500.5-0.6 90 1.00-1.25 1250.6-0.7 120 1.25-1.50 1000.7-0.8 150 1.50-1.75 750.8-0.9 180 1.75-2.00 500.9-1.0 210 2.00-2.25 25

4 G 17 10 0.00-0.10 140F 34 0.10-0.20 120E 51 0.20-0.30 100D 68 0.30-0.40 80C 85 0.40-0.50 60B 102 0.50-0.60 40A 119 0.60-0.70 20

5 0.2-0.3 25 11 G 150.3-0.4 50 F 300.4-0.5 75 E 450.5-0.6 100 D 600.6-0.7 125 C 750.7-0.8 150 B 900.8-0.9 175 A 105

6 G 12 12 G 12F 24 F 24E 36 E 36D 48 D 48C 60 C 60B 72 B 72A 84 A 84

53

54

possible levels of discomfort the DM can assign in moving from the best

to the worst levels of each criterion (Gershon et al., 1982). With this

understanding, the discordance index, D(m,n), can be defined as:

D(m,n) = (Max(f(n,i)-f(m,i)) / R* , for i=1,2,...,N (4)

where f(n,i) is the evaluation of alternative n with respect to

criterion i, and R* is the largest of the N criterion scales. Thus the

normalized discordance interval is calculated for each criterion where

alternative n is preferred to alternative m, and the largest normalized

discordance interval of these criteria is defined as the discordance

coefficient for alternatives in and n (Goicoechea et al., 1982;

Szidarovszky et al., 1986).

To synthesize both the concordance and discordance matrices and

determine an outranking relationship among the nondominated

alternatives, threshold values (p,q), both between 0 and 1, are selected

by the decision maker. P specifies the minimum limit of concordance

level whereas q defines the maximum level of discordance the decision

maker is willing to accept. A value of p=1 represents full concordance

while a value of q=0 indicates no discordance at all. The result of

ELECTRE I is an outranking relationship that provides a partial ordering

of the alternative systems. Such results of the problem under

consideration in this research will be presented in section 5.1.1.

4.1.2. ELECTRE II.

ELECTRE II is an extension of ELECTRE I which may provide a

complete ordering of the nondominated set of alternatives (Jacquet-

Lagreze, 1974; Goicoechea et al., 1982; Duckstein and Gershon, 1982;

Duckstein et al., 1984; Szidarovszky et al., 1986). This ordering is

accomplished by the construction of outranking relationships based on

the DM's preference structure.

As in the case of ELECTRE I, alternative m is preferred to

alternative n (i.e. m outranks n) if and only if the required

concordance and discordance conditions are both satisfied. According to

Duckstein et al. (1983), and Szidarovszky et al. (1986), these necessary

conditions are the fulfillment of: 1. the test of concordance, so as to

be above the minimum level of acceptability of an alternative; and 2.

the test of non-discordance, so as to be below the upper limit of non-

acceptability of an alternative, for every criterion i. The concordance

space is determined, as in ELECTRE I, using equation (3). The

discordance space Di associated with criterion i, on the other hand, can

be figured out using equation (5).

(f(m,n),f(n,i))cD i <==> f(n,i)-f(m,i) > q (5)

where the terms f(m,i) and f(n,i) are as described in section 4.1.1.,

and 0 < q < 1 is a threshold value restricting the discordance space.

Therefore, in order for m to be preferred to n, (f(m,i),f(n.i)fDi); that

is the converse to equation (5) must be true.

a. Ranking relationships. Unlike in the case of ELECTRE I,

multiple levels of concordance and discordance conditions are specified

in ELECTRE II in order to construct two different outranking

relationships, that is, a strong relationship, R s , and a weak

55

56

relationship, Rw , from which a strong graph, G s , and a weak graph Gw ,

respectively are constructed. The strong preference graph results from

the use of stringent threshold values (p,q) that is, the DM selects a

high level of concordance and a low level of discordance. For the weak

preference graph, the DM relaxes his/her threshold values (lower p,

higher q). The relaxed threshold values represent lower bounds on

system performance that the DM is willing to accept (Gershon et al.,

1982).

The ranking relationships are determined using the multiple

concordance and discordance levels provided. For the concordance

space, these levels are given as p* 0, p, p - in decreasing order to

represent, say, 'high', 'average', and 'low' levels, such that

1 > p* > p

0 > p- > 0.5. Likewise, the discordance space can be

* 0indicated as D i and D i to stand for 'low' and 'average' conditions,

* 0which are respectively controlled by the parameters q i and qi in

equation (5) (Duckstein et al., 1983). Any pair of system (m,n) can

thus belong to three discordance spaces:

0 01. f(n,i)-f(m,i) < qi : low discordance (f(m,i),f(n,i))1Di

0 *2. qi < f(n,i)-f(m,i) < q i : average discordance

(f(m,i),f(n,i))cD; andf 4

*3. qi < f(n,i)-f(m,i) : high discordance (unacceptable)

(f(m,i),f(n,i))cD.1

57

Now, given the above relationships, the strong and weak outrankings are

defined as follows:

1. m strongly outranks n if:

mR sn <==> ( E 4 > E wj )fi (C(m,n) > pES 1 (m,n) itS2 (m,n)

and (f(m,i),f(n,i))eD; andi

or

mR sn <==> ( E > z wj )fi(C(m,n) >icS i (m,n) icS2 (m,n)

and (f(m,i),f(n,i))1D;)

where S i (m,n)-(iimPn} and S2 (m,n)=IiimEn} and all other terms are as

described above (Goicoechea et al.,1982; Duckstein et al., 1983).

2. m weakly outranks n if:

taR w n <==> ( E 4 > E VIT )11 (C(m,n) > p-icSi (m,n) icS2 (m,n)

and (f(m,i,f(n,i))ED;)

or

mRw n <==> ( E wI > E w.T )(1 (C(m,n) picS i (m,n) icS2 (m,n)

and (f(m,i),f(n,i))eD; and fiD*i

o

The possible outranking relationships are summarized in Table 8

(Duckstein et al., 1983; Szidarovszky et al., 1986).

The two binary relations determined, in 1. and 2. are

respectively the R s and R w described above from which the corresponding

acyclic graphs, G s and G w are constructed. Using these graphs, the

(9)

(10)

(12)

Table 8. Type of Outranking as a Function of Concordance andDiscordance Levels.

concordance level

high average low

lowdiscordance

strong(equation(10))

strong(equation(10))

weak(equation(11))

averagediscordance

strong(equation(9))

weak(equation(12))

alternatives are ranked with respect to the criteria provided. The

ranking procedure follows a certain sequence of steps (Goicoechea et

al., 1982; Szidarovszky et al., 1986).

b. Ranking procedure. The complete ranking procedure follows

three consecutive steps: forward ranking, reverse ranking and average

ranking (Duckstein and Gershon, 1982; Duckstein et al., 1983).

(i). Forward ranking: In this stage a subgraph of G s (the set of all

alternative systems) is selected and denoted as Y(k). The set of

preferred alternatives, A(k), is chosen from Y(k) and the forward

ranking (v') is obtained by using the following steps:

1. Start with k=1 and Y(1)=Y (the nondominated set of alternatives in

Gs ).

2. Select all nodes of Y(k) not having a precedent (that is, the

alternatives not outranked by others) and denote this by C(k).

3. Next, use Gw (the graph of weak outranking) to remove as many ties

58

59

as possible between systems in C(k). To do this identify all nodes in

C(k) that are joined by an arc in Gw and represent these nodes by U.

4. Select all nodes in U not having a precedent in Gw. Denote this set

as B.

5. Define A(k) as A(k) = (C(k)-U)U B

where C(k)-U ={xixe C(k),xcUI

6. Rank every alternative x CA(k) by setting v'(x) = k.

7. Identify Y(k+1)=Y(k)-A(k) and delete all arcs emanating from A(k).

This eliminates alternatives that have been ranked from repeated

consideration in the forward ranking process.

8. If Y(k+1) is an empty set, then all the representative elements in

the reduced graph of R s have been ranked. If Y(k+1) is not empty set,

then set k=k+1 and go to step 2 above (Goicoechea et al., 1982;

Duckstein et al., 1983).

(ii). Reverse ranking: This procedure embodies the above process and

consists of three steps:

1. Reversing the directions of the arcs in Gs and G.

2. Determining a ranking, a(x), for each alternative x in the same way

as was done in the forward ranking, but replacing a(x) for v'(x) in

step 6.

3. Re-establishing the correct ranking order using the relationship:

v"(x) = 1 'I- amax a(x), Vxa

(13)

where X is the set of all nondominated alternatives and

amax = Maxxexa(x).

60

(iii). Average ranking: This is the final ranking, V obtained from v'

and v" using the ranking function (Duckstein et al., 1983; Szidarovszky

et al., 1986):

V(x) = (v'(x) + v"(x))/2 + 0.5 (14)

where 17, v' and v" are integers.

The ranking procedure for the alternative wastewater management systems

is similar to that of Duckstein and Gershon (1983).

4.1.3. Multicriterion g-analysis I.

Like ELECTRE, multicriterion Q-analysis is a technique for

modeling discrete multi objective problems from the viewpoint of multiple

criteria. The criteria can be non-commensurable, quantitative or

qualitative in scale. This shows that MCQA is convenient for analyzing

the evaluation matrix of Table 5.

a. Payoff and preference matrices. In multicriterion Q-

analysis, the elements of the evaluation matrix, can be defined to

represent the relations between finite sets, the set of alternatives,

X =.(x(i)li=1,....,I1 and the set of criteria, Y =-ry(i)ii=1,---,J}

(Atkin, 1974; Johnson, 1981). In order to make the evaluation matrix

easier to map into a preference matrix, all the entries on the

evaluation matrix are transformed into dimensionless quantitative values

making the payoff matrix of Table 9. The allocation of these values is

based on the range of scales selected for each criterion (Table 6). To

express it algebraically the payoff matrix may be represented as

D =-[d(i,j)1i=1,...,I;j=1,...,J1 (15)

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61

62

where D is the payoff matrix, d(i,j) is the element of the matrix, I is

the number of alternatives and J is the number of criteria used. The

payoff matrix, D is, then, mapped into a preference matrix, U(D) having

values in the interval [0,1] using a standardization technique (Voogd,

1983) that can be expressed as:

U(d(i,j))=(d(i,j)-min(d(i,j)))/(max(d(i,j))-min(d(i,j))) (16)

for i=1,...,I and j=1,...,J

where U(d(i,j)) is a value in the interval [0,1], and min(d(i,j)) and

max(d(i,j)) are the minimum and the maximum entries for each j in the

preference matrix.

b. Incidence matrix. In order to perform Q-analysis upon it,

the preference matrix is transformed into a binary matrix, B(k) using an

equally-spaced threshold or slicing parameters, s(k), k=1,...,K. This

assigns a value of 1 to entries in the matrix which are greater or equal

to s(k), otherwise, the entries get a value of 0. In this way K

incidence matrices are constructed such that to the k-th slicing

parameter, s(k), there corresponds the incidence matrix

B(k) =-[b(i,j)li = 1, = 1, for 0<s(k)<1 (17)

This incidence matrix reflects which alternatives satisfy given criteria

at a particular slicing level s(k). The decision maker's preference

function over the criteria is embedded in the set of weights the DM

assigns the criteria.

63

Now, for each of the K complexes, a Q-analysis may be performed

(Kempf et al., 1979) in order to determine the alternative with the

highest dimensional level (in its structure vector) for high values of

the slicing level s(k) (Hiessl et al., 1985). The structure vector

indicates the q-connectivity of the alternatives with respect to the

given criteria.

c. q-connectivity. To describe q-connectivity, let the

simplicial complex Kx (Y;L) have vertices and simplices consisting,

respectively, of the set of elements Y and X as pointed out above.

Then, the element of 1 in the derived incidence matrix indicates that

X(i) is L-related to Y(j). The geometric dimension of a simplex is

equal to the number of vertices minus one. In p-simplex, if p stands

for the dimension of the simplex, the number of vertices is p+1. If the

roles of X and Y are exchanged through a transpose relation such that

the elements of X representing the vertices while the elements of Y

becoming the simplices, then a conjugate complex K y(X;L-1 ) is generated.

An intuitive way to view these simplicial complexes is as a

collection of simplices joined together through a sharing of vertices

via the faces. The number of vertices shared by two simplices

constitutes a natural connectivity concept, the complexes themselves

being the logical extension of the two dimensional graphs. Examination

of how the simpl ices of the complex K x(Y;L) are connected can bring

about the characteristics of the complex structure. This can be

addressed through formal definitions of the relation of q-connection on

the simplices of a complex.

64

Two simplices are considered to be q-near in K if and only if

they share a q-dimensional face in K. Likewise, two simplices Or and

G s in a complex K are joined by a chain of connection if and only if

there exists a finite sequence of simplices { a such that

• o r is a face of a 2. o ce is a face of °s a a and1

0 a share a face of dimension a m m = 1,..,p-1; andm+1

4. q = minir,13 1 ,f3 2 ,..,f3,si (Johnson, 1981).

It can be shown that q-connectedness (C q) on the simplices of K

is an equivalence relation (i.e. it is symmetric, reflexive and

transitive) and the equivalence classes are the elements of the quotient

set Kq /C

q and they are called q-connected components of K (K is the set

of simplices in K with dimension greater or equal to q). The number of

distinct classes at each dimensional level will determine the entries in

the structure vector Q. From the Q vector, the chains of connections q

at each dimensional level can be determined (Casti, 1979; Casti et al.,

1979; Johnson, 1981; Duckstein, 1983; Featherkile and Duckstein, 1986).

The determination of the components of q, and the listing of the numbers

Q0,Q 1 ,—.,Q p , where p=dimK, is the Q-analysis of K (Kempf et al., 1979;

Johnson, 1981). This provides a preference order on the alternatives

from which a final ranking is obtained with the help of indices.

d. Alternative or project selection. A project satisfaction

index (PSI) is introduced, at this level, to intrinsically characterize

each alternative. Toward this end, the following equation (18) which

65

defines the complex V(i,k) for each alternative x(i) at each s(k) level,

is presented (Duckstein, 1984; Hiessl et al., 1985):

V(i,k) = w(j(1))+w(j(2))+....+w(j(q+1)) (18)

for j=1,2,....,J and w(j(.)) = j(.)-th element of w.

The right hand side of equation (18) is the sum of the weights

corresponding to the criteria satisfied at slicing level s(k).

Geometrically, this term of the equation corresponds to a simplex with

the vertices y(j(1)),...., y(j(q+1)). From this the PSI of alternative

x(i) is defined as

PSI(i) = z s(k)V(i,k)

(19)k=1

According to this index, an alternative will have a high PSI if it is

rated high on several criteria. The value of PSI is therefore, to

indicate the DM's level of satisfaction with a given alternative without

any comparison with the other alternatives in the system.

Inter-alternative comparison is effected with the introduction

of another index called project comparison index (PCI). This index is

based on the q-connectivity between the various alternatives. To

determine the index Hiessl et al. (1985), defined qmax(i,k) to be the

highest q-level of alternative x(i) and qmin(i,k) to be the level at

which x(i) is for the first time in the same equivalence class as

another alternative. The difference of these two terms:

Aq(i,k) = qmax(i,k) - qmin(i,k) (20)

66

is, then used to derive the expression of the PCI for alternative x(i)

which can be written as

KPCI(i) = z s(k). Aq(i,k)

(21)k=1

There may be a conflict between these two different indices, the PSI,

representing the intrinsic value of the alternatives individually (a

value-type concept), while the PCI indicates the relative ranking of the

alternatives (an outranking type concept).

A means to resolve the problem is by combining the PSI and PCI

through a Lp-norm to generate a new index known as the first type

project rating index,(PRI-I) (Hiessl et al., 1985). For this purpose,

the first two indices are normalized to the interval [0,1] using:

PSIN(i) = PSI(i)/PSIMAX and PCIN(i) = PCI(i)/PCIMAX

where PSIMAX and PCIMAX are the highest values the indices PSI and PCI,

respectively, are expected to assume. The PRI-I for alternative x(i),

is, then, expressed as

PRI-I(i)= (11-PSIN(i)1P+11-PCIN(i)1 1 /P) ; 1 < p < œ (22)

This method constitutes a concordance analysis with a tradeoff between

the value type index, PSI and the outranking-type, PCI (Siskos et al.,

1983; Duckstein et al., 1984). The interpretation of this index for the

different values of p is similar to that of the Lp-norm in compromise

programming (Duckstein and Opricovic, 1980). When p=2, for example,

PRI-I corresponds to the Euclidean distance between the actual

67

alternative vector (PSIN(i),PCIN(i)) and the ideal alternative vector

(1,1) (Hiessl et al., 1985). At any rate, the alternative with the

least PRI-I values is most preferred according to this index.

There is an apparent weakness with the use of mere concordance

analysis for the fact that an alternative X(i i ) may be rated above

alternative X(i 2) although it is characterized by at least one criterion

j such that z(i i ,j) is unacceptably worse than z(i 2 ,j). The situation

can be avoided by utilizing an outranking technique that includes the

concept of discordance as in ELECTRE (Benayoun et al., 1966; Grama and

Hansen, 1983; Hiessl et al., 1985).

4.1.4. Multicriterion Q-Analysis II.

MCQA-II is an outranking type which, in addition to the steps

followed in MCQA-I, includes the concept of discordance analysis.

Discordance, in this case, refers to undesirable rating of an

alternative with respect to any criterion for a given slicing level

s(k). The effect of this discordance condition on ranking the

alternatives is analyzed using a project (alternative) discordance-index

(PDI). Analogous to the PCI, the PDI is defined using the complementary

Aincidence matrix B(k) instead of original concordance incidence matrix

B(k) (Duckstein, 1984; Hiessl et al., 1985).

'B(k) = 1 - B(k) (23)

Under these circumstances, the PDI of alternative x(i) is described by

KPDI(i) = E s(k). 4(i,k) (24)

k=1

0.where Aq(i,k) is defined analogously to Aq(i,k).

68

Again as with PSI and PCI, a normalized PDI in the interval

[0,1], PDIN is defined using PDIN(i)=PDI(i)/PDIMAX, where PDIMAX is the

highest value PDI(i) can take for the problem under consideration.

PDIN(i) and the normalized forms of PSI(i) and PCI(i) are then combined

in the form of an Lp-norm to obtain a second type of project index (PDI-

II) for a complete ordering of the alternatives. This PRI-II for

alternative x(i) can be defined (Duckstein, 1984; Hiessl et al., 1985)

as

PRI-II(i)=[11-PSIN(i)1411-PCIN(i)141PDIN(i)1 13] 1113 (25)

for 1 < p <

The ideal value in equation (25) is zero which is the minimum

distance possible to the ideal point (1,1,1). This value represents a

trade off between the three indices PSI, PCI, and PDI. As a result, it

can be considered to have a better ordering outcome than any one or a

combination of the first two indices. In any case, for p=1, the Lp-norm

(25) corresponds to the sum of absolute value deviations, while for p=2

it corresponds to the Euclidean distance (Hiessl et al., 1985).

4.2. Distance-Based Techniques.

Both compromise programming (CP) and cooperative game theory

(CGT) are distance-based techniques in which the solutions are

determined in reference to some point in space. This point is mostly

referred as the ideal point in CP while it is usually called the 'status

quo' point in CGT.

69

4.2.1. Compromise programming.

This technique is designed to identify solutions which are

closest to an ideal point (Figure 6) by some distance measure (Zeleny,

1973, 1974, 1982; Starr and Zeleny, 1977; Duckstein and Opricovic, 1980;

Szidarovszky et al., 1986; Tecle and Fogel, 1986). An ideal solution,

in general, can be defined as the vector f*=[fl ,f2 ,...,f0 where the 4

are the solutions to the problem stated as Maxx f i (x), In a

discrete setting such as the case problem under consideration, however,

the ideal solution is defined as the vector of best values selected from

the payoff matrix. Such a payoff matrix is shown in Table 9. The

vector of worst values represents the minimum objective function values

denoted as el * (Table 6). These values are valuable in determining the

degree of closeness of an alternative to the ideal solution.

One of the most commonly used measures of closeness is a family

of Lp metrics (Duckstein and Opricovic, 1980; Zeleny, 1982; Goicoechea

et al., 1982), that can be expressed as

min [L(x) = E wY(4-fi(x)) 13] 1 / 13]i=1

subject to x c X

where the weights w1>0 indicate the relative importance of the

objectives to the DM. For p = ., the largest of the deviations

completely dominates the distance measure (Duckstein et al., 1980;

Tecle and Fogel, 1986). Consequently equation (26) reduces to the

expression:

min [L(x) = max w 1 (4-f1(x)), for i=1,2,...,N]

(26)

(27)

f*i *, f 2**)

status quopoint

f l

Thus, in the process, a double weighting scheme exists in the

terms of w- and p. For values of p>2, a distance quite different from

the geometric straight line is expected in terms of distance as a

measure of human preference. The measure of human preference is

multidimensional and not limited to the interval of geometrically

intuitive measure of 1 < p < 2 (Zeleny, 1982).

Figure 6: Illustration of two distance-based techniques:compromise programming and cooperative gametheory.

70

71

In this study, since the criteria of the objective functions are

not measured in commensurable terms (Tables 3 and 5), relative rather

than absolute deviations are used to determine the distance measure. In

this case, the compromise solution with respect to p can be expressed as

min [L(x) = [ z wY((4-fi(x)/(4-4 * )) 13114]i=1

(28)

**where f = minXf(x) i=1,2,—,N is the minimum objective functioni

value in terms of criterion i as pointed out above (Table 6).

The set of all compromise solutions for a particular set of

weights (w l ,w 2 ,—,w N) and for all 1 < p < . constitute a compromise

set. In this paper only three points of the compromise set, that is,

those corresponding to p=1,2 and . are calculated, and the results are

discussed in the next chapter.

4.2.2. Cooperative Game Theory.

The second distance-based multicriterion decision making

technique used in this study is cooperative game theory (CGT). Game

theory, in general, is a mathematical study of conflict resolution.

Cooperative game theory, on the other hand, is one aspect of game theory

in which the participants have the opportunity to communicate with one

another and form binding and enforceable agreements among themselves.

Such an agreement results in the formulation of a payoff matrix. In

this study, the attributes of each alternative with respect to each

criterion represent the payoff matrix (Table 9).

72

A number of solution schemes have been proposed to n-person

cooperative game problems (Rapoport, 1970; Szidarovszky et al., 1978;

Guiasu et al., 1980; Colman, 1982). Most of these solution concepts are

usually based on the subjective choice of weights, bounds and/or

distances. The solution concept in this paper is similar to the one

followed in Szidarovszky et al., 1982; Tecle and Fogel, 1986). It is

based on a certain set of axioms and the subjectivity of the DM in

accepting or rejecting the axioms and determining the 'status quo' point

(Figure 6). The status quo point in a cooperative game (CG) can be

taken as the vector of payoffs which the n-players can be respectively

sure of obtaining even if no cooperation among themselves exists. To

say it differently, the status quo point is a disagreement payoff

vector, f** , in the payoff space p (Rapoport, 1976). With this

assumption Nash developed a solution procedure for two-player bargaining

games (Nash, 1950, 1953). Harsanyi (1977) extended the Nash procedure

to an n-player game (n > 3). The resulting Nash-Harsanyi model can be

expressed (Goicoechea et al., 1982; Szidarovszky et al., 1984) as

Max 7 (f - f**i ) wi

i=1(29)

subject to f i >ei * and fa', and f**= [f** f** w i is the weight

for criterion i=1,2,—.,N.

This equation can be derived in one or two ways. It can be

arrived at using Zeuthen's bargaining principle which states that the

next concession always comes from the objective having the least risk in

a conflict (Zeuthen, 1930). It can also be derived from satisfying the

73

well known Nash's axioms (Goicoechea et al., 1982; Szidarovszky et al.,

1982; Tecle and Fogel, 1986).

The idea is that if the players agree to the axioms as general

principles then they can apply a bargaining procedure that satisfies the

axioms in all situations in order to get a "satisficing" solution.

Mathematically such a bargaining procedure can be defined using the

vector-valued function as a mapping from f* and P to some point f** in

P, that is, (f** ,P) = f*. It has been proven that this procedure and

Zeuthen's bargaining procedure arrive at the same solution (Harsanyi,

1977) which can be obtained using equation (29) (Szidarovszky et al.,

1984). This equation is used to obtain the CGT solution to the problem

under consideration, and the results are discussed in the next chapter.

CHAPTER 5

APPLICATION OF MODELS AND ANALYSIS OF RESULTS

After formulating the problem in a multicriterion context and

describing the mathematical programming procedures used, the next

logical step was to combine these two procedural classes to find a

numerical solution to the problem. Each of the six MCDM techniques

described above was used separately to determine the solution. This was

done through the use of six different computer algorithms, one for each

of the six techniques. The quantified payoff matrix (Table 9) was used

as a homogeneous input into the programs. Criterion weights and other

model-specific parameters were also imputed as needed into the

different programs. The nature of these parameters have been discussed

along with their respective techniques.

For a better understanding of the techniques' absolute

performances, their individual applicability was first singularly

analyzed. Sensitivity analyses were performed on each technique to

determine any changes in the DM's preference structure and other

parameters on the final outcome. Comparison among the techniques

reveals their individual relative applicability on the case study.

5.1. Application of Models, Solutions and Sensitivity Analyses.

Even though all the techniques utilized in this study are

designed to determine solutions to multicriterion problems, their

74

75

methods of applicability can be widely divergent. Such variations can

range from single parameter procedure in cooperative game theory to

multiparameter procedure in ELECTRE II and from the analytical approach

in compromise programminy to the synthetic nature of the solution

technique in MCQA. Further classification of these similarities and

differences is provided through a comprehensive discussion of the

applicability of each technique to the problem under consideration.

5.1.1. ELECTRE I.

a. Application of ELECTRE I. ELECTRE I can be appropriately

used for the purpose of eliminating dominated alternatives and partial

ordering of the nondominated ones. In this study, the technique was

applied to the evaluation matrix of Table 5. Equations (3) and (4)

were used to respectively calculate the concordance (Table 10) and

discordance (Table 11) matrices. The criterion weights and scales, used

in deriving, respectively, the concordance and discordance matrices are

taken from Table 6. Only the first set of weights in the table were

used in this technique. In any case, the weights are assumed to

represent the DM's preference structure, while the scales reflect the

relative difference between the best and the worst for each criterion

with respect to the other criteria.

The concordance and discordance matrices were assessed between

any pair of alternatives for all criteria resulting in square matrices

(Tables 10 and 11). A synthesis between these two matrices was then

made using the threshold values (p,q) resulting in the composite graph

of Figure 7. As previously pointed out p represents the minimum

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Lc) Ls) 0.) c:) r-. CS) Lc) (.c) r-- r-- Lc) CPI cc) dro C r-4 01 LII r-4 r-- r-- Lf) LC> r-4 Cr) r-4CD Lc) r-4 r-4 Q r-4 CI" CI- 01 Cr) LC) r-4 cI r--• • • • • • • • • • • • • •

CV re) di- 1.1-) I--- 00 (7) c) csj celr-#

CT)

CO

LID

LC)

CY)

78

acceptable concordance level while q stands for the maximum acceptable

discordance level. Thirty one different pairs of (p,q) values were

imputed into a computer program (Appendix C) to determine the effect of

varying threshold values upon the wastewater treatment alternatives that

pass screening for acceptability. As a result, there were a total of 31

different trials performed.

Figure 7. Composite graph of ELECTRE I used to obtain the kernel(nondominated alternatives) for (p,q) values of (0.7,0.2).

b. Results of ELECTRE I. The results for twenty of the trials

are shown in Table 12 (while 31 of them are represented in Appendix C).

From the table, it can be observed that alternative 10, facultative

79

Table 12. ELECTRE I Results.

thresholdvalues

p 9

selectedalternatives

thresholdvalues

p 9

selectedalternatives

.3 .1 6,10,13,15 .8 .2 6,10,13,15

.4 .1 6,10,13,15 .8 .3 6,10,13

.5 .1 6,10,13,15 .8 .4 10,13

.6 .1 6,10,13,15 .8 .5 10,13

.6 .3 10,11 .8 .6 10,13

.7 .1 6,10,13,15 .9 .1 3,6,10,12,13,15

.7 .2 6,10,11,13 .9 .2 6,10,12,13,15

.7 .3 10,11 .9 .3 6,10,12,13,15

.7 .4 10,11 .9 .4 6,10,12,13,15

.8 .1 6,10,13,15 .9 .5 6,10,12,13,15

lagoons supplemented by infiltration algae removal and nutrient removal

facilities (20 times); alternative 13, facultative lagoons plus land

application (17 times); alternative 6, facultative lagoons with

filtration algae removal facilities (14 times); followed by alternative

15, oxidation ditches with land application (11 times) are the

alternatives most often preferred. The preference frequency of every

one of the set of alternatives considered for the 31 pairs of (p,q)

values used is shown in the bar chart of Figure 8. Again, alternative

10 has 97% frequency of occurrences in all the kernels obtained followed

by alternatives 13, 6, 15, and 11 the frequency of which are 87%, 74%,

58% and 48% respectively. Alternative three is the least preferred of

them all. It should be noted at this point that while ELECTRE I can

determine the preferred alternatives, it does not provide their complete

0

6

5

I 11111F1111 2 3 4 5 6 7 El 9 10 1 1 12 13 14 15

Alternative in Kernel

80

rankings. Therefore, the frequency of occurrences above does not

indicate the most preferred alternative(s).

Figure 8. Occurrence frequency of alternatives in a kernel out of totalof 31 trials using ELECTRE I.

c. Sensitivity Analysis of ELECTRE I. Sensitivity analyses are

usually performed to test model robustness with respect to variations in

weights and scales (Gershon et al., 1982; Alley, 1982). Gershon et al.

(1982) found that ELECTRE I was fairly robust with respect to changes in

weights and was not significantly affected with scale changes in their

application to river basin planning. As a result, weights and scale

12

15

.2 .3 .4 .5Discordance level G

81

changes are assumed to have the same effect in the present application

of ELECTRE I as in Gershon et al.

In this study, changes in the pair of threshold values (p,q)

were found to affect significantly the outcome of ELECTRE I application

to the problem under consideration (Figure 9). The charts in this

figure and the results in Table 12 illustrate that the selection process

Figure 9. Number of alternatives selected (in a kernel) with respect todifferent pairs of combinations of the thresholds p and qvalues.

becomes sharper as the difference between threshold values p and q

becomes smaller at higher values of p except when p = 0.9. The number

82

of nondominated alternatives is narrowed down to two when 0.6 < p < 0.9

and 0.3 < q < 0.6 except for (p,q) = (0.8,0.3) as shown in Table 12 and

Figure 9. Therefore, the "best" choices for the threshold values p and

q can be considered to be between the above respective intervals as

these intervals seem to satisfy the purpose of the threshold values

(p,q) in the process. This purpose is nothing other than narrowing

down, as much as possible, the number of satisficing alternatives

selected.

One complicating observation in defining the purpose of the

threshold values, is the switching effect they have on the nondominated

alternatives at different levels of the concordance index, p within the

desired interval level. This is observed in Table 12 when the non-

dominated pair of alternatives changed from 10 and 11 at 0.6 < p < 0.7

to alternatives 10 and 13 at p = 0.8. Here alternative 13 which was

dominated at the lower values of p becomes nondominated as p becomes

tighter. The reverse situation takes place for alternative 11. The

reason behind this discrepancy is that the threshold-values do not

reflect the absolute magnitude of each element in the respective matrix

than the range of interval value. In this case, when p is assigned a

value of 0.7, alternative 10 with a value of 1.00 and alternative 11

with a value of 0.772 in the concordance matrix were dominating each

other forming a nondominated cycle in the composite graph of Figure 7.

That means all values above p=0.7 were considered equivalent.

Therefore, in order to conform with the above definition of the purpose

of the threshold values, the value of p must assume a higher possible

83

value within the desired interval. A better value of p in this case is

0.8. As a result, the appropriate pair of nondominated alternatives

that should be selected are considered to be 10 and 13.

5.1.2. ELECTRE II.

The motivation for using ELECTRE II is to present a complete

ranking of the alternative wastewater treatment schemes. This was not

possible to do using the ELECTRE I method above.

a. Application of ELECTRE II. ELECTRE II was directly applied

to the payoff matrix of Table 9. This was done to preserve the

homogeneity of inputs to the various techniques used, otherwise the mix

of ordinal and cardinal data matrix of Table 5 could have been used as

in Gershon et al. (1982). Along with the payoff matrix the following

groups of characteristics were considered in applying the ELECTRE II

method to the problem under consideration:

1. Levels of concordance: (p - ,p0 ,p

* ) 1 = (0.5,0.75,0.90) and

(p - ,p0 ,p

* ) 2 = (0.50,0.667,0.75).

2. Levels of nondiscordance condition: (q0 ,q

* ) 1 = (0.2,0.6) and

0 *(q ,q ) 2 = (0.1,0.5).

3. The four sets of weights in Table 6 were used to assess the

sensitivity of the technique to changes in weights.

4. Two sets of discordance scale types (the best and the worst types of

Table 6) were used to determine the sensitivity of the method to

changes in scale type. The best values-column of Table 6

84

constitutes set I while the worst values-column in the table is

referred to as set 2 scale type.

A total of 32 different cases of sensitivity analyses can be

made to determine the effect of every combination of the above

characteristics on the model. But for the purpose of this study only

seven cases which are considered to be sufficient to demonstrate the

effect of each characteristic on choosing the most satisficing

alternative wastewater treatment scheme were provided. The

characteristic make-up of the seven cases is shown in Table 13.

Table 13. Characteristic Makeup of the Seven Cases for SensivityAnalysis of ELECTRE II application.

Case Weight Concordance Discordance Discordance

SetLevel Level scale type(c10 , q ) Set

1 1 (.50,.75,.90) (.2,.6) 1

2 2 (.50,.75,.90) (.2,.6) 1

3 3 (.50,.75,.90) (.2,.6) 1

4 4 (.50,.75,.90) (.2,.6) 1

5 1 (.50,.75,.90) (.1,.5) 1

6 1 (.50,.67,.75) (.2,.6) 1

7 1 (.50,.75,.90) (.2,.6) 2

To evaluate the above seven cases, the computational techniques

discussed in 4.1.2. were followed step by step. The process of

computation, however, was done both using the computer and by hand. The

first was used to determine the strong, G s and the weak, Gw graphs while

the hand calculation technique (Appendix D) was used to synthesize the

two graphs in order to determine the complete outranking relationships

among the alternatives. For illustrative purposes a strong and weak

graphs for case 1 are provided. Figure 10 demonstrates the reduced

strong graph while Figure 11 shows the reduced weak graph. All cycles

(closed paths) are eliminated in both figures.

,,,,i A. •

iliimitim.A.eiNivirPort

c'gqw1L40F.- 7ote:.X1 stands for

alternatives 6,7,8,9and 12

Figue 10. Reduced graph of the strong relationship of ELECTRE IIapplication.

b. Results of ELECTRE II Application. Table 14 shows the

computed ranking of the different wastewater treatment alternatives in

each of the seven cases of Table 13. On an average and/or median basis,

the alternatives assuming the first through the fourth ranking are

85

86

X1 stands foralternatives 6,7,8,9and 12

Figure 11. Reduced graph of the weak relationship of ELECTRE IIapplication.

consecutively alternative 10 consisting of facultative lagoons plus

filtration algae removal and nutrient removal facilities, alternative

13, which supplements facultative lagoons with land application,

alternative 11 which includes aerobic lagoons, filtration algae removal

and nutrient removal facilities, and alternative 6 which is facultative

lagoons with filtration algae removal facilities only. Comparison of

the rankings in the individual cases of Table 14 shows that the

technique is relatively robust with respect to changes in weights and

the concordance and discordance threshold values. In all cases except

87

Table 14. Ranking of Alternatives Using ELECTRE II for the SevenDifferent Cases.

Cases

Alternatives 1 2 3 4 5 6 7

1 9 7 5 7 10 10 42 8 8 6 7 9 9 33 10 10 7 8 11 11 54 6 4 4 4 6 7 35 7 6 5 5 7 8 46 5 4 3 3 4 5 17 5 5 4 4 5 5 28 5 4 4 3 4 5 39 5 4 5 3 5 6 4

10 1 2 2 1 1 1 211 2 3 4 2 2 2 312 5 7 4 4 5 5 413 3 1 1 1 2 3 214 4 4 4 3 3 4 315 8 9 7 6 8 9 4

case 7 (Table 14), the first ranking was exchanged between alternatives

10 and 13. Case 7 with set 2 scale type seems to be significantly

different from the others. Here alternative 6, facultative lagoons with

filtration algae removal facilities is ranked first.

c. Sensitivity Analysis of ELECTRE II. As already pointed out

above with the requirement for the different kinds of parameters such as

the 3 levels of concordance, two sets of weights for the discordance

space, sets of scale types and the weights assigned to each criteria,

carrying a sensitivity analysis test on every possible combination of

these characteristics would be cumbersome, time consuming and expensive.

To overcome the problem in this study, sensitivity analysis test on the

88

parameters were limited to few representative conditions (Table 13), the

results of which are briefly discussed below.

1. Changes in sets of weights: Columns one through four are the

computed rankings for the 4 different sets of weights (Table 6) and were

determined by keeping all other parameters the same in all four runs.

The weights were selected so as to give more emphasis to water quality

in case 1, equivalent weights to both quality and cost criteria in

case 2, more weights to cost than quality in case 3, and equal weights

to all criteria in case 4. The results of these cases demonstrate that

ELECTRE II is moderately sensitive to changes in weights when dealing

with closely comparable alternatives. In case 1, alternative 10 is

ranked first while it is second to alternative 13 in both the second and

third cases. Alternative 13 is ranked third in case 1. In the fourth

case, however, both alternatives 10 and 13 are equally ranked first. In

general, barely noticeable sensitivity with respect to weights permeates

throughout the whole range of the alternatives as shown in Tablel 14.

2. Changes in discordance level: The level of discordance (q* ,q

0 ) in

case 1 was changed from (0.2,0.6) to (0.1,0.5) to make case 5

(Table 13). Except for a slight variation in some rankings, the

technique appears to be robust with respect to change in the levels of

discordance (Column 5, Table 14).

3. Changes in concordance level: To determine the sensitivity of the

ELECTRE II technique to changes in the levels of concordance parameters,

the (p - ,p0 ,p

* ) values of case 1 were changed from (0.50,0.75,0.90) to

(0.50,0.667,0.75). The results shown in column 6 of Table 14 indicate

89

that the technique is also quite robust with respect to changes in the

levels of concordance at least for the case under consideration.

However, since such a change tightens the concordance condition, it may,

in some other cases, affect the outcome of ELECTRE II application

(Duckstein and Gershon, 1983).

4. Scale changes: Of all the different cases tried, the ELECTRE II

technique seems to be more sensitive to scale changes as shown in column

7 of Table 14. In this case, the set of scales in case 1 was changed by

the set of worst values in column 3 of Table 6 all other parameters

remaining the same as in case 1 of Table 13. As already pointed out in

section 5.1.2., part b, when the lower set of scales were used

alternative 6 was ranked first while 10 and 13 assumed second. Quality

wise, alternative 6 has lower treatment capacity than alternatives 10,

13 and some of the other treatment alternatives but has much lower cost

(Table 5 and Appendix E) and the latter attribute seems to outweigh the

first in this case. Besides, from the DM's point of view a small scale

type value would tend to incorporate more the influence of any criterion

that would yield a large value of (f(n,i)-f(m,i)) (Goicoechea et al.,

1982). Thus the high ranking of alternative 6 with respect to the cost

criteria of capital cost and 0 and M cost may explain the reason for the

superior ranking of the alternative in this particular case.

5.1.3. Multicriterion Q-Analysis I.

a. Application of Multicriterion Q-Analysis I. Multicriterion

Q-analysis is applied to the payoff matrix of Table 9. This payoff

90

matrix was transformed into a preference matrix having values in the

interval [0,1] using the procedures described in 4.1.3. After that, a

vector of slicing parameters having values ranging from 0.05 to 0.95 and

increasing at a constant increment of 0.05 was specified. Then, to make

the preference matrix suitable for polyhedral structure analysis, it was

mapped into binary incidence matrices, one matrix for each slicing

parameter used in the process. For illustrative purposes, the incidence

matrix obtained using the slicing parameter 0.90 is shown in Table 15.

Table 15. Binary Incidence Matrix of Slicing Parameter Value of 0.9.

alternatives(simplices)

criteria (vertices)

1 2 3 4 5 6 7 8 9 10 11 12

1 0 0 0 0 1 0 0 0 1 1 0 12 0 0 0 0 1 0 0 0 1 0 0 03 0 0 0 0 0 0 0 0 0 0 0 04 0 0 0 0 0 0 0 0 0 1 0 05 0 0 0 0 0 0 0 0 0 0 0 06 0 0 0 0 1 0 0 0 1 1 0 07 0 0 0 0 0 0 0 0 0 0 0 08 1 0 0 0 0 0 1 0 0 0 0 09 1 0 0 0 0 0 1 0 0 0 0 0

10 1 1 1 0 1 1 1 0 0 0 0 011 1 1 1 0 0 1 1 0 0 0 0 012 1 1 1 0 0 0 0 1 0 0 0 013 0 0 0 1 1 1 1 0 0 1 0 114 0 0 0 1 0 1 1 0 0 0 0 015 0 0 0 1 0 0 0 0 0 0 0 0

In Q-analysis, the elements of the matrix are said to represent

the relations between vertices (criteria) and simplices (alternatives).

In this case, we can have almost 15 p- simpl ices with a possible maximum

91

dimension of 15 (i.e. p < 15). A collection of the simplices joined

together through sharing of vertices via the faces forms a simplicial

complex denoted K x(Y;L) with y representing the set of simplices

(alternatives) as pointed out in 4.1.3. The geometric representation of

this complex is shown in Figure 12. Figure 13 is another simplicial

Figure 12. Connectivity structure of the simplicial complex Kx(Y;L).

92

1

Figure 13. Connectivity structure of the conjugate simplicial complexKy(X;L-1 ), that is the inverse of Figure 12.

complex which is known as the conjugate complex of Kx(Y;L). This

conjugate complex is nothing but the inverse of the simplicial complex

and is represented as Ky(X;L -1). The two figures clearly portray the q-

connectivity between the sequence of simplices in their respective

93

complexes. They also help in graphically visualizing the rankings among

the different wastewater treatment alternatives.

The ranking of the alternatives was made through the

introduction of the indices, PSI and PCI and an L p-norm index, PRI-I,

First type Project Rating Index which is a synthesis of the first two

indices (equation 22). As previously pointed out, PSI is a value-type

index which characterizes each alternative in an intrinsic manner

without comparing it to the other alternatives, while PCI is an

outranking type of index based on the q-connectivity between

alternatives. The values of PSI and PCI for the different alternatives

were computed using the equations (19) and (21) respectively. The

normalized form of the first set of weights in Table 6 were used to

determine the PSI values in equations (18) and (19).

b. Results of MCQA I Application. The results of applying MCQA

I to the case study are shown in Table 16. The last column in the Table

Table 16. Results of Q-Analysis.

slicinglevel

q-dimensionvector

Q-structurevector

highest q-levelchoice

0.90 (5 4 3 2 1 0 -1) (2 2 4 1 1 1 1) (10,13)0.80 (8 7 6 5 4 3 2 1 0) (1 2 2 1 11 1 2 1) (13)0.70 (8 7 6 5 4 3 2 1) (1 2 2 111 1 2) (13)0.60 (8 7 6 5 4 3) (1 3 3 3 1 1) (13)0.50 (8 7 6 5 4 3) (3 3 1 3 1 1) (7,10,13)0.40 (10 9 8 7 6 5) (2 1 1 1 1 1) (6,7,10,11)0.30 (11 10 9 8 7 6 5) (111 1 1 1 1) (6,7,10)0.20 (11 10 9 8 7) (1 1 111) (4,5,6,7,8,10)0.10 (11 10 9 8 7) (11 11 1) (4,5,6,7,8,10)

.05.10.15.20.25.30.35.40.45.50.55.60.65.70.75.80.85.90.95

94

consists of the alternatives selected at the highest dimensional level

for each slicing parameter in column 1. Columns 2 and 3 show the

structure vector which indicates the q-connectivity of the alternatives

with respect to criteria taken globally. According to these results, the

best choice or choices varies with the value of the slicing parameter

selected. As the value of the slicing parameter becomes bigger, the

number of best choices becomes narrower until it becomes one,

alternative 13 when the slicing parameter value lies between 0.55 and

0.80 (Figure 14 and Appendix E). This Figure and the Appendix were

7

IDQ) 6

5

ro

. 4

34-

o

* 2C32:

0

Slicing parameter

Figure 14. Illustration of the number of selected alternatives versesslicing parameters - the lower the slicing parameter theless the selectivity of alternatives.

95

included to illustrate further the optimal range of slicing parameters

to choose. In this application that range appears to be between 0.55

and 0.80.

Since the above procedure did not produce a complete ranking of

the alternatives, the latter was determined using the indices PSI, PCI

and PRI-I as shown in Table 17. Except for the value-type index, PSI

Table 17. Ranking of Alternatives Using MCQA-I Techniques.

alternatives PSI

PRI-I

PCI P=1 P=2 P=3

1 11 6 9 5 52 13 8 13 14 133 15 4 14 12 74 10 8 12 13 125 14 8 15 15 146 6 8 8 10 107 12 7 10 6 68 5 8 7 9 109 8 8 11 11 11

10 1 2 2 2 211 3 8 4 7 812 7 3 3 3 313 2 1 1 1 114 4 8 5 8 915 9 5 6 4 4

which ranked alternative 10 first and 13 second, both the outranking

type, PCI and the L p-norm, PRI-I indices ranked alternative 13 as first

and alternative 10 as second. The latter remains the same for p values

of 1, 2 and 3. The sequences of the lower rankings also remained the

same in all columns except under the PCI column in which 8 of the

96

alternatives assumed the same lowest ranking, and under the PSI and the

p=1 column of PRI-I where the rankings of the first and fourteenth

alternatives are exchanged when compared with those under p=2 and p=3 of

PRI-I column, that is alternative 14 is fourth ranked with PSI and fifth

ranked with p=1 of PRI-I while the fifth rank for p=2 and 3 of PRI-I

goes to alternative 1.

C. Sensitivity Analysis of MCQA I. A number of sensitivity

analysis measures can be taken to test MCQA I technique robustness with

respect to varying parameter values, such as weight sets, slicing

parameter sets, and the p values in the L p metrics. Since PSI is a

function of both the criterion set of weights and a given set of slicing

parameters, a change of these parameters would be expected to affect the

preference rating outcome of the index. Previous work by Hiessl et al.

(1985) claimed that the closer the slicing parameter applied to a

certain maximum density, the better the choice would be. In this study

no attempt was made to either verify the above claim or test the

sensitivity of the technique to varying sets of weights.

As already pointed out in part a above, however, the q-analysis

made for 19 different slicing parameters did produce different outcomes

as shown in Figure 14 and Table 16 (the latter for 9 trials). In

addition, varying the p values in the L metrics from 1 to 3 was found

to have little effect on the rankings of the alternatives (Table 17).

97

5.1.4. Multicriterion Q-Analysis II.

a. Application of MCQA II. MCQA II is an extension of MCQA I.

It only differs from the latter by incorporating the concept of

discordance in ranking the alternatives. This means, in order to apply

this technique to the problem under consideration, a project discordance

index, PDI (equation 24) was defined using an incidence matrix described

in 4.1.3. In this application, therefore, the normalized forms of the

PSI and PCI values obtained in MCQA I along with that of PDI were used

in equation (25) to determine the complete outranking index, Second type

Project Rating Index (PRI-II). Three different sets of alternative

rankings were calculated; one for each of the three p-values of the L p-

norm (Appendix F).

b. Results of MCQA II Application. The calculated bases for

ranking the different wastewater treatment alternatives are shown in

Appendix F. The last three columns in the table of the Appendix are the

Project Rating Index II values for three Lp-norm cases (p=1,2,3), while

the first two columns are respectively the same PSI and PCI values

obtained for MCQA I. Column three, on the other hand represents the

project discordance index determined for use in MCQA II. The values for

each alternative in each of the last three columns of the Table in

Appendix F represent a synthesis of the corresponding three indices

values shown in the first three columns for each p value. Thus, the

PRI-II values represent the bases upon which the final rankings of the

alternatives in Table 18 are based. According to Table 18, alternative

13, facultative lagoons with land application and alternative 10,

98

Table 18. Ranking of Alternatives Using MCQA-II Techniques.

alternatives PSI PCI PD1

PRI-II

P=1 P=2 P=3

1 11 6 2 7 4 42 13 8 1 10 13 113 15 4 8 15 15 134 10 8 1 8 10 105 14 8 4 13 14 126 6 8 1 5 8 87 12 7 1 6 5 58 5 8 1 4 7 89 8 8 5 12 11 10

10 1 2 1 2 2 211 3 8 3 3 6 712 7 3 6 11 3 313 2 1 1 1 1 114 4 8 4 9 9 915 9 5 7 14 12 6

facultative lagoons with filtration algae removal and nutrient removal

facilities are respectively the first and second most preferred

alternatives for all three p values. Other rankings differ with p

values. Alternative 11, aerobic lagoons with filtration algae removal

and nutrient removal facilities is third for p=1 while it assumes the

sixth and seventh rank for p=2 and 3 respectively. Alternative 3,

oxidation ditches is the least preferred wastewater treatment scheme for

all three values of p.

c. sensitivity Analysis of MCQA II. It would be possible to do

sensitivity analyses on MCQA II with respect to changes on a number of

parameters such as weights, slicing parameters, p values of the Lp

99

metrics, and the selection of the "ideal" points. But since the purpose

of this study was not an exhaustive treatise on use of this technique, a

constant set of criterion weights, slicing parameters and ideal points

were considered. The work of Hiessl et al. (1985), however, can be

referred to for the model's sensitivity to different sets of slicing

parameters. In that study, the optimum slicing set was found to be one

corresponding to the set of all different values in the preference

matrix. Such an optimum set was used in this study as it represents the

best use of available information on the problem. On the other hand,

the values of p in the L metrics were varied from 1 to 3 to test theirP

effect on the final outcome. As pointed out in part b above, except in

the highest two, there appears to be slight variation in the other

rankings, particularly when the values for p=1 are compared with either

for p=2 or 3 (Table 18).

5.1.5. Compromise Programming.

In this section, the application of the distance-based

technique, CP and the solutions obtained under varying relevant

parameters are discussed. The parameters varied were criterion weights

and the p-values of the L p-metrics. This was done to test the

sensitivity of the CP technique to different values of these parameters.

a. Application of Compromise Programming. This technique was

applied to the payoff matrix of Table 9. Since this technique requires

defining an "ideal point", the vector of best values for all criteria in

Table 9 were assumed to represent the ideal point while the minimum

values for each criterion stand for the vector of worst values. These

100

two vectors which are shown in columns 2 and 3 of table 6, respectively,

were bases for constructing the payoff matrix from the evaluation matrix

of table 5. The four different sets of weights in Table 6 were used to

test the CP techniques sensitivity to changes in criterion weights.

These pieces of information were used in equation (28) to determine the

distance of each alternative from the "ideal point" for each set of

weights. The actual computation was performed for each p=1,2 and .

using a computer algorithm in a HP 1000.

b. Results of Compromise Programming Application. Table 19

summarizes the rankings of the wastewater treatment alternative schemes

obtained for weight sets 1 to 4 and p=1,2 and .. On an average or

median basis, it is observed that alternative 10 which consists of

facultative lagoons supplemented by filtration algae removal and

nutrient removal facilities is the most preferred alternative. This

actually happened with respect to all the three p values and using

weight sets 1, 2 and 4. For weight set 3 and p values 1 and 2,

alternative 13, facultative lagoons plus land application is the most

preferred one while for the same weight and p=., alternatives 6 and 7,

facultative lagoons plus filtration algae removal and aerobic lagoons

plus filtration algae removal facilities, respectively are ranked first.

The last alternatives are also ranked first along with alternative 10

for the combination of weight set 4 and p=.. Rankings 2 to 4 are mostly

interchanged between alternatives 6, 11 and 13 (Table 19). Figure 15 is

presented to illustrate the rank of high-ranking alternatives under the

different runs using the CP technique. Four of these alternatives have

Table 19. Alternative Ranking Using Compromise Programming forthe 4 Sets of Weights and p=1,2 and ..

Weight Sets

alternatives

1 2 3 4

p-values p-values p-values p-values

12 . 1 2 . 12 - 12 -

1 14 14 7 13 14 5 13 11 6 13 15 32 13 13 6 14 13 5 14 10 6 14 14 33 15 15 7 15 15 4 15 15 9 15 13 34 1084 632 5 4 3 7725 12 11 4 10 8 2 10 7 4 12 10 26 553 4 2 2 321 5 3 17 963 8 5 2 731 8 5 18 444 5 4 3 897 4 2 29 8104 9 10 4 11 12 8 983

10 111 1 1 1 255 1 1 111 222 3 7 4 488 3 4 312 7 12 4 11 11 4 9 13 9 10 11 313 335 2 6 5 112 2 6 314 675 7 9 5 664 6 9 315 11 9 5 12 12 5 12 14 10 11 12 3

ranked first at least once in the 12 runs in Table 19. Alternatives 1,

2 and 3 are the least preferred wastewater treatment schemes.

Alternative 1 is the existing wastewater treatment method in the study

area.

C. Sensitivity Analysis of Compromise Programming. As pointed

out above, various compromise solutions to the problem were obtained

with respect to changes in the sets of criterion weights and the p

values of the L metrics to portray the model's sensitivity to these

101

1

2

3

4

5

6

0

0

9

10

11

12

13

14

15

1 2 va 1 .2 1$4 1 2 1,0

1 2 3

Weight sets

1 2 'go p

4

102

Most preferred Alternatives 7Is- - - - - - - 10 - - - 10%3 - JO

• " .,

- •,6/I • •

•C... —.6„ ID

: A•••-6< 7

a , • / lo.. — g. I • .. .. .

\ : '. i 7 \ /V :}

Figure 15. Graphical illustration of the most and least preferredalternatives using compromise programming.

parameters. It is observed that when more weight was given to quality

weight set 1, alternative 10 followed by alternative 11 are most

preferred. Both these alternatives have tertiary level treatment

capacities. The first choice is the same when quality, cost and land

1 03

criteria were all given highest weights in weight set 2, but the second

choice varied from alternative 13 for p=1 to alternative 6 for p=2 and

p=.. Alternative 13 also has a tertiary treatment capacity which

involves land application. The latter, however, may have some

groundwater polluting effect. Alternative 6, on the other hand, has

less than tertiary treatment capacity. It does not possess nutrient

removal capacity and is cheapest of the 3 top ranked alternatives under

weight set 2 while alternative 13 costs in between the other two

alternatives. The model's sensitivity to weight becomes more apparent

when more weights was given only to cost in weight set 3 (column 6 of

Table 6). Under this circumstances, alternative 13 is ranked first for

both p=1 and 2 and second for p=œ while alternative 6 and 7 rank first

for p=.. In this case, alternative 10 is ranked second when p=1 but

fifth for both p=2 and .. When all criteria were equally weighted in

weight set 4, alternative 10 is again ranked first for all 3 values of

p. Alternatives 6 and 7 are also ranked first when p=.. The second

ranking goes to alternative 13 for p=1 and alternative 8 when p=2 and ..

An overall observation of table 19 shows that ranking using the minimax

solution for p=œ seems to be based on cost criteria while the rankings

for p=1 seem more influenced by quality criteria since maximum damage is

related to environmental indices. Rankings for p=2, on the other hand,

appears to be closer to being equally influenced by both types of

criteria. Duckstein and Opricovic (1980) observed the same trend in

their multiobjective river basin management study.

104

5.1.6. Cooperative Game Theory.

a. Application of cooperative Game Theory (CGT). This is the

last technique applied to select the most preferred wastewater treatment

alternative in this case study. Like compromise programming, it is a

distance-based technique. But the distance in this case is the maximum

geometric distance from a "bad" point in the feasible region, the status

quo point. The worst values for each criterion shown in column 3 of

Table 6 were assumed to represent the status quo vector in this

application. A normalized form of the set of weights in column 4 of

table 6 were taken to represent the criterion weights in the model.

Then the elements of the payoff matrix of table 9 were imputed as

decision variables into the CGT algorithm of equation (29) to determine

the solutions to the problem under consideration.

b. Results of CGT Application. A distance from the status quo

point for each alternative wastewater treatment scheme was obtained

using a computer. Then, the results were ranked with respect to their

distance measure, the largest one ranking first while the smallest

becoming the last. Accordingly, alternative 10 was ranked first while

alternatives 6, 7 and 8 became the second, third and fourth choices

respectively. The lowest rank obtained in this method was seventh and

it consisted of 9 different alternatives while the first through the

sixth ranks consist only one alternative each as shown in Figure 16.

In comparison with the other techniques applied, no sensitivity

analyses on CGT with respect to changes in parameters such as the

1 0

9

6

7

8

9

10

CID 7a)›

•t-1 A4-)CC1 Cd• P4• 5

r1 Q4.-1

• 4

4c4-1O 3

•z° 2

O2 3 4 5

Rank

II II II II II1

criterion weights or selection of the status quo points were made in

this study.

105

Figure 16. Ranking specificity of cooperative game theory.

5.2. Comparative Evaluation of Techniques and their Application Results.

The six techniques applied in this study have many

characteristics that are quite different from one technique to the

other. As already pointed out some of the techniques are outranking

types like ELECTRE I and II which are strictly designed to deal with

discrete problems while compromise programming and cooperative game

theory are distance based techniques that can be applied to continuous

problems. These and other kinds of differences between the techniques

106

can generate contrasting effects on their applicability to empirical

problems (Goicoechea et al., 1982; Fandel and Spronk, 1985; Szidarovszky

et al., 1986) such as the under consideration. As a result, a

comparative evaluation of these techniques specifically with respect to

their respective performances to determine the solutions in this case

study is in order. Likewise, comparison of the preferred alternatives

in terms of their timeliness, ability to solve existing and future

wastewater management problems and other pertinent attributes they may

have is also discussed below.

5.2.1. Comparative Discussion of Model Performances.

In this section, comparisons of the six different techniques

with respect to their efficiency and ease of applicability on one hand,

and outcome specificity and stability on the other are discussed. Ease

of applicability of a technique is seen both in terms of adaptability to

a particular MCDM problem such as the case study under consideration and

in terms of how readily it can be accepted and understood by a novice

user, while application efficiency refers to the timing of performance,

that is, how fast a solution can be obtained usin9 the technique in

terms of computer time or man-hour or both. Outcome specificity, on the

other hand, describes how sharply the technique can rank individual

alternatives separately, while outcome stability refers to consistency

of results obtained under different conditions. The latter is another

way of describing the degree of technique robustness with respect to

changes in parameters.

107

In terms of ease of applicability and understandability by a lay

analyst, the distance-based techniques, CP and CGT seem to be preferable

to the outranking types of ELECTRE and MCQA techniques which require

more matrices transformations, many parameter inputs and complex

computer programming algorithms (Casti et al., 1979; Duckstein and

Gershon, 1983; Hiessl et al., 1985). Among the distance-basea types, CP

may be more readily understood and accepted than CGT as the latter has

to satisfy a number of axioms before it can be adopted (Szidarovszky et

al., 1984; Szidarovszky et al., 1986; Tecle and Fogel, 1986).

The results for the individual model applications (Tables 12, 14

and 16 to 19) show that there are variations among the capacities of the

techniques to rank alternatives completely. ELECTRE I which only

identifies partial outranking relationships among alternatives may be

ranked last in terms of alternative ranking specificity while MCQA I and

II followed by CP may be ranked high. When p=œ and w(i)=1 for Vi, CP

becomes minimax problem and the degree of ranking specificity on the

alternatives is lower. In terms of this ranking specificity criterion

CGT, may be ranked average as it only managed to sequentially order the

first through sixth ranked alternatives while obscuring the ranking for

the remaining nine (Figure 16).

Like outcome specificity, the stability of model application

results varied under changing parameters from technique to technique as

pointed out in the individual model sensitivity analysis discussion.

All the six techniques, however, were not comparatively subjected to

sensitivity analysis with respect to changes in their specific

108

parameters and weights. Sensitivity analysis with respect to both

criterion weights and p values of the L metrics were made in CP

application, effects of variations of criterion weights, concordance and

discordance levels on final results were checked in ELECTRE II, while no

sensitivity analysis was made in CGT. P values of the L p metrics were

varied in MCQA I and II and only the effect of the threshold values

(p,q) were tested in ELECTRE I. Under these conditions, a global inter-

model comparison with respect to outcome stability among all

alternatives cannot be appropriate. Comparison of final alternative

rankings in ELECTRE II and CP in which relatively comparable sensitivity

analyses were made show the rankings to be slightly unstable with

respect to changes in the parameters. Similar comparisons of the

rankings under PRI-I and PRI-II for p = 1, 2 and 3 obtained using MCQA

I (Table 17) and MCQA II (Table 18) respectively, indicate the latter

technique to have lower outcome stability than the first. The addition

of PDI in MCQA II, thus decreased the stabilities of the alternative

rankings with respect to changes in p values.

Table 20 shows the rankings of the first through fourth ranked

alternatives determined using all the six techniques applied in this

study. This table reveals alternatives 10 and 13 to be the most

preferred wastewater treatment schemes. Alternative 10 is ranked first

using ELECTRE I, ELECTRE II, compromise programming and cooperative game

theory while it is ranked second using MCQA I and MCQA II. Alternative

13, on the other hand, is ranked first using the last two techniques and

becomes second with ELECTRE I, ELECTRE II and when p=1 in CP. These

109

Table 20. First Through Fourth Ranked Alternatives in Each of the SixModels.

Ranking

Techniques 1 2 3 4 Comment

ELECTRE I 10(97%) 13(87%) 6(74%) 15(58%) % of occurrencein 31 trials

ELECTRE II 10 13 11 6,14 median of 7 trialsMCQA-I:PSI 10 13 11 14

PCI 13 10 12 3PRI-I:p=1 13 10 12 11

p=2 13 10 12 15

P=3 13 10 12 15MCQA-II:PRI-II:

p=1 13 10 11 8p=2 13 10 12 1

P=3 13 10 12 1CompromiseProgramming

P=I 10 13 11 6 median or meanof 4 trials

p=2 10 6 13 7,8II

p=œ 10 6,7 13 8CooperativeGame Theory 10 6 8 7

choices are in agreement with those suggested using cost-effectiveness

analyses methods by Arthur Beard Engineers, Inc. (1982 and 1984). In

that work, these two alternatives were indicated as option 3 and option

4 (Appendix G).

5.2.2. Comparative Discussion of Most Preferred Alternatives.

As already pointed out in 5.2.1., the most preferred

alternatives are alternative 10 composed of facultative lagoons plus

110

filtration algae removal and nutrient removal facilities, and

alternative 13, consisting of facultative lagoons and land application

schemes. Comparison of these two techniques with respect to their

performances and cost criteria can be made using the evaluation matrix

of Table 5. In this table, the criterion scores for each of these two

alternatives are shown under columns A10 and A13 respectively.

Either of these alternative treatment schemes is considered to

handle 11.5 mgd of wastewater influent, meet 1984 National Pollutant

Discharge Elimination System (NPDES) effluent limitations and provide

nutrient removal acceptable to meet nutrient limitations required by

Arizona Department of Health Services (ADHS) and/or U.S. Environmental

Protection Agency. For these purposes, each alternative wastewater

treatment system is considered to have tertiary treatment capacity.

There may, however, be a limited risk of some pollutants reaching

shallow groundwater level in the land application system. This is the

reason for assigning lower values to the pollution vulnerability and

water quality criteria in alternative 13.

Considering resource utilization in the two alternatives,

alternative 10 will need deepening of existing lagoons to allow sludge

accumulation and installation of mechanical aeration, while alternative

13 will need about 2000-2500 acres of land for disposal of the effluent

on a year around basis. In alternative 13 most of the effluent will be

used for growing some sort of crops or irrigating golf courses. 1200

acres of the available land for land application is grandfathered land

while the rest can be obtained by negotiating with neighboring farmers

1 11

and the approval of the Arizona State legislature for converting non-

grandfathered to grandfathered land. Grandfathered land is land that

has been irrigated since 1975. The costs for the alternative schemes

will be a capital cost of 5,800,000 dollars and an annual 0 and M cost

of 1,680,000 dollars for alternative 10 compared with corresponding

costs of 3,950,000 dollars and 1,091,000 dollars for alternative 13

(Arthur Beard Engineers, Inc, 1982, 1984). In addition to this

alternative 10 with its biological, chemical and mechanical processes

would require more expert manpower than alternative 13.

Thus, the choice between these two almost equally often selected

alternatives using the six techniques will be based on the tradeoffs

between quality and cost in the one hand, and land requirements and

expert manpower needs on the other. Having the choice narrowed from 15

to two alternatives, it would not be difficult for the DM to pick one of

them. At any rate, considering the existing wastewater problems

discussed in chapter 2, it is necessary and timely that a wastewater

treatment that can satisfy the stated requirements be initiated. It is

important to remember the fact that land application of wastewater

requires some land which is physically there, but not legally available

for irrigation purposes according to the 1980 Arizona Groundwater law.

The selection of alternative 13, therefore, will depend upon state

legislation to change the additional land needed for irrigation from

non-grandfathered to grandfathered status. If this process is slow

and/or negotiating with the farmers to acquire the land become stalled

then alternative 10 should be selected.

CHAPTER 6

SUMMARY AND CONCLUSIONS

Wastewater is both a social product and a social nuisance that

must be dealt with in the "best" possible way. Best is taken in the

sense of environmental quality, technical feasibility, resources

availability, economic viability, socio-political acceptability and

post-treatment usability. Thus it is a complex activity that needs to

satisfy a host of constraints and meet a number of objectives. In

addition many different kinds of wastewater management alternatives may

be available. For these reasons the use of multicriterion decision

making (MCDM) techniques to select the most appropriate treatment

alternative would he necessary. In addition, the use of many kinds of

MCDM techniques may enhance the selection process. There is likely to

he more confidence in a wastewater management system which is preferred

using many techniques rather than one. This is one of the reasons for

using the six techniques in this study. Another one is to determine the

acceptability of each of the techniques to select appropriate wastewater

management scheme from a number of competing ones.

6.1. Summary

The problem of wastewater management was carefully defined in

terms of system components such as objectives, specifications, criteria,

criterion scores and alternative actions in accordance with the

112

113

multicriterion "optimization" approach. The acquired data was evaluated

and then integrated into the system as expected criterion scores for

each alternative action. As such an evaluation matrix of alternatives

versus criteria array was constructed. The matrix consisting of both

ordinal and cardinal data was mapped into a wholly quantitative matrix,

the payoff matrix, to make it globally suitable for application of all

six techniques. The elemental values of the payoff matrix are based on

a seven point scale division of the range between the maximum and

minimum scale values assigned each criterion (Table 6) and the position

in tnat range the corresponding value in the evaluation matrix takes.

The mathematical procedure of each MCDM technique was arranged

into a computer algorithm to compute separately a solution to the

problem. ELECTRE I and ELECTRE II methods directly produce,

respectively partial ordering and complete ordering of the nondominated

alternatives; in CP, MCQA I, MCQA II and CGT, however, alternatives are

ranked according to their distances from a reference point. In the

first three cases, the reference point is an ideal point, and the

minimum the distance of an alternative from the ideal point, the higher

(better) its rank would be. In the case of CGT, on the other hand, the

reference point is usually the lowest point in the feasible region known

as the status quo point, and rankings are assigned highest to lowest in

accordance with their maximum distance from that point.

Sensitivity analyses were made on each model except cooperative

game theory to test their robustness with respect to changes in

criterion weights and other model-specific parameters. The results show

114

that some of the models are not quite insensitive to parameter changes.

The preferred alternatives in ELECTRE I and MCQA, for example, changed

with the (p,q) and slicing parameter values respectively, while a slight

exchange on the ranking order with respect to criterion weight was

observed in both ELECTRE II and compromise programming. Some

sensitivities due to changes in the p values of the L p-metrics were also

observed in compromise programming, MCQA I and MCQA II. In spite of

these phenomena, however, the techniques, in general, appear to he

consistent in selecting the most and least preferred alternative

wastewater treatment schemes. The most preferred ones are alternatives

10 and 13 both of which uses facultative lagoons for their primary and

secondary treatment levels hut vary widely in their advanced treatment

stages. Alternative 10, includes filtration algae removal and nutrient

removal facilities while alternative 13 uses land application to have

tertiary level of wastewater treatment performances. The least

preferred alternatives on the other hand, are determined to he 3 and 5

as shown in the last column of Table 21. Alternative 3 is oxidation

ditches while alternative 5 is aerated lagoons plus chemical algae

removal facilities. As shown in Table 21 all alternatives except MCQA I

and cooperative game theory have alternative 3 as the least preferred

one while MCQA I determined this to be alternative 5. Cooperative game

theory, on the other hand did not differentiate among the less preferred

nine alternatives (Figure 16).

115

Table 21. Effects of Model Sensitivity Analysis with Respect toParameters and Model-Wise Preferred Alternatives.

Parameters Used MostPrefer-red

LeastPrefer-red

Alter- Weight Slicing p-value Concor- Discor- alter- alter-native Para-

meterof L n-metrics

dance dance native native

ELECTRE I -- NA NA VA VA 10 3ELECTRE II SA NA NA R R 10 3MCQA I VA R MA NA 13 5MCQA II -- VA SA NA MA 13 3CP MA NA SA NA NA 10 3CGT NA NA NA NA 10 .rr

Note: NA = Not applicableSA = Slightly affectedVA = Very much affected

R = Robust (not affected)MA = Moderately affected

= No sensitivity analysiswas done

6.2. Conclusions

The following concluding remarks can be made concerning the

lessons learned in this study:

1. Multicriterion analysis makes it possible to study the

systematically the complex relationships among the basic components of a

problem which can be described in terms of a set of criteria and

alternative schemes. In this case study, a multicriterion formulation

of the problem is provided to make the problem suitable for application

of mathematical MCDM techniques. The techniques in return are used to

reduce the set of criteria into few indices to obtain a preference

ordering of the alternative schemes. The structural relationship

between criteria and alternative actions has proven to be the essential

116

stage upon which the six multicriterion decision making techniques were

applied in order to yield individual solutions to the problem under

consideration.

2. With respect to data input type, both CP and CGT require a

cardinal scale, while ELECTRE I, ELECTRE II, MCQA I and MCQA II can he

used to analyze discrete MCDM problems with non-commensurable multiple

ordinal criteria.

3. A comparison of the results obtained showed that all six

techniques can be conveniently used to determine the preference ordering

of a competing finite number of alternative schemes in the wastewater

management problem, even though these techniques may require different

procedures to accomplish the same task. ELECTRE I and ELECTRE II use

pairwise comparisons among alternatives to rank them; compromise

programming incorporates preferences under the form of weights to

determine solutions in terms of L distances; MCQA I and MCQA II

combines q-connectivity, and outranking relationships among alternatives

to get solutions in L p distance form, while cooperative game theory uses

geometric distance as its objective function to arrive at the solution.

4. In ELECTRE I, ELECTRE II, CP, MCQA I and MCQA II, the DM

can choose the weights, specify the value of the metric parameter p in

the last three techniques, select the threshold values of p and q in the

first two, and provide the slicing parameter vector s(k) in the last two

techniques. In CGT, however, the optimum solution is uniquely

determined once the DM has accepted the axioms and chosen the °status

quo' point.

117

5. Table 21 describes the effects of a limited sensitivity

analyses on the results of the problem considered with respect to

parameter and weight changes. In general, except for MCQA I and MCQA II

with respect to slicing parameter changes, and for ELECTRE I with

respect to changes in the threshold parameters p and q values, the

techniques appear to be fairly robust. Similar conclusions were made in

previous studies for ELECTRE I and ELECTRE II in Gershon et al. (1982),

Duckstein and Gershon (1983), for MCQA I and MCQA II in Hiessl et al.

(1985) and for compromise programming in Duckstein and Opricovic (1980)

and Tecle and Fogel (1986).

6. Given the criterion set used in this study (Table 3 and

Table 5) the most preferred wastewater management schemes from the

viewpoint of all six techniques are alternatives 10 and 13 (see last

column of Table 21). According to ELECTRE I, ELECTRE II, CP and CGT,

alternative 10, which consists of facultative lagoons with filtration

algae removal and nutrient removal facilities, is the most preferred.

Alternative 13, consistiny of facultative lagoons and land application

activities, on the other hand, is preferred by both MCQA I and MCQA II

techniques.

7. A choice between the above two alternatives or a compromise

between them will depend on the tradeoffs the DM is willing to make as

described at the end of section 5.2.2.

To end the concluding remarks, it is possible to formulate a

complex problem with non-commensurable, discrete objectives in ways

suitable for application of different types of MCDM techniques.

118

Furthermore it can safely be argued that arriving at the same solution

using different techniques not only proves the applicability of each

technique to the problem under consideration but also enhances the

credibility of the final solution at least from the analyst's point of

view. In this study all six MCDM techniques utilized were consistent in

selecting the two top-ranked alternative treatment schemes. These

results conform to alternatives preferred in previous Engineering

studies (Appendix G).

APPENDIX A

DESIGN PARAMETERS OF THE EXISTINGNOGALES INTERNATIONAL WASTEWATER TREATMENT PLANT

Design year, population served, average daily flow, andBOD per capita, are in accordance with International Boundaryand Water Commission Minute 227, dated September 5, 1967.

1. Design year2. Population served3. Average daily flow

4. Average daily flow5. Peak flow rate (plant effluent)6. Ratio of peak flow rate to

average daily flow (planteffluent)

7. Raw sewage BOD8. Raw sewage BOD9. Raw sewage suspended solids10. Raw sewage temperature range

PLANT CCMPONENTS:

1. Aerated lagoonsDesign flaw, averageDetention time design flowVolume of lagoonsNumber of lagoonsVolume each lagoonDesign water depthNominal surface area per lagoonApplied BOD at design flowApplied BOD at design flowApplied BOD at design flowDesign BOD removalMinimum D. O. concentrationin aerobic layerMethod of aerationNumber of aeratorsFreeboardSide slopesBenn width at ton

1980102,0008.20 mgd (12.68 cfs)

(25.17 a-f/day)80 gallons/capita/day10.66 mgd (16.49 cfs)

130%0.17 pounds/capita/day250 ppm250 ppm60°F. to 75°F. (15°C. to 24°C.)

8.20 mgd5 days126 acre feet263 acre feet10 feet7 acres17,340 lbs/day1,240 lbs/ao/daY3.16 lbs/1,000 cu.ft./day13,870 lbs/day

2 ppmSurface aerators, floating typeEight, each 60 hp3 feet3:115 feet nun.. 100 feet max.

119

120

2. Stabilization pondsDesign flowApplied BOD at design flow

Applied BOD at design flowApplied BOD at design flowTotal surface area of pondsDesign water depthVolume of ponds at 3 feet depthVolume of ponds at 5 feet depthDetention time at design flawDetention time at design flowNumber of pondsSurface area, each pondPond flow arrangementNUmber of ponds first stagePond surface area first stageApplied BOD to first stage at

design flowNumber of ponds second stagePond surface area second stageFreeboardSide slopesBerm width at top

3. Chlorine contact basinDesign flowPeak flaw rateNumber of basinsDetention time at peak flow rateDesign chlorine residual at peak

flowVolume of basinDepthWidthLengthForm of Chlorine appliedPoint of application

4. Chlorination facilitiesNumber of ChlorinatorsMaximum rated capacity each

chlorinatorMaximum dosage rate per

chlorinator at peak flowrate (10.66 mgd)

Nominal dosage rate anticipatedfor 1 ppm residual in effluent

8.20 mgd50 lbs/acre/day

20% of raw sewage BOD3,470 lbs/day69 acres3 feet nominal (5 feet maximum)207 acre feet345 acre feet8 days (at 3 feet depth)14 days (at 5 feet depth)323 acres2 stage (series flow)246 acres

75 lbs/acre/day123 acres3 feet3:115 feet min., 100 feet max.

8.20 mgd10.66 mgd (16.49 cfs)115 minutes

1 ppm14,850 c.f.6 feet30 feet82.5 feetGas in solutionDiffuser at basin

2 (one active, one standby)

2,000 lbs/day

22.5 ppm

7.5 to 10 ppm

5. Inlet flow measuring facilitiesFlow measuring deviceSize of throatInstrunentation

Location of receiver

6. Outlet flow measurement facilitiesFlow measuring deviceSize of throatInstrumentation

Location of receiver

Parshall flume2 feetTransmitter, with receiver to

totalize, indicate, and recordControl and Maintenance Building

Parshall flume2 feetTransmitter, to chlorinators for

proportioning chlorine dosageto effluent. Receiver to totalize,indicate, and record

Chlorinator Building

121

NOTE: From Carobo report - 1979.

APPENDIX B

ESTIMATED COST CRITERIA FOR EACH CONSIDERED ALTERNATIVE WASTEWATERTREATMENT SCHEME (IN 1988 DOLLARS)

Total Capital Capital AnnualAlternative Capital Annual Owning Cost Owning

Scheme Cost and Oand M Cost /1000 gals. /1000 gals.

Al 2.250x106 0.860x106 0.54 0.21A2 2.560x106 1.530x106 0.61 0.37A3 7.710x106 1.580x106 1.85 0.38A4A5

3.360x106 1.174x106 0.80

3.660x106 1.804x10 6 0.870.280.44

A6 2.900x106 0.984x10 6 0.69 0.24A7 3.194x106 1.615x10 6 0.77 0.39A8 6.302x106 1.850x106 1.51 0.45A9 6.600x106 2.540x106 1.58 0.61A10 5.540x106 1.670x10 6 1.40 0.40All 6.140x106 2.285x106 1.47 0.55Al2 8.140x106 2.010x10 6 1.95 0.48A 1 3 3.940x106 1.070x106 0.94 0.26A 14 4.330x106 1.740x10 6 1.01 0.42A1 5 9.380x106 1.790x106 2.25 0.43

Note. Based on Information in Arthur Beard Engineers, Inc. (1984).

122

APPENDIX C

OPTIMAL (P,Q) VALUE DETERMINATION AND SENSITIVITY OF ELECTRE IWITH RESPECT TO CHANGES IN (P,Q) VALUES

Threshold Parameters Kernel

.3 .1 6,10,13,15

.3 .2 4,5,6,7,8,9,10,11,13,14

.3 .3 all hut 3

.4 .1 6,10,13,15

.4 .2 4,5,6,7,8,9,10,11,13,14

.4 .3 all hut 3

.4 .4 all hut 3

.5 .1 6,10,13,15

.5 .2 4,5,6,7,8,9,10,11,13,14

.5 .3 all hut 3

.5 .4 all hut 3

.5 .5 0

.6 .1 6,10,13,15

.6 .2 4,5,6,7,10,11,13,14

.6 .3 10,11

.6 .4 8,9,10,11,12,13,14,15

.7 .1 6,10,13,15

.7 .2 6,10,11,13

.7 .3 10,11

.7 .4 10,11

.8 .1 6,10,13,15

.8 .2 6,10,13,15

.8 .3 6,10,13

.8 .4 10,13

.8 .5 10,13

.8 .6 10,13

.9 .1 3,6,10,12,13,15

.9 .2 6,10,12,13,15

.9 .3 6,10,12,13,15

.9 .4 6,10,12,13,15

.9 .5 6,10,12,13,15

Lower p - Willing to accept less preferred alternatives.Higher q - Willing to accept alternatives at higher discomfort

or dissatisfaction.

123

APPENDIX D

ILLUSTRATION OF THE STEP BY STEPRANKING PROCEDURE IN ELECTRE II

This illustration demonstrates the derivation of the actual

alternative rankings shown in column 2 of Table 14. To carry out this

procedure, the strong and weak graphs of Figures 10 and 11, respectively

must be a priori determined.

A. Forward Ranking

Initial Valuek=1, Y(1)=G5

a. Iteration One.(1) C=[10,13] (nodes in G, without precedent)(2) U=[13] (nodes in C reated through Rw )(3) B=[10] (nodes in U without precedent in Gw )(4) A(k)=A(1)=(C- U) UB=[10](5) Ranking: v'(10)=1(6) k=k+1=1+1=2(7) Y(k)=Y(2)=Y(1)-A(1)=[1,2,3,4,5,X1,11,13,14,15]Af

Hence continue to iteration two.

b. Iteration Two.(1) C=[11,13](2) U=[13](3) B=[11](4) A(k)=A(2)=(C-U)U B=[11](5) Ranking: v'(11)=2(6) k=k+1=2+1=3(7) Y(k)=Y(3)=Y(2)-A(2)=[1,2,3,4,5,X1,13,14,15]Af

Hence continue to iteration three.

c. Iteration Three.(1) C=[13,X1](2) U=[X1](3) B=[13](4) A(k)=A(3)=(C-U) UB= [ 13](5) Ranking: v'(13)=3

124

(6) k=k+1=3+1=4(7) Y(k)=Y(4)=Y(3)-A(3)=[1,2,3,4,5,X1,14,15]#0

Hence continue to iteration four.

d. Iteration Four.(1) C=[14,X1](2) U=[Xi](3) B=[14](4) A(k)=A(4)=(C-U)U B=[14](5) Ranking: v'(14)=4(6) k=k+1=4+1=5(7) Y(k)=Y(5)=Y(4)-A(4)=[1,2,3,4,5,X1,15]A4

Hence continue to iteration five.

e. Iteration Five.(1) C=[15,X1](2) U= [ 15](3) B=[X1](4) A(k)=A(5)=(C-U)UB=[X1 X1=6,7,8,9,12](5) Ranking: vi(XI)=V 1 (6)=V i (7)=v 1 (8)=v l (9)=v'(12)=5(6) k=k+1=5+1=6(7) Y(k)=Y(6)=Y(5)-A(5)=[1,2,3,4,5,15 ]N

Hence continue to iteration six.

f. Iteration Six.(1) C=[4,15](2) U= [ 15](3) B=[4](4) A(k)=A(6)=(C-U)UB=[4](5) Ranking: v'(4)=6(6) k=k+1=6+1=7(7) Y(k)=Y(7)=Y(6)-A(6)=[1,2,3,5,15] 3 if

Hence continue to iteration seven.

9. Iteration Seven.(1) C=[5,15](2) U=[01(3) B=[5,15](4) A(k)=A(7)=(C -U)UB= [ 5,15](5) Ranking: v'(5)=v 1 (15)=7(6) k=k+1=7+1=8(7) Y(k)=Y(8)=Y(7)-A(7)=[1,2,3]Aii

Hence continue to iteration eight

h. Iteration Eight.(1) C=[2,3](2) U=[3](3) B=[2](4) A(k)=A(8)=(C-U)U B=[2](5) Ranking: v'(2)=8

125

(6) k=k+1=8+1=9(7) Y(k)=Y(9)=Y(8)-A(8)=[1,3] � g

Hence continue to iteration nine.

i. Iteration Nine.(1) C=[1,3](2) U=[3,3](3) B=[1](4) A(k)=A(9)=(C -U) U B=[1](5) Ranking: v'(1)=9(6) k=k+1=9+1=10(7) Y(k)=Y(10)=Y(9)-A(9)= [3 ]gi

Hence continue to iteration ten.

j. Iteration ten.(1) C= [3](2) U=[ø](3) B=[3](4) A(k)=A(10)=(C-U) U B=[3](5) Ranking: v'(3)=10(6) k=k+1=10+1=11(7) Y(k)=Y(11)=Y(10)-A(10)=0

Since Y(k)=[0], forward ranking iteration stops.

B. Reverse Ranking

In determining the reverse ranking of the alternatives, the

following steps are followed (Duckstein and Gershon, 1983):

(1) Reverse the direction of the arcs in the Gs and Gw figures of 10

and 11 respectively. This is tantamount to transposing both the

strong (R s) and the weak (Rw) relationship matrices.

(2) Using the same procedure as in the forward ranking v', the

following ranking a(x) is obtained:

Node 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ranking a(x) 2 3 1 5 4 6 6 6 6 10 9 6 8 7 2

126

127

(3) Then, the ranking a(x) is reversed using equation (13), with

amax=1°' to obtain v"(x) such that

Node 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

v"(x) 9 8 10 6 7 5 5 5 5 1 2 5 3 4 9

C. Average Ranking

The final ranking of the alternatives is then deterined using

equation (14) as shown below:

Node 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

vl(x) 9 8 10 6 7 5 5 5 5 1 2 5 3 4 7

v"(x) 9 8 10 6 7 5 5 5 5 1 2 5 3 4 9

7(x) 9 8 10 6 7 5 5 5 5 1 2 5 3 4 8

APPENDIX E

SENSITIVITY ANALYSIS OF MCQA WITH RESPECT TOCHANGES OF SCALAR SLICING PARAMETER S(K)

slicinglevel

q-dimensionvector

Q-structurevector

highest y-levelchoice

0.95 (5 4 3 2 1 0 -1) (2 2 4 1 1 1 1) (10,13)0.90 (5 4 3 2 1 0 -1) (2 2 4 1 1 1 1) (10,13)0.85 (5 4 3 2 1 0 -1) (2 2 4 1 1 11) (10,13)0.80 (8 7 6 5 4 3 2 1 0) (1 2 2 1 1 1 1 2 1) (13)0.75 (8 7 6 5 4 3 2 1) (1 2 2 1 1 1 1 2) (13)0.70 (8 7 6 5 4 3 2 1) (1 2 2 1 1 1 1 2) (13)0.65 (8 7 6 5 4 3) (1 2 2 3 1 2) (13)0.60 (8 7 6 5 4 3) (1 3 3 3 1 1) (13)0.55 (8 7 6 5 4 3) (1 3 3 3 1 1) (13)0.50 (8 7 6 5 4 3) (3 3 1 3 1 1) (7,10,13)0.45 (8 7 6 5 4 3) (3 3 1 3 11) (7,10,13)0.40 (10 9 8 7 6 5) (2 1 1 11 1) (6,7,10,11)0.35 (10 9 8 7 6 5) (2 1 1 1 1 1) (6,7,10,11)0.30 (11 10 9 8 7 6 5) (1 1 1 1 1 1 1) (6,7,10)0.25 (11 10 9 8 7 6 5) (1 1 1 11 11) (6,7,10)0.20 (11 10 9 8 7) (11 11 1) (4,5,6,7,8,10)0.15 (11 10 9 8 7) (1 1 1 1 1) (4,5,6,7,8,10)0.10 (11 10 9 8 7) (11 1 1 1) (4,5,6,7,8,10)0.05 (11 10 9 8 7) (1 1 1 11) (4,5,6,7,8,10)

128

APPENDIX F

RESULTS OF MCQA I AND MCQA II IN TERMS OF DISTANCEFROM THEIR RESPECTIVE REFERENCE POINTS OF (1,1) AND (1,1,1)

A. Distance from point (1,1,1) with p = 1.0:

PROJ. PSI PCI PDI PRI-I PRI-II

1 3.700 2.700 1.800 1.307 1.4202 3.388 .000 .000 1.506 1.5063 1.908 3.550 16.050 1.520 2.5204 3.863 .000 .000 1.437 1.4375 3.139 .000 3.500 1.543 1.7616 4.845 .000 .000 1.294 1.2947 3.661 2.100 .000 1.347 1.3478 4.870 .000 .000 1.290 1.2909 4.265 .000 4.900 1.378 1.684

10 6.862 8.150 .000 .537 .53711 6.005 .000 2.100 1.125 1.25612 4.474 4.800 7.550 1.075 1.54613 6.567 17.600 .000 .043 .04314 5.074 .000 3.500 1.261 1.47915 3.958 2.750 9.450 1.267 1.856

129

B. Distance from point (1,1,1) with p = 2.0:

PROJ. PSI PCI PDI PRI-I PRI-II

1 3.700 2.700 1.800 .964 .9702 3.388 .000 .000 1.121 1.1213 1.908 3.550 16.050 1.076 1.4694 3.863 .000 .000 1.091 1.0915 3.139 .000 3.500 1.138 1.1586 4.845 .000 .000 1.042 1.0427 3.661 2.100 .000 .997 .9978 4.870 .000 .000 1.041 1.0419 4.265 .000 4.900 1.089 1.112

10 6.862 8.150 .000 .537 .53711 6.005 .000 2.100 1.008 1.01612 4.474 4.800 7.550 .806 .93313 6.567 17.600 .000 .043 .04314 5.074 .000 3.500 1.033 1.05615 3.958 2.750 9.450 .944 1.113

C. Distance from point (1,1,1) with p = 3.0:

PROJ. PSI PCI PDI PRI-I PRI-II

1 3.700 2.700 1.800 .890 .8902 3.388 .000 .000 1.042 1.0423 1.908 3.550 16.050 .960 1.2354 3.863 .000 .000 1.027 1.0275 3.139 .000 3.500 1.051 1.0546 4.845 .000 .000 1.008 1.0087 3.661 2.100 .000 .922 .9228 4.870 .000 .000 1.008 1.0089 4.265 .000 4.900 1.018 1.027

10 6.862 8.150 .000 .537 .53711 6.005 .000 2.100 1.001 1.00112 4.474 4.800 7.550 .753 .81013 6.567 17.600 .000 .043 .04314 5.074 .000 3.500 1.006 1.00915 3.958 2.750 9.450 .878 .959

130

aiO

131

.-4

C •))

T-4

Cn

Cy)

(r)

Cy"

CY)

Cv)

g-1

Cn1

CL)> 0..-- t/) QV) 0+-) 0 •r- 0 -I-- 4L-r0 0 .00

4-) 0 00 MI 4-,

01 S- CT) -0 •,-= co CU 0:5 •r•• CIC.) -J cc _J XMS 0LI-

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