Feb 2, 2007 - Table 2.4.1 - Water quality standards for different water-uses*..................................... 213 ... Table 4.1.10 – Regional compatibility index and plan ranks . ...... By considering the above notations, a decision support system for compatibility ...... 4 New Brunswick 0.922 0.570 0.680 0.526 0 0.183 1.821 4.702.
DEVELOPMENT OF FUZZY MULTIPLE-ATTRIBUTE DECISION AID METHODOLOGIES FOR ENERGY-ENVIRONMENTAL POLICY ANALYSIS AND ASSESSMENT
A Thesis
Submitted to the Faculty of Graduate Studies and Research in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Engineering University of Regina
by Fuzhan Nasiri
Regina, Saskatchewan February 2007
Copyright 2007: F. Nasiri
UNIVERSITY OF REGINA FACULTY OF GRADUATE STUDIES AND RESEARCH SUPERVISORY AND EXAMINING COMMITTEE Fuzhan Nasiri, candidate for the degree of Doctor of Philosophy, has presented a dissertation titled, Development of Fuzzy Multiple-Attribute Decision Aid Methodologies for Energy-Environmental Policy Analysis and Assessment, in an oral examination held on February 02, 2007. The following committee members have found the dissertation acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material.
External Examiner:
Dr. Liping Fang, Ryerson University
Supervisor:
Dr. G. H. Huang, Faculty of Engineering
Committee Member:
Dr. Mehran Mehrandezh, Faculty of Engineering
Committee Member:
Dr. JingTao Yao, Department of Computer Science
Committee Member:
Dr. Adisorn Aroonwilas, Faculty of Engineering
Chair of Defense:
Dr. Dongyan Blachford, Faculty of Graduate Studies and Research
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Abstract This dissertation introduces innovative decision aid methodologies for energyenvironmental policy analysis, based on theory of multiple-attribute decision-making and fuzzy set theory. A number of challenging areas of energy-environmental policy analysis; groundwater remediation, surface water quality, waste recycling, electricity generation and transportation planning, will be the focus of this dissertation. In the groundwater remediation policy area, emphasis is placed on compatibility assessment. The purpose of such an analysis is to identify the most compatible remediation plan for a contaminated site. In this thesis, a decision support model is designed for the prioritization of remediation plans based on their computed compatibility index. Then, factorial, regional and overall compatibility indicators are computed; by comparing the generated indicators, remediation policies are prioritized. In the transportation planning policy area, the study is built on the need to incorporate environmental consequences of transportation activities in planning, to come up with a model capable of considering environmental requirements, despite that the major objective within the planning infrastructure is to minimize the cost of transportation. In doing so, a multiple-objective optimization model with cost and environmental objectives has been developed to obtain policy-based planning solutions. With the implementation of a post-optimization assessment, the optimality of each solution will be evaluated based on decision-makers’ interests, and local and global environmental requirements. This process will determine the preferred strategy of transportation. A decision-making framework in waste recycling policy analysis through performance assessment is also presented. Two categories of performance indicators, efficiency and
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effectiveness, are developed. Aggregation of these indicators, in relation to their associated criticalities, will provide a number of environmental performance indices to represent the status of the environmental performance. In the policy analysis through water quality assessment, a comprehensive, but easy to use tool for the assessment and evaluation of water quality policies is developed based on the concept of water quality index. Considering the complexities involved, to get such an index, there is a need for a methodology to not only structure and identify information relevant to the problem, but also help users reach a decision. In this regard, a fuzzy multiple-attribute decision aid is proposed, to provide an outline for prioritization of alternative plans, on the basis of their resulted improvements in water quality index. This dissertation also proposes an integrated model for capacity planning in electricity generation considering a number of energy and environmental target policies. The model utilizes a multiple-criteria linear programming to incorporate cost and environmental objectives into the planning. Optimization of this model provides different planning scenarios. To determine the best compromise plan, a post-optimization assessment based on concept of fuzzy memberships is then developed. To provide sufficient background for constructed methodologies, a comprehensive literature review on both multiple-attribute decision analysis and fuzzy set theory are presented. This will be accompanied by further review on contributions in using fuzzybased multiple-attribute decision analysis techniques as well as review and background on energy-environmental policy analysis areas.
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Acknowledgements I would like to express my gratitude to Dr. Gordon Huang, my supervisor, whose invaluable guidance and support has provided me with a unique opportunity to pursue my PhD studies. I am very much grateful to Dr. Gerald Fuller and Dr. Norma Fuller for their quality input, technical guidance, and kind help. Special thanks are reserved for my friends at the Environmental Informatics Laboratory, Dr. Imran Maqsood, Ms. Anastassia Manuilova, Mr. Hengliang Li, Dr. Bing Chen, and Dr. Yuefei Huang, and many others at the institution for their collaboration, assistance and support. I would like to extend my thanks to Dr. Mehran Merandezh, Dr. Jing Tao Yao and Dr. Andy Aroonwilas for serving on my supervisory committee, and providing careful suggestions that were helpful in improving this dissertation. I am also very thankful to the anonymous reviewers of my peer-reviewed journal papers for their very insightful comments and suggestions that were very helpful in improving the presentation of my research in this dissertation. I thankfully acknowledge the Faculty of Graduate Studies and Research and the Faculty of Engineering at the University of Regina, for providing research scholarships, teaching assistantships, and research awards during the course of my Ph.D. studies.
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Dedication I am very grateful to my parents and my only brother that their support and understanding have made this journey possible. I would especially like to present this dissertation to my wife and lover, Fereshteh, whose presence in my life is the reason behind my achievements.
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Table of Contents
Abstract .............................................................................................................................. iii Acknowledgements............................................................................................................. v Dedication .......................................................................................................................... vi List of Figures ................................................................................................................... xii List of Tables ................................................................................................................... xiv List of Tables ................................................................................................................... xiv CHAPTER 1
– INTRODUCTION............................................................................ 1
CHAPTER 2
– LITERATURE REVIEW ............................................................... 8
2.1.
Multiple-Attribute Decision-Making (MADM) ................................................. 8
2.2.
Classical MADM Methods ............................................................................... 10
2.2.1.
Dominance Method............................................................................... 10
2.2.2.
Max-Min Method .................................................................................. 11
2.2.3.
Max-Max Method.................................................................................. 11
2.2.4.
Conjunctive Method .............................................................................. 12
2.2.5.
Disjunctive Method ............................................................................... 12
2.2.6.
Lexicographic Method .......................................................................... 13
2.2.7.
Linear Assignment Method ................................................................... 13
2.2.8.
Simple Additive Weighting Method....................................................... 14
2.2.9.
TOPSIS.................................................................................................. 15
2.2.10.
Weighted Product Method .................................................................... 17
2.2.11.
Distance from Target Method............................................................... 18
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2.3.
Fuzzy Multiple-Attribute Decision-Making ..................................................... 19
2.3.1.
Fuzzy set theory..................................................................................... 19
2.3.2.
Decision-making in a Fuzzy Environment ............................................ 25
2.3.3.
Fuzzy MADM Methods ......................................................................... 26
2.3.3.1.
Fuzzy Simple Additive Weighting Method ........................................ 27
2.3.3.1.1. Baas & Kwakernaak’s Approach................................................... 28 2.3.3.1.2. Kwakernaak’s Approach................................................................ 30 2.3.3.1.3. Dubois & Prade’s Approach .......................................................... 32 2.3.3.1.4. Cheng & McInnis’s Approach ....................................................... 34 2.3.3.1.5. Bonissone’s Approach ................................................................... 35 2.3.3.2.
Analytic Hierarchy Process (AHP)..................................................... 37
2.3.3.3.
Fuzzy Max-Min Method ..................................................................... 40
2.3.3.4.
Fuzzy TOPSIS Method ....................................................................... 41
2.3.3.5.
Fuzzy Rank Ordering Methods........................................................... 45
2.3.3.5.1. Distribution-based Approaches...................................................... 45 2.3.3.5.1.1. Centroid Value ........................................................................ 45 2.3.3.5.1.2. Mean-Spread Values ............................................................... 46 2.3.3.5.1.3. Preference Degree ................................................................... 47 2.3.3.5.1.4. Fuzzy Reasoning Method ....................................................... 48 2.3.3.5.1.5. Haming Distance..................................................................... 49 2.3.3.5.1.6. Left-Right Scores .................................................................... 51 2.3.3.5.2. α-Cut-based Approaches................................................................ 53 2.3.3.5.2.1. Upper Point Method................................................................ 53
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2.4.
2.3.3.5.2.2.
Dominance Index .................................................................... 54
2.3.3.5.2.3.
Height of Fuzzy Set Method ................................................... 57
2.3.3.5.2.4.
Degree of Optimism Method .................................................. 58
Energy-Environmental
Policy
Analysis
and
Assessment:
Areas
and
Foundations..................................................................................................... 59 2.4.1.
Groundwater Remediation Policy Area: Compatibility Assessment .... 59
2.4.2.
Transportation Policy Area: Environmental Optimality Assessment... 61
2.4.3.
Waste Recycling Policy Area: Performance Assessment...................... 64
2.4.4.
Surface Water Policy Area: Water Quality Assessment ....................... 68
2.4.5.
Electricity Generation Policy Area: Capacity Planning EnergyEnvironmental Assessment.................................................................. 73
CHAPTER 3 3.1.
- METHODOLOGY ......................................................................... 78
Groundwater Remediation Policy Area: Compatibility Assessment................ 78
3.1.1.
Effect Sets.............................................................................................. 79
3.1.2.
Joint Effect Sets..................................................................................... 80
3.1.3.
Regional Compatibility ......................................................................... 83
3.1.4.
Overall Compatibility ........................................................................... 85
3.2.
Transportation Policy Area: Environmental Optimality Assessment............... 87
3.2.1.
Upper and Lower Solutions .................................................................. 89
3.2.2.
Target Solutions .................................................................................... 90
3.2.3.
Optimality Assessment .......................................................................... 92
3.3.
Waste Recycling Policy Area: Performance Assessment................................. 97
3.3.1.
Environmental Benefits......................................................................... 97
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3.3.2.
Performance Indicators ........................................................................ 98
3.3.3.
Performance Scenarios and Indices ..................................................... 99
3.3.4.
Performance Assessment .................................................................... 102
3.4.
Surface Water Policy Area: Water Quality Assessment................................. 105
3.4.1.
Quality Factors ................................................................................... 105
3.4.2.
Quality Indicators and Quality Attributes .......................................... 105
3.4.3.
Scoring Process .................................................................................. 107
3.4.4.
Importance of Factors......................................................................... 107
3.4.5.
Reach Quality Index............................................................................ 109
3.4.6.
Prioritizing per Reach......................................................................... 109
3.4.7.
Reach Importance Degree .................................................................. 110
3.4.8.
River Quality Index ............................................................................. 111
3.4.9.
Prioritizing per River Quality............................................................. 112
3.5.
Electricity Generation Policy Area: Capacity Planning Energy-Environmental Assessment.................................................................................................... 113
3.5.1.
Optimization........................................................................................ 116
3.5.2.
Post-optimization Assessment............................................................. 118
CHAPTER 4
– CASE STUDY .............................................................................. 122
4.1.
Groundwater Remediation Policy Area: Compatibility Assessment.............. 122
4.2.
Transportation Policy Area: Environmental Optimality Assessment............. 127
4.3.
Waste Recycling Policy Area: Performance Assessment............................... 131
4.4.
Surface Water Policy Area: Water Quality Assessment................................. 136
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4.5.
Electricity Generation Policy Area: Capacity Planning Energy-Environmental Assessment.................................................................................................... 141
CHAPTER 5
– CONCLUSIONS .......................................................................... 147
5.1.
Discussion....................................................................................................... 147
5.2.
Summary of Achievements............................................................................. 154
5.3.
Recommendation for Future Research ........................................................... 157
References....................................................................................................................... 159 Figures............................................................................................................................. 184 Tables.............................................................................................................................. 211
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List of Figures
Figure 2.3.1 – Triangular Fuzzy Set ............................................................................... 185 Figure 2.3.2 – Trapezoidal Fuzzy Set ............................................................................. 186 Figure 2.3.3 – Fuzzy-Crisp Approach............................................................................. 187 Figure 2.3.4 – Fuzzy-Fuzzy Approach ........................................................................... 188
~ Figure 2.3.5 – Estimation of membership distribution for U i ....................................... 189 Figure 2.3.6 – Bonissone trapezoidal fuzzy number....................................................... 190 Figure 2.3.7 – An h-level decision-making problem ...................................................... 191 Figure 3.1.1 – Compatibility assessment decision support system................................. 192 Figure 3.2.1 – Optimality assessment decision aid for transportation planning............. 193 Figure 3.3.1 – Environmental performance assessment decision aid ............................. 194 Figure 3.3.2 – Membership distributions for fuzzy weights........................................... 195 Figure 3.4.1 - Fuzzy river-pollution decision support expert system ............................. 196 Figure 3.5.1 - Integrated capacity planning decision aid model ..................................... 197 Figure 4.1.1 - Remediation regions and contamination sources ..................................... 198 Figure 4.1.2 – Fuzzy distributions for effect linguistics ................................................. 199 Figure 4.1.3 – Fuzzy distributions for weights ............................................................... 200 ~
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Figure 4.1.4 - Joint effect sets for (a) set time JE i ,1 ( r ) and (b) durability JE i , 2 ( r ) ............ 201 ~
Figure 4.1.5 – Regional compatibility (score) sets COM i (r ) .......................................... 202 ~
Figure 4.1.6 – Overall compatibility (score) sets COM i ( r ) ............................................. 203 Figure 4.2.1 – Feasible transportation network for packed products.............................. 204
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Figure 4.2.2 – Feasible transportation network for bulk products.................................. 205 Figure 4.3.1 – Efficiency and Effectiveness indices....................................................... 206 Figure 4.4.1 - Membership distributions for quality attributes ....................................... 207 Figure 4.4.2 - Membership distributions for importance degrees................................... 208 ~
Figure 4.4.3 - Reach quality index set QI r and the optimal index QIr * ......................... 209 ~
Figure 4.4.4 - Reach weight sets Wr ............................................................................... 210
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List of Tables Table 2.3.1 – A multiple-attribute decision-making table with fuzzy values................. 212 Table 2.4.1 - Water quality standards for different water-uses*..................................... 213 Table 3.3.1 – Importance of environmental benefits ...................................................... 214 Table 3.3.2 –Waste recycling environmental performance indicators ........................... 215 Table 3.3.3 - α-level cuts for fuzzy sets associated with linguistic terms ...................... 216 Table 4.1.1 – Proposed grouting technologies................................................................ 217 ~
Table 4.1.2 – Contaminant effect sets ( E ni , j ) for n1 = 1, 2, … 10................................... 218 1
Table 4.1.3 – Soil types................................................................................................... 219 ~
Table 4.1.4 – Soil effect sets ( E ni , j ) for n2 = 1, 2, … 9 .................................................. 220 2 Table 4.1.5 – Soil-Contamination distribution in regions............................................... 221 Table 4.1.6 – Definitions of linguistic attributes ............................................................ 222 Table 4.1.7 – Soil-Contaminant joint effect sets JE~i , j ( r ) ............................................... 223 Table 4.1.8 – Joint effect indicators JE i*, j ( r ) and plan ranks per factor........................... 224 Table 4.1.9 – Relative importance of compatibility factors ( W~ j ) .................................. 225 Table 4.1.10 – Regional compatibility index and plan ranks ......................................... 226 Table 4.1.11 – Relative region criticalities ..................................................................... 227 Table 4.1.12 – Overall compatibility index and plan ranking ........................................ 228 Table 4.2.1 – Distance information for feasible routes (km).......................................... 229 Table 4.2.2 – Cost (€/ton) information for feasible routes ( C ijrm ) ................................. 230 Table 4.2.3 – CO2 emission (kg)* information for feasible routes ( E ijrm (1) )................ 231
xiv
Table 4.2.4 – SO2 emission (kg)* information for feasible routes ( E ijrm (2) ).................. 232 Table 4.2.5 – NO x emission (kg)* information for feasible routes ( E ijrm (3) ) ................ 233 Table 4.2.6 – Energy consumption (GJ)* information for feasible routes ( E ijrm (4) )...... 234 Table 4.2.7 – Single objective optimization with upper and lower solutions................. 235 Table 4.2.8 – Policy packages and their target values ( l ep ) ........................................... 236 Table 4.2.9 – Policy-based optimal solutions ( X ijrm ) .................................................... 237 Table 4.2.10 – Policy-based objective values ( f ( p ) ) and the associated λp levels........ 238 ~ Table 4.2.11 – Policy-based optimality solutions ( f * ).................................................. 239
Table 4.2.12 – Local and global weights of objectives ( w ) on a scale of 1 to 5 ............ 240 Table 4.2.13 – Optimality index ( I p ) of policy-based solutions ................................... 241 Table 4.3.1 – Beverage container recycling systems...................................................... 242 Table 4.3.2 – General information of provinces ............................................................. 243 Table 4.3.3 – Provincial beverage container waste generation (tons/year) .................... 244 Table 4.3.4 – Provincial beverage container waste recycling (tons/year) ...................... 245 Table 4.3.5 – Provincial beverage container waste recycling cost ................................. 246 Table 4.3.6 – Average unit weight of containers (gr/unit).............................................. 247 Table 4.3.7 – Environmental benefits of recycled containers......................................... 248 Table 4.3.8 – Annual environmental benefits of BCR programs ................................... 249 Table 4.3.9 – Normalized (dimensionless) performance indicators ............................... 250 Table 4.3.10 – α-level cuts for efficiency index sets (α=0.00) ...................................... 251 Table 4.3.11 – α-level cuts for local effectiveness index sets (α=0.00)......................... 252
xv
Table 4.3.12 – α-level cuts for global effectiveness index sets (α=0.00) ...................... 253 Table 4.3.13 – Pair-wise relations for efficiency indicators ........................................... 254 Table 4.3.14 – Pair-wise relations for effectiveness indicators (local scenario) ............ 255 Table 4.3.15 – Pair-wise relations for effectiveness indicators (global scenario) .......... 256 Table 4.3.16 – Performance ranks .................................................................................. 257 Table 4.4.1 - Measurements of quality factors ............................................................... 258 Table 4.4.2 - Water-uses in river reaches ....................................................................... 259 ~
Table 4.4.3 - Quality attributes assigned by experts ( S i, r )............................................. 260 ~
Table 4.4.4 - Importance of quality factors in each reach ( Wi, r ) ................................... 261 Table 4.4.5 - Representative values for importance degrees .......................................... 262 Table 4.4.6 - Quality score memberships and importance degrees ................................ 263 Table 4.4.7 - Improvements in reach quality index ........................................................ 264 Table 4.4.8 - Plan effectiveness indicator ( + QIr * ).......................................................... 265 Table 4.4.9 - Priorities per reach..................................................................................... 266 Table 4.4.10- Reach weights........................................................................................... 267 Table 4.4.11 - Priorities for the river .............................................................................. 268 Table 4.5.1 – Current electricity generation capacities by source (GW)........................ 269 Table 4.5.2 – Annual demand, peak demand and peak reserve ratio measures.............. 270 Table 4.5.3 – Characteristics of electricity generation sources ...................................... 271 Table 4.5.4 – Scheduled retirement of electricity generation capacity until 2020 (GW) 272 Table 4.5.5 – Potential capacity development for hydro and wind sources (GW) ......... 273 Table 4.5.6 – Possible routes of electricity export.......................................................... 274 Table 4.5.7 – Energy and emission intensity of electricity generation sources.............. 275
xvi
Table 4.5.8 – Energy-emission policies and their associated targets for 2020 ............... 276 Table 4.5.9 – Integrated capacity planning scenarios and objective values for 2020..... 277 Table 4.5.10 – Integrated capacity planning solutions and optimality of associated ..... 278 Table 4.5.11 – Integrated capacity planning solutions and optimality of associated ..... 279 Table 4.5.12 – Preferred planning strategies .................................................................. 280 Table 4.5.13 – Optimal power generation plan (GWh) for 2020.................................... 281 Table 4.5.14 – Optimal capacity development and additional retirement (GW) for 2020 ................................................................................................................................. 282 Table 4.5.15 – Electricity exports (GWh) in optimal scenarios for 2020....................... 283 Table 4.5.16 – Overview of optimal share of power generation sources for 2020......... 284
xvii
Chapter 1 – INTRODUCTION
Stop and think for a moment about our most basic needs for life: Air, without air we would not survive for more than a few minutes. Water, without water our life would be measured in only a few days. Food, without food our survival would be at most a couple of months. The environment supplies all these basic needs. If the air, land or water that sustains us becomes polluted or damaged, our health or survival could be jeopardized. In today’s world, although in many societies there have been major improvements in public health and growing concern for the environment, industrial pollution emissions to the air, water and land continue to increase. As a result, environmental problems, such as deforestation, soil erosion, and the expansion of desert areas, are aggravated. These problems have severely threatened the health of human beings, as well as the global sustainability of resources for future generations. To prevent, control or treat these environmental issues, many environmental protection policies have been proposed at all levels, from the general public to national governments and international agencies, especially in the final decades of the 20th century (Rubin 2001). During this time, there have been major challenges in dealing with the environmental protection policies. First of all, the priorities and preferences for environmental protection always vary from community to community and from country to country. For instance, a nation struggling to provide its citizens with the basic necessities of life is not likely to be as worried about wilderness preservation as a wealthy nation. At different times, even in wealthy societies, environmental protection policies have been assigned different ranks in communities and on the government agenda. 1
Moreover, environmental protection policies are broad and complex issues with some equal impact worldwide, involving not only technical efforts in the area of engineering, but also a host of many different professions and stakeholders, such as city and regional planners, government regulatory agencies, and other special interest groups responsible for the review and approval of policies. This complexity makes the decision-making becomes time-consuming, and on some occasions impossible. Finally, each environmental policy mainly consists of two different levels of activities; policy analysis and policy implementation. Failure in either of them could result in an unsuccessful protection policy. Policy analysis is looking for analytical evaluation of policies in terms of economical, technological and societal factors, and/or environmental risks and impacts. Policy implementation includes actions required to run a policy. This contains policy prerequisites, and policy structure in terms of scope-based executive programs or trends. The problem in the analytical evaluation of policies, or policy analysis, is that in most of the environmental policies we encounter a lot of parameters and factors that have to be measured. In many cases identifying all factors that affect a particular evaluation process can be difficult. Even if the evaluation is carried out on the basis of the most available factors, it could be cumbersome due to the complexity of problems caused by trade-offs between factors, and/or personal values or preferences of analysts and decision-makers. Also, gathering information is usually a time consuming and obscure process, which may lead to uncertain and vague data. The more pressing is the fact that we are not always able to provide ways to establish quantitative policy analysis procedures (such as
2
monetary analysis), due to the qualitative nature of some parameters and therefore, we need to provide frameworks to effectively analyze the policies in such circumstances. By considering these challenges, we need innovative tools not only for effective decision-making, but also for structuring and identifying information relevant to each problem. This is where the decision-making theory provides assistance; proposing a multiple-criteria analysis. Multiple-Criteria Decision-Making (MCDM) was introduced as a promising and important field of study in the early 1970s. Since then the number of contributions to theories and models has continued to grow at a steady rate. A number of surveys (Bana e Costa and Vincke 1990) show the vitality of the field and the multitude of methods that have been developed. On the other hand, in order to treat uncertain, vague and qualitative information required for environmental policy analysis, fuzzy set theory is applied to recognize the inherent fuzziness of such a decision-making. This application became possible when Bellman and Zadeh in 1970 and Zimmermann in 1978 introduced fuzzy set theory concepts into the field of multiple-criteria analysis. They cleared the way for a new family of methods to deal with problems that had previously been inaccessible, and unsolvable with standard MCDM techniques. The applications of classical and fuzzy multiple-criteria analysis models have been developed in different areas of environmental engineering and management (Sobral et al. 1981, Horsak and Damico 1985, Wenger and Ronge 1987, Anandalingam and Westfall 1988, Julien and Byer 1990, Smith 1992 and 1994, Yin et al. 1999 and Cheng et al. 2002 and 2003). This dissertation introduces methodologies for energy-environmental policy analysis based on the theory of multiple-attribute decision-making and fuzzy set theory. It is
3
focused on a number of challenging areas of environmental policy analysis, including groundwater remediation, surface water quality, waste recycling, electricity generation and transportation planning. To illustrate the applicability of each methodology, adequate real-world case studies have been provided. These innovative methodologies, and accompanied case studies are presented in Chapter 3 and Chapter 4, respectively. In the groundwater remediation policy analysis, emphasis is placed on compatibility assessment. Compatibility assessment targets the interactions between remediation technologies and site characteristics, such as the types of active contaminants and their concentrations, soil composition and geological features, etc. The purpose of such an analysis is to identify the most compatible remediation plan for the contaminated site. In this dissertation, a decision support model is introduced for the prioritization of remediation plans based on their computed compatibility index. As this model receives data in terms of linguistic judgments and experts’ opinions, fuzzy set theory is used to deal with such qualitative assessments. In this process, the concept of compatibility is broken down into measurable factors. Then by using a multiple-attribute decision-making outline, factorial, regional and overall compatibility indicators are computed for each plan. Finally, by comparing these generated indicators, remediation policies are ranked. In the policy analysis for transportation planning, the study is built on the realization that despite the major objective in planning is to minimize the total cost of transportation, there is a need to consider the environmental consequences of transportation activities, and to develop a model capable of considering environmental requirements. In doing so, in this dissertation a multiple-objective optimization model with cost and environmental objectives has been developed. Then, policy-based planning solutions of the model are
4
examined in terms of optimality. In this way, by doing a post-optimization assessment, the optimality of each solution will be evaluated based on decision-makers’ interests along with local and global environmental requirements. This will determine the best compromise strategy of transportation. In optimality assessment, fuzzy set theory is applied to interpret the optimality levels of objectives by using the concept of membership function, which generates optimality index of alternatives. Decision-makers in the realm of environmental policy, environmental assessment and transportation planning will find the model, the assessment methodology and the associated real-world case study helpful in practices for joint consideration of cost and environmental objectives. In the waste recycling policy analysis, a methodological framework for environmental performance assessment of waste recycling programs is presented. Two categories of indicators are then developed: The efficiency indicators that compare the environmental achievements of a program with the required expenditures (benefits per unit cost), and the effectiveness indicators that compare the environmental benefits of a program with the amount of generated wastes (benefits per unit waste). Aggregation of these indicators, in relation to their associated criticalities, will give us a number of environmental performance indices to represent the status of the environmental performance. This score-based assessment has two major advantages; it takes complex scientific information and synthesizes it in a way that makes it easily understandable for non-experts while in comparison with other environmental performance assessment methods, it is not computationally intensive. In this aggregation, the importance values (criticalities) are often expert based uncertain judgments, which are defined according to
5
the objective of performance assessment. Therefore, a fuzzy multiple-attribute analysis can be employed to express these judgments by fuzzy sets and to formulate the weighted aggregation process. For case study, environmental performance of the provincial beverage container recycling programs in Canada has been studied, which illustrates the applicability of the proposed methodology. In the policy analysis through water quality assessment, this dissertation starts from the fact that water quality management policies, which are proposed to prevent, control or treat environmental problems related to quality of water, are broad and complex issues. There are various types of water resources, different water-uses and several decision parameters with several levels of decision-makers involved. Moreover, there are a lot of strategies and technologies available to be applied for water quality management, and so environmental decision-makers need to evaluate and to prioritize them in order to choose the best possible plan for each particular problem. To provide a comprehensive but easy to use tool in the assessment and evaluation of water quality policies, concept of water quality index (WQI) has been developed by Horton in 1965. Due to the above-mentioned complexities, to get this index, there is a need for a methodology to not only structure and identify information relevant to the problem, but also help users reach a decision. Designing a multiple-attribute decision aid model, which makes expert knowledge be available to non-expert users, can do this. In doing so, there are qualitative or linguistic assessments in the index making process. Again, fuzzy set theory can be applied to recognize this inherent fuzziness. Briefly, in this regard a fuzzy multiple-attribute decision aid is proposed to compute the water quality index and to provide an outline for prioritization of alternative plans based on the amount of improvements in WQI. At the
6
end, applicability and usefulness of the proposed methodology is revealed by a case study. For the energy/environmental policy analysis in electricity generation, this dissertation proposes an integrated model for capacity planning in electricity generation. It utilizes a multiple-criteria linear-programming model to incorporate cost and environmental objectives into the planning. To treat the uncertainties embedded in definition of model parameters, concept of decision-maker’s degree of optimism will be used. The model optimization provides different planning scenarios. To determine the best compromise plan, a post-optimization assessment based on fuzzy set theory concepts is developed. The proposed methodology is employed for a medium-term capacity planning in Canadian electricity generation sector. The results predict a major capacity growth for natural gas facilities accompanied by retirement of most coal-burning facilities. Chapter 2 provides sufficient background for design and utilization of presented energy-environmental policy analysis methodologies, which will be presented in Chapter 3. Chapter 2 consists of a comprehensive literature review on both multiple-attribute decision analysis, and fuzzy set theory. This will be accompanied by a further review on contributions using fuzzy-based multiple-attribute decision analysis techniques. Then, areas of energy-environmental policy analysis along with their foundations of assessment will be explored. Chapter 4 illustrates the applicability of the proposed policy analysis methodologies through a number of real-world case studies. Finally, concluding remarks of this dissertation are presented in Chapter 5.
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Chapter 2 – LITERATURE REVIEW
2.1. Multiple-Attribute Decision-Making (MADM) In classical decision theory, a decision-making problem is defined as a process of evaluation with three steps (Neufville 1990a): 1) Structuring the problem: Objectives and Constraints 2) Defining choices based on joint consideration of objectives and constraints 3) Identifying an optimal strategy over all possible choices
In such a problem, there are two categories of knowledge that should be addressed (Rijckaert et al. 1988): declarative knowledge and procedural knowledge. The declarative aspect describes ‘what’ objects (facts, terms, concepts, etc.) are used by decision-makers, and it describes the relationships between these objects. The procedural aspect contains information on how these objects can be used to infer new calculations and ultimately arrive at a decision. This part is expressed as the ‘how-to’ knowledge. To work through these categories of knowledge in real-world problems where decision-makers encounter a vast number of parameters and complex relations, decision aid methodologies have been implemented. Quality of decisions is maintained by processing the required data, the inferences and the presentation of final results. A decision aid model should be expandable and modifiable according to the scale of problems. It is necessary for a decision aid model to be able to work with human natural
8
language in order to interact seamlessly with users and human experts. It should be able to reason under conditions of uncertainty and insufficient information. In some categories of decision-making problems, we seek an optimal choice based on a single evaluation attribute such as cost, revenue, risk, etc. But in most of the real-world problems, the concentration is on decision-making with several criteria. Using a decision aid methodology could help decision-makers to manage the complexities arise from the involvement of multiple evaluation criteria. The area of multiple-criteria decision-making has grown significantly in recent past (Hwang and Lin 1987, Munda 1995, Asgharpour 1998). Generally, this area consists of two major fields: •
Multiple-Objective Decision-Making (MODM) (Michnik and Trzaskalik 2002), works on continuous decision spaces, primarily on mathematical programming with several objective functions.
•
Multiple-Attribute Decision-Making (MADM) (Yoon and Hwang 1995), focuses on problems with discrete decision spaces. MADM methods choose an optimal alternative from a set of alternatives with respect to several evaluation attributes with different weights.
A classical multiple-attribute decision-making process consists of two stages (Zimmermann 1990): Aggregation of attribute attainments of each decision alternative with respect to attribute weights, and rank ordering of alternatives based on this aggregation. A decision aid model for multiple-attribute analysis is required to (a) clearly identify decision attributes and/or alternatives, (b) assign an importance degree (if applicable) to
9
these attributes, (c) define the attainments of alternatives for each attribute, (d) aggregate the attainments of each alternative with respect to attribute weights, which provides a utility degree for each alternative and (e) compare and rank the alternatives based on their utility degrees. In the light of the above facts, in the following section, an overview of classical multiple-attribute decision-making methods is presented. It should be mentioned that in this dissertation ‘i’ and ‘j’ represent alternatives and attributes of the decision making process (unless otherwise noted).
2.2. Classical MADM Methods 2.2.1. Dominance Method Logic: An alternative is dominated if there is another alternative better than it in one or more attributes and equals it in the others (Yoon and Hwang 1981). Procedure (for ‘m’ alternatives): 1- Compare the first two alternatives. Discard the dominated one. 2- Compare the remained alternative with the third one. Discard the dominated one. 3- Come to the fourth alternative and so on. 4- After (m-1) steps, the non-dominated alternative is determined. Assumptions: None. Advantages: Simple and easy to understand.
10
Disadvantages: Some dominated alternatives, which are discarded, may possibly be the better overall than some of the remained alternatives. It is quite possible that in each pairwise comparison neither of alternatives is dominated.
2.2.2. Max-Min Method Logic: The overall performance of an alternative is determined by its poorest attribute (MacCrimmon 1968). Procedure: 1- For each alternative, define its poorest attribute value. 2- Select the alternative with the best value on the poorest attribute. Assumptions: All attributes must be measured on a common scale (unit less). Advantages: Simple and easy to understand. Disadvantages: Only one attribute is used to represent an alternative.
2.2.3. Max-Max Method Logic: An alternative is evaluated by its best attribute value (Yoon and Hwang 1981). Procedure: 1- For each alternative, identify its best attribute value. 2- Select the alternative with the maximum best attribute value. Assumptions: All attributes must be measured on a common scale (unit less). Advantages: Simple and easy to understand. Disadvantages: Only one attribute is used to represent an alternative.
11
2.2.4. Conjunctive Method Logic: An alternative is rejected if it does not meet the minimum acceptable level for all of attributes (Dawes 1964). Procedure: 1- A minimum acceptable level is assigned to each attribute. 2- For each alternative, compare the attribute values with their acceptable levels. 3- If the alternative does not satisfy all acceptable levels, it is rejected. Assumptions: A minimum acceptable level must be defined for each attribute. Advantages: Simple and easy to understand. Disadvantages: An alternative with just one unacceptable attribute is discarded even if it has better values for the other attributes.
2.2.5. Disjunctive Method Logic: An alternative, which at least one of its attribute values is better than the desirable level, is acceptable (Dawes 1964). Procedure: 1- A desirable level is defined for each attribute. 2- For each alternative, compare the attribute values with their desirable levels. 3- If the alternative satisfies at least one of the desirable levels, it is acceptable. Assumptions: A desirable level must be defined for each attribute. Advantages: Simple and easy to understand.
12
Disadvantages: An alternative that is good in all attributes but not with values better than desirable levels is rejected; meanwhile, an alternative with just one exceptional value but poor situations for the others is accepted.
2.2.6. Lexicographic Method Logic: In some decision-making problems, a single attribute seems to predominate (Yoon and Hwang 1981). Procedure: 1- Compare all alternatives with respect to the most important attribute. Select the alternative with the best value on that attribute. 2- If there are several alternatives with the best value, compare them with respect to the next most important attribute. 3- This comparison procedure continues until only one alternative is left or until all attributes have been considered. Assumptions: The attributes must be ranked in terms of importance. Advantages: Simple and easy to understand. Disadvantages: The tradeoff among attributes (overall evaluation based on confluence of attainments and weights) is not considered.
2.2.7. Linear Assignment Method Logic: An alternative, which has many high ranked attributes, is ranked high (Bernardo and Blin 1977).
13
Procedure: 1- Rank the alternatives for each attribute. 2- Assign an importance weight to each attribute. 3- Score of an alternative is the sum of importance weights of attributes where that alternative has the highest rank. 4- The alternative with the highest overall score has the highest rank. 5- To find the rank of other alternatives, repeat the procedure without the first ranked alternative. Assumptions: An importance weight must be defined for each attribute. Advantages: Simple and easy to understand. It requires less effort in data collection as it uses ordinal data. Disadvantages: The actual cardinal difference between attainments of alternatives on each attribute is not considered.
2.2.8. Simple Additive Weighting Method Logic: The overall score of an alternative is the weighted sum of its attribute values (Yoon and Hwang 1981). Procedure: 1- Compute weighted average of attribute attainments for each alternative. 2- Rank the alternatives based on the above computed scores. Assumptions: The attainments and their weights must be both numerical and comparable.
14
Advantages: Simple, easy to understand and most widely used method. It considers the tradeoff among attributes (overall evaluation based on confluence of attainments and weights). Disadvantages: None.
2.2.9. TOPSIS Logic: In TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) alternatives are ranked based on their distance from an ideal solution and a negative-ideal solution (Yoon and Hwang 1981). Procedure: 1- For decision matrix D = [ xij ]m × n (matrix of attainments for ‘m’ alternative with respect to ‘n’ attributes) calculate the normalized (unit free) decision matrix
D N = [rij ]m× n . The normalized value rij is defined as:
rij =
xij ∑im=1 xij
2
i = 1, 2, ..., m , j = 1, 2, ..., n
(2.2-1) where xij is the attainment of attribute ‘j’ by alternative ‘i’ 2- Compute weighted normalized decision matrix [Vij ]m× n . The weighted normalized value Vij is defined as:
15
Vij = w j rij
(2.2-2) where w j is the importance weight of attribute ‘j’ 3- Determine the ideal A + and negative ideal A− solutions:
A+ = (V1+ ,V2+ , ... Vn+ ) = [V j+ ]1×n
(2.2-3) A− = (V1− ,V2− , ... Vn− ) = [V j− ]1×n
(2.2-4) where V j+ and V j− are defined as the followings:
V j+ = Max(Vij )
for
i = 1, 2, ..., m
(2.2-5) V j− = Min(Vij )
for
i = 1, 2, ..., m
(2.2-6) 4- Compute separation measures (distance to ideal and negative-ideal solutions):
S i+ = ∑ nj =1 (Vij − V j+ ) 2
(2.2-7)
16
S i− = ∑ nj =1(Vij − V j− ) 2
(2.2-8) 5- The relative closeness to the ideal solution is defined as: RCi =
S i− S i− + S i+
(2.2-9) 6- Rank the alternatives in descending order of their relative closeness. Assumptions: The attainments and their weights must be both numerical and comparable. Advantages: Simple and easy to understand. It considers the tradeoff among attributes (overall evaluation based on confluence of attainments and weights). Disadvantages: None.
2.2.10.Weighted Product Method Logic: In order to penalize alternatives with poor attribute attainments more heavily, a product instead of a sum of values is proposed with the attribute importance weights as the exponents (Easton 1973). Procedure: 1- Raise each attribute attainment of an alternative to a power equal to the importance weights of that attribute. Then multiply the results over all attributes for that alternative. 2- Rank the alternatives based on the computed products.
17
Assumptions: The attribute values and their weights must be both numerical and comparable. Advantages: Simple and easy to understand. It considers the tradeoff among attributes (overall evaluation based on confluence of attainments and weights). Disadvantages: None.
2.2.11.Distance from Target Method Logic: In some problems, there is a target value considered for each attribute. Therefore, the attribute value must be compared with that target. The alternative, which has a shorter distance from the target alternative, is ranked higher (Easton 1973). Procedure: 1- For each alternative, compute the distance from target alternative:
d i = ∑ nj =1 w 2j ( xij − t j ) 2
i = 1, 2, ..., m
(2.2-10) where t j is the target value of attribute ‘j’ 2- Rank the alternatives based on their distance (from target) values. Assumptions: A target value must be defined for each attribute. The attribute values and their weights must be both numerical and comparable. Advantages: Simple and easy to understand. It considers the tradeoff among attributes (overall evaluation based on confluence of attainments and weights). Disadvantages: None. 18
2.3. Fuzzy Multiple-Attribute Decision-Making 2.3.1. Fuzzy set theory There are two major types of methodologies to deal with uncertainties; a stochastic approach when the uncertainty originates from the randomness of parameters, and a fuzzy logic approach when we face vague parametric values based on imprecise, illdefined information, and subjective opinions or judgments. L.A. Zadeh introduced fuzzy set theory in 1965. It is a complement for the traditional divalent “0 or 1” or “true or false” logic, as it pronounces degrees of belief between “0 and 1” or “true and false”. Theoretical foundations of fuzzy set theory have been derived from theory of sets in mathematics:
A classical set is defined as a collection of elements: A = {x | x ∈ A} . Therefore, for each element x , we can say that x ∈ A OR x ∉ A ( x belongs to A or not). It means that membership of x in A can be 1 or 0. Also a subset B of a classical set A ( B ⊆ A ) is defined as ∀ x ∈ B ⇒ x ∈ A .
A fuzzy set is also a collection of elements, which belong to it with a degree of membership. This degree is called membership function. In this way a fuzzy set is defined as follows (Zadeh 1965):
~ A = {( x, µ A~ ( x)} (2.3-1) 19
~ We can say that x belongs to A with a degree of µ A~ ( x) ∈ [0, 1] . On the other hand, x ~ ~ does not belong to A with a degree of 1 − µ A~ ( x) . A is called a normal fuzzy set if
~ ∃ x ∈ A with µ A~ ( x) = 1 (Zimmermann 1990).
~
In this sense, a subset of fuzzy set A is defined as (Zadeh 1965): ~ ~ ~ ( x ) ≤ µ ~ ( x) B ⊆ A ⇔ ∀x : µ B A
(2.3-2) Support of a fuzzy set is a classical crisp set, which contains elements of that fuzzy set (Zimmermann 1990):
~ S ( A) = {x | µ A~ ( x) > 0} (2.3-3) α – level set or α – cut of a fuzzy set is a classical crisp set, which contains elements of
that fuzzy set with a membership function of at least α (Zimmermann 1990):
~ Aα = {x | x ∈ A, µ A~ ( x) ≥ α } (2.3-4) ~ Aα′ is called a strong α – cut of the fuzzy set A if (Zimmermann 1990):
~ Aα′ = {x | x ∈ A, µ A~ ( x) > α } (2.3-5)
20
~ A fuzzy set A is called convex if its membership function is a convex function
(Zimmermann 1990): ~ ∀x1 , x2 ∈ A, λ ∈ [0,1] ⇒ µ A~ (λx1 + (1 − λ ) x2 ) ≥ Min( µ A~ ( x1 ), µ A~ ( x2 )) (2.3-6) ~ A convex fuzzy set A is called triangular if it has a triangular membership function
(Figure 2.3.1) with the following illustration (Zimmermann 1990): ~ A = (a, m, b)
(2.3-7) ~ A convex fuzzy set A is called trapezoidal if it has a trapezoidal membership function
(Figure 2.3.2) with the following illustration (Zimmermann 1990):
~ A = (a, b, c, d )
(2.3-8) Operations with fuzzy sets are defined via their membership functions. Assume that ~ ~ A = {( x, µ A~ ( x)} and B = {( x, µ B~ ( x)} are fuzzy sets:
~ ~ ~ Intersection of fuzzy sets A and B is a fuzzy set C defined as (Zimmermann 1990):
~ ~ ~ C = A ∩ B = {( x, µ C~ ( x)) | µ C~ ( x) = µ A~ ( x) ∧ µ B~ ( x) = Min( µ A~ ( x), µ B~ ( x))}
(2.3-9) ~ ~ ~ Union of fuzzy sets A and B is a fuzzy set C defined as (Zimmermann 1990):
21
~ ~ ~ C = A ∪ B = {( x, µ C~ ( x)) | µ C~ ( x) = µ A~ ( x) ∨ µ B~ ( x) = Max( µ A~ ( x), µ B~ ( x))}
(2.3-10) ~ Complement of fuzzy set A is a fuzzy set defined as (Zimmermann 1990):
~ ~ ~ ~ A =⊄ A = A′ = {( x, µ A~ ′ ( x)) | µ A~ ′ ( x) = 1 − µ A~ ( x)} (2.3-11) ~ ~ ~ ~ Cartesian product of fuzzy sets A1 , A2 , …, An is a fuzzy set C called Cartesian fuzzy set (Zimmermann 1990):
~ ~ ~ ~ C = A1 × A2 × ... × An = {( y, µ C~ ( y )) | y = ( x1 , x2 ,...x n ), µ C~ ( y ) = Min( µ A~ ( x1 ), µ A~ ( x 2 ), ..., µ A~ ( x n ))} 1
2
n
(2.3-12) ~ mth power of a fuzzy set A is defined as a fuzzy set (Zimmermann 1990):
~ A m = {( x, µ A~ m ( x) | µ A~ m ( x) = [ µ A~ ( x)]m } (2.3-13) ~ ~ ~ Bounded sum of fuzzy sets A and B is a fuzzy set C defined as (Zimmermann 1990):
~ ~ ~ C = A ⊕ B = {( x, µ C~ ( x)) | µ C~ ( x) = Min(1, µ A~ ( x) + µ B~ ( x))}
(2.3-14)
22
~ ~ ~ Bounded difference of fuzzy sets A and B is a fuzzy set C defined as (Zimmermann 1990): ~ ~ ~ C = A ΘB = {( x, µ C~ ( x)) | µ C~ ( x) = Min( µ A~ ( x) + µ B~ ( x) − 1,0)}
(2.3-15) Functions on fuzzy sets are defined via their elements and membership functions. Let ~ A = {( x, µ A~ ( x)} be a fuzzy set and f be a real valued function ( f : ℜ → ℜ ):
~ ~ ~ Extension Principle: f (A) is a fuzzy set which maps from A into a fuzzy set B (Zadeh
1965, Negoita and Ralescu 1975, Dubois and Prade 1980): ~ ~ f : A → B ~ ~ B = f ( A ) = {( y, µ f ( A~ ) ( y )) | y = f ( x), µ f ( A~ ) ( y ) = Sup ( µ A~ ( x))} f −1 ( y ) (2.3-16) ~ ~ ~ ~ ~ ~ Cartesian function f ( A1 , A2 , …, An ) on fuzzy sets A1 , A2 , …, An is defined as (Zimmermann 1990):
~ ~ ~ ~ f : ( A1 , A2 , ..., An ) → B ~ ~ ~ ~ B = f ( A1 , A2 , ..., An ) = {( y, µ B~ ( y ))} (2.3-17) where y = f ( x1 , x 2 , ..., x n ) and µ B~ ( y ) = Sup ( Min( µ A~ ( xi ))) i −1 f
( y)
Therefore, the real valued function f ( x, y ) = x o y ( o is an algebraic operation like +, ~ ~ ~ , ∗ , ÷ , etc.) on fuzzy sets A1 and A2 defines a fuzzy set B (Zimmermann 1990):
23
~ ~ ~ B = f ( A1, A2 ) = {( y, µ B~ ( y )) | y = x1 o x2 , µ B~ ( y ) = Sup {Min( µ A~ ( x1), µ A~ ( x2 ))}} 1 2 f −1 ( y )
(2.3-18) Maximizing set of a real valued function f which is bounded from below by “L” and from above by “U” is defined as (Zadeh 1972):
f ( x) − L ~ M f = {( x, µ M~ ( x)) | µ M~ ( x) = } f f U −L
(2.3-19) ~ ~ Fuzzy maximum of a real valued function f : A → B is defined as (Dubois and Prade
1980):
Ma~x f = {( z , µ Ma~x f ( z )) | z = Max( y ), µ Ma~x f ( z ) = Min( µ f ( A~ ) ( y ))
(2.3-20) ~ ~ Fuzzy minimum of a real valued function f : A → B is defined as (Dubois and Prade
1980):
~ M i n f = {( z , µ M~i n ( z )) | z = Min( y ), µ M~i n ( z ) = Min( µ f ( A~ ) ( y )) f
f
(2.3-21)
24
2.3.2. Decision-making in a Fuzzy Environment It is possible that all, or some of parameters of a decision-making problem (objectives/constraints or attainments/weights) have imprecise values. To treat such uncertainties, in 1970 Bellman and Zadeh introduced fuzzy set theory to the area of decision-making. According to them, a decision is defined as the confluence of objectives and constraints (Zimmermann 1990):
~ ~ Let G j = {( x, µ G~ ( x)) | x ∈ X } and Ci = {( x, µ C~ ( x)) | x ∈ X } be fuzzy sets of j
i
objectives ( j = 1, 2, ..., n ) and constraints ( i = 1, 2, ..., m ) of a problem in a decision space of X , respectively. A decision is then defined as:
~ D = {( x, µ D~ ( x)) | x ∈ X , µ D~ ( x) = [⊗ µ C~ ( x)] ⊗ [⊗ µ G~ ( x)]} i
i
j
j
(2.3-22) where ⊗ , ⊗ and ⊗ are aggregation operators (Mesiar and Fuller 1997, Slowinsky 1998). i
j
Bellman and Zadeh suggest that aggregation operators in (2.3-22) can be represented by fuzzy intersection to express the confluence and the joint consideration of objectives and constraints:
~ ~ ~ ~ ~ ~ ~ D = G1 ∩ G2 ∩ ...Gn ∩ C1 ∩ C 2 ∩ ...C m (2.3-23)
25
and correspondingly form (2.3-9):
µ D~ ( x ) = Min ( µ G~ ( x ), µ G~ ( x ), ... µ G~ ( x ), µ C~ ( x ), µ C~ ( x ), ..., µ C~ ( x )) 1
2
1
n
2
m
(2.3-24) They indicated that the “Min” interpretation of the aggregation might have to be modified depending on the context of problem. Therefore, decision-makers in a fuzzy environment should seek a suitable definition for the confluence (aggregation) based on the context of problem, and should find a reasonable way to select the optimal choice on the basis of aggregated values.
2.3.3. Fuzzy MADM Methods Bellman and Zadeh’s definition was a starting point for applying fuzzy multiple-criteria analysis in various decision-making domains. Fuzzy multiple-attribute decision-making methods are justified by the understanding that the attributes and/or their attainments by alternatives cannot be defined or judged crisply, but only as fuzzy values. There are two directions to deal with such fuzziness: Fuzzy-to-Crisp and Fuzzy-to-Fuzzy approaches. The Fuzzy-to-Crisp approach proposes methodologies to convert fuzzy data to crisp values to use them in a classical MADM model (Figure 2.3.3). Fuzzy-to-Fuzzy proposes algorithms to develop MADM methodologies capable of using fuzzy data (Figure 2.3.4). The former approach is less complex and needs less computation as Chen and Hwang theorized in their studies (Chen and Hwang 1992). But conversion of fuzzy data to a crisp value is at risk for remarkable degrees of error as we substitute an uncertain interval of values, with a representative crisp value. The latter approach with a more complicated model capable of aggregating fuzzy values should be designed. In this case, the final 26
evaluation results of decision-making process could be presented by fuzzy sets, which have to be ordered to determine the priority of alternatives. A fuzzy MADM model, with fuzzy values for attainments and attribute weights, is ~ generally formulated based on an outline presented in Table 2.3.1. In this table W j and
~ ri, j are the importance (weight) of attribute ‘j’ and the attainment of attribute ‘j’ by the alternative ‘i’, respectively ( j = 1, 2, ..., n and i = 1, 2, ..., m ), expressed via convex fuzzy sets as follows:
~ W j = {( y j , µW~ ( y j ))} j
(2.3-25) ~ ri, j = {( xij , µ ~r ( xij ))} i, j (2.3-26)
On the basis of the above theoretical foundations, in the following sections, fuzzy multiple-attribute methodologies will be overviewed.
2.3.3.1. Fuzzy Simple Additive Weighting Method
In fuzzy MADM, similar to a classical one, a utility function is generated to aggregate the attribute attainments of alternatives. This utility function is represented by the weighted average of attainments:
27
n ~ ~ ~ ∑ j =1 w j rij Ui = ~ ∑ nj =1 w j
(2.3-27) ~ ~ and ~ As w rij are convex fuzzy values, U i will be a convex fuzzy number j (Zimmermann 1990). The problem is how to compute this function with fuzzy weights and fuzzy attainments. There are a number of algorithms proposed to deal with such a problem:
2.3.3.1.1. Baas & Kwakernaak’s Approach ~ To find U i for i = 1, 2, ..., m (Baas and Kwakernaak 1977): 1- Consider an α-cut, for example: α = α 0 2- Define values of y j and xij for which:
µ w~ j ( y j ) = µ ~rij ( xij ) = α 0 ∀i, j (2.3-28) Based on the shape of membership functions (triangular, trapezoidal, etc.), we will have different values of y j and xij from (2.3-28). Thus, there will be different values of ui for α = α 0 :
28
ui (α 0 ) =
[α =α 0 ] ~[α =α 0 ] rij ~[α =α 0 ] ∑ nj =1 w j
~ ∑ nj =1 w j
(2.3-29) 3- Set: ~ U i L (α 0 ) = Min{ui (α 0 )}
(2.3-30) and ~ U iU (α 0 ) = Max{ui (α 0 )}
(2.3-31)
4- By the above values and selecting different levels of α , we can estimate a ~ membership distribution of µU~ (ui ) for U i (Figure 2.3.5). Therefore: i
~ U i = {(ui , µU~ (ui ))} i
(2.3-32) Note - This method is simple and easy to understand but it needs huge amount of ~ calculations for large-scale decision-making problems. For triangular fuzzy numbers w j ~ and ~ rij , we may have up to 4n tries in computing (2.3-29) to find U i L (α 0 ) and ~ U iU (α 0 ) .
29
2.3.3.1.2. Kwakernaak’s Approach
This algorithm was proposed as an extension to the previous method to reduce the number of computation (Kwakernaak 1979).
~ To find U i for i = 1, 2, ... m : 1- Consider an α-cut, for example: α = α 0 ~ and ~ 2- For α = α 0 define α-cut sets of w rij ( j = 1, 2, ..., n ): j
~ W j (α 0 ) = {( y j , µW~ ( y j )) | µW~ ( y j ) ≥ α 0 } j
j
(2.3-33) ~ ri, j (α 0 ) = {( xij , µ ~ri , j ( xij )) | µ ~ri , j ( xij ) ≥ α 0 }
(2.3-34) 3- Define the following values:
~ (α )) y j + = Max( w j 0 (2.3-35) ~ (α )) y j − = Min( w j 0 (2.3-36) xij + = Max(~ rij (α 0 )) (2.3-37)
30
xij − = Min(r~ij (α 0 )) (2.3-38) 4- Sort each of the above values from their minimum to maximum as:
m1− ≤ m2− ≤ m3− ≤ ... ≤ mn−
( m1− = Min( xij − ) ≤ ... ≤ mn− = Max( xij − ) ) j
j
(2.3-39) m1+ ≤ m2+ ≤ m3+ ≤ ... ≤ mn+
( m1+ = Min( xij + ) ≤ ... ≤ mn+ = Max( xij + ) ) j
j
(2.3-40) z1− ≤ z 2− ≤ z 3− ≤ ... ≤ z n−
( z1− = Min( y j − ) ≤ ... ≤ z n− = Max( y j − ) ) j
j
(2.3-41) z1+ ≤ z 2+ ≤ z 3+ ≤ ... ≤ z n+
( z1+ = Mi n ( y j + ) ≤ ... ≤ z n+ = Max ( y j + ) ) j
j
(2.3-42) 5- Then: − − ∑ j mk− z + + ∑ n ~L k =1 k k = j +1 mk z k U i (α 0 ) = Min + − j ∑ kj =1 z k + ∑ nk = j +1 z k
(2.3-43) + + ∑ j m+ z − +∑ n ~U k =1 k k k = j +1 mk z k U i (α 0 ) = Max − + j ∑ kj =1 z k + ∑ nk = j +1 z k
(2.3-44)
31
6- Using the above values and selecting different levels of α , we can estimate a ~ membership distribution of µU~ (ui ) for U i similar to Bass and Kwakernaak method. i
Note – This method is more complicated than Baas & Kwakernaak method but needs remarkably less computation, especially in large-scale problems. For triangular fuzzy ~ ~ ~ and ~ numbers w rij , we may need up to C2nn−−1 2 tries to find U i L (α 0 ) and U iU (α 0 ) j
values, which is noticeably less than Bass and Kwakernaak method.
2.3.3.1.3. Dubois & Prade’s Approach
The first three steps of this method are just like Kwakernaak method. The other steps are as follows (Dubois and Prade 1982):
4- Sort xij − and xij + values from their minimum to maximum as:
m1− ≤ m2− ≤ m3− ≤ ... ≤ mn−
( m1− = Min( xij − ) ≤ ... ≤ mn− = Max( xij − ) ) j
j
(2.3-45)
m1+ ≤ m2+ ≤ m3+ ≤ ... ≤ mn+
( m1+ = Min( xij + ) ≤ ... ≤ mn+ = Max( xij + ) ) j
j
(2.3-46)
32
~ 5- Convert W j (α 0 ) = [ y j − , y j + ] to a normal interval [ p j − , p j + ] by the following transformations:
p −j = y −j ( y −j + ∑ k ≠ j y k+ ) (2.3-47) p +j = y +j ( y +j + ∑ k ≠ j y k− ) (2.3-48)
6- Then:
~ U iL (α 0 ) = (∑ hj −=11 p +j m −j ) + z h mh− + (∑ nj = h +1 p −j m −j )
(2.3-49) ~ U iU (α 0 ) = (∑ tj−=11 p −j m +j ) + zt mt+ + (∑ nj = t +1 p +j m +j )
(2.3-50)
where h , z h , t and zt can be defined as:
(h, z h ) = {(h ∈ {1,2, ..., n}, z h = 1 − (∑ hj −=11 p +j + ∑ nj = h +1 p −j )) | z h ∈ [ p h− , p h+ ]}
(2.3-51) (t , z t ) = {(t ∈ {1,2, ..., n}, z t = 1 − (∑ tj−=11 p −j + ∑ nj = t +1 p +j )) | z t ∈ [ pt− , pt+ ]}
(2.3-52) 33
~ ~ 7- By using U iL (α 0 ) and U iU (α 0 ) values and choosing different levels of α , we can
~ estimate a membership distribution µU~ (ui ) for U i . i
Dubois and Prade have shown that there are unique values of h and t , which can be obtained from (2.3-51) and (2.3-52).
Note – This method is much more complex than both previous methods but it needs less ~ and ~ effort in calculations. For triangular fuzzy numbers w rij , we may need up to j ~ ~ 2(n − 1) tries to find h and t to obtain U i L (α 0 ) and U iU (α 0 ) values, which is less than
both previous methods.
2.3.3.1.4. Cheng & McInnis’s Approach The first four steps of this method are just like previous method. Then, the algorithm comes to the following steps (Cheng and McInnis 1980): 5- Set:
∑ bm ∑ b n
~L U i (α 0 ) =
j =1 n
j
− j
j
j =1
(2.3-53)
∑ am ∑ a n
~U U i (α 0 ) =
j =1 n
j =1
j
+ j
j
(2.3-54)
34
where a j and b j are defined as : a n = y n+
(2.3-55)
y+ j aj = y −j
+
if
∑ kj +=1n a k m k ∑ kj +=1n a k
≤
+ + + ∑ kj +=1n a k m k + y j m j
( ∑ kj +=1n a k ) + y +j
otherwise
(2.3-56)
y +j bj = − y j
if a j = y −j otherwise (2.3-57)
6- Using the above values and selecting different levels of α , we can estimate a ~ membership distribution µU~ (ui ) for U i similar to Bass and Kwakernaak method. i
~ and ~ Note - For triangular fuzzy numbers w rij , we need up to n − 1 tries to find j ~ ~ U i L (α 0 ) and U iU (α 0 ) values, which is far less than all previous methods.
2.3.3.1.5. Bonissone’s Approach Bonissone assumed that every fuzzy number could be estimated by a convex trapezoidal membership distribution (Bonissone 1982). He illustrated a trapezoidal fuzzy number ~ M as the following (see Figure 2.3.6):
35
~ M = ( a , b, α , β )
(2.3-58) where this number will be a triangular fuzzy number if a = b .
~ ~ Bonissone showed that for fuzzy numbers M = (a, b, α , β ) and N = (a ′, b′, α ′, β ′) we can
write:
~ ~ M + N = (a + a ′, b + b′, α + α ′, β + β ′)
(2.3-59) ~ ~ M − N = (a − b′, b − a′, α + β ′, β + α ′)
(2.3-60) ~ ~ M ⋅ N = (aa′, bb′, aα ′ + αa′ − αα ′, bβ ′ + b′β + ββ ′)
(2.3-61) a b aβ ′ + bα bα ′ + a ′β ~ ~ , ) M ÷N =( , , b′ a′ b′(b′ + β ′) a ′(a ′ − α ′) (2.3-62)
~ and ~ rij by trapezoidal (or triangular) Therefore, if we estimate fuzzy numbers of w j distributions, utility value in (2.3-27) can be obtained using (2.3-59) to (2.3-60) definitions. 36
~ and ~ Note - For trapezoidal (or triangular) fuzzy numbers w rij , this method shows the j best performance in terms of amount of calculations, but the problem is the fact that Bonissone’s estimations are limited to trapezoidal fuzzy numbers. This may limit the applicability of the method and may impose some degrees of error.
2.3.3.2. Analytic Hierarchy Process (AHP)
In some multiple-attribute decision-making problems, we are not able to form a decision matrix like Table 2.3.1 due to hierarchical interrelations of attributes. The solution process for these types of problems is a hierarchy decision structure called the AHP (Analytic Hierarchy Process) method (Saaty 1977 and 1978). Figure 2.3.7 shows an AHP model for a problem with ‘h’ levels. The top level consists of a single element. Each element of a given level dominates or covers (serves as a property or a purpose for) some, or all of the elements in the level immediately below. Each Ar , k is a pair-wise comparison matrix, comparing elements of level r + 1 to obtain their relative weights with respect to the purpose from the element ‘k’ in the adjacent higher level. The process is repeated up the hierarchy. The resulted priorities are composed to obtain an overall priority vector (consists of utility functions) for the impact of the lowest elements (level ‘h’) on the top element of the hierarchy, which represents the decision, by successive weighting and composition. In a fuzzy environment, which the relative weights (comparisons) are fuzzy numbers, the decision-making problem by AHP is formulated as follows (Buckley 1985):
37
1- Form the positive comparison matrix Ar , k which shows the relative weights of ‘nr+1’ attributes of the level ‘r+1’ with respect to element ‘k’ in the dominant level (level r):
a~11 a~ 21 Ar , k = M ~ anr +11
a~12 a~
22
M a~
nr +1 2
K K K
a~1nr +1 a~
2nr +1 = [a~ij ]nr +1×nr +1 M a~nr +1nr +1
(2.3-63) k = 1, 2, ..., nr
r = h − 1, h − 2, ..., 2, 1
where a~i, j is a fuzzy number which shows relative weight of elements ‘i’ and ‘j’ and nr is the number of elements at level ‘r’
2- Compute the geometric mean for each row of the above matrix:
1 ~ Zi = [a~i1 o a~i 2 o ...a~inr +1 ] nr +1
i = 1, 2, ..., nr +1 (2.3-64)
where ‘ o ’ represents fuzzy multiplication
3- Then, Relative score of each attribute ‘i’ (in level ‘r+1’) in satisfying element ‘k’ of the dominant level ‘r’ is calculated as (attainment of ‘k’ by ‘i’):
38
~ Zi ~ ~ ~ ~ (r ) −1 ~ sik = Z i o ( Z1 + Z 2 + ... + Z n r +1 ) = ~ ~ ~ Z1 + Z 2 + ... + Z n r +1
(2.3-65)
4- Repeat this process for all values of ‘k’ to get the score matrix of level ‘r’ based on level ‘r+1’:
Sr = [~ sik( r ) ]nr +1×nr
r = h − 1, h − 2, ..., 2, 1
(2.3-66)
5- Find the above matrix for all values of ‘r’. 6- Then the utility matrix of level ‘L’ based on the top-level element (decision) is defined as:
~ U L = S L −1S L − 2 ...S1 = [U i( L) ] 1 ×nL
L = 1, 2 ..., h i = 1, 2, ..., n L
(2.3-67)
~ 7- Thus, U h = [ U i( h) ] will give us the utility (fuzzy) values of level ‘h’ (the lowest level) elements based on the top-level element (decision).
Note- The multiplication of matrices with fuzzy elements is the same as that of regular matrices. Therefore, we will have multiplication and summation of fuzzy numbers, which
39
can be obtained by using the proposed outlines for multiplication and summation of fuzzy values in section 2.3.3.1. It should be mentioned that the AHP method needs a huge amount of calculations and data collection efforts, especially in large-scale problems.
2.3.3.3. Fuzzy Max-Min Method
This category of methods is based on Bellman and Zadeh definition for ‘decision’ reviewed in section 2.3.2.
Let X = {x1 , x 2 , ..., x m } be a set of alternatives. Each goal is represented by a fuzzy set ~ G j for j = 1, 2, ..., n . The importance (weight) of goal ‘j’ is expressed by w j . The
attainment of goal ‘j’ by alternative xi is illustrated by the membership function of xi in ~ fuzzy set G j :
~ G j = {( xi , µ G~ ( xi )) | i = 1, 2, ..., m} j
(2.3-68) Decision is a fuzzy set defined as (Yager 1978):
~w ~ ~ ~w D = G1 1 I G2 2 I ...Gnwn (2.3-69) where from (2.3-9) and (2.3-13):
40
wj
µ D~ ( xi ) = Min([ µG~ ( xi )] j
j
)
(2.3-70) The alternatives are ranked in descending order of the above membership degree.
Note - This method provides a simple logic-driven procedure to aggregate attribute attainments of alternatives and to rank the alternatives based on such an aggregation. Using the weights as exponents has effect of making the membership function of decision ~ set D being more determined by more important attributes. However, attribute weights
are not considered to be fuzzy sets and should be crisp values.
2.3.3.4. Fuzzy TOPSIS Method
Negi modified classical TOPSIS method (section 2.2.9) to become capable of accepting fuzzy values (Negi 1989):
~ xij ]m× n , calculate the normalized (unit free) matrix 1- For given decision matrix D = [ ~ ~ D N = [~ rij ]m× n . The normalized fuzzy value ~ rij is defined as:
xij aij bij cij d ij ~ ~ (U ) = ( (U ) , (U ) , (U ) , (U ) ) dj cj bj aj x j ~ rij = x (j L) a (jL) b (jL) c (jL) d (jL) ~ , , , ) ~ =( d ij cij bij aij xij
j is a benefit attribute
j is a cos t attribute (2.3-71)
i = 1, 2, ..., m , j = 1, 2, ..., n
41
where ~ xij is the attainment of attribute ‘j’ by alternative ‘i’ assumed to be a trapezoidal fuzzy number (2.3-8):
~ xij = (aij , bij , cij , d ij ) (2.3-72) and
~ x (jU ) = Max( ~ xij ) = (a (jU ) , b (jU ) , c (jU ) , d (jU ) ) i
(2.3-73) ~ x (j L) = Min( ~ xij ) = (a (jL) , b (jL ) , c (jL) , d (jL ) ) i
(2.3-74)
where (a (jU ) , b (jU ) , c (jU ) , d (jU ) ) = ( Max(aij ), Max(bij ), Max(cij ), Max(d ij )) i
i
i
i
(2.3-75) (a (jL) , b (jL ) , c (jL ) , d (jL ) ) = ( Min(aij ), Min(bij ), Min(cij ), Min(d ij )) i
i
i
i
(2.3-76)
~ 2- Compute the weighted normalized decision matrix V = [v~ij ]m× n . The weighted
normalized fuzzy value v~ij is defined as:
42
~ ~ v~ij = w j rij (2.3-77) ~ is the importance (weight) of attribute ‘j’ assumed to be a trapezoidal fuzzy where w j number: ~ = (α , β , γ , δ ) , j = 1, 2, ..., n w j j j j j (2.3-78) Therefore: bij cij d ij aij ( , , , α β γ (U ) j (U ) j (U ) j (U ) δ j ) cj bj aj d j v~ij = ( L) b (jL) c (jL ) d (jL) aj α j, β j, γ j, δ j) ( cij bij aij d ij
for benefit attributes
for cos t attributes (2.3-79)
~ ~ 3- Determine the ideal A + and negative ideal A − solutions:
~ A + = ( v~1+ , v~2+ , ... , v~n+ ) = [ v~ j+ ]1× n (2.3-80) ~ A − = ( v~1− , v~2− , ... , v~n− ) = [ v~ j− ]1× n (2.3-81) where v~ j+ and v~ j− are defined as the followings:
43
v~j+ = Max(v~ij )
for
i = 1, 2, ... , m
(2.3-82) v~ j− = Min(v~ij )
for
i = 1, 2, ... , m
(2.3-83) To find v~ j+ and v~ j− , we need to compare ‘m’ fuzzy sets: v~ij ( i = 1, 2, ..., m ). To do this comparison, we can use one of fuzzy rank-ordering methods introduced in section 2.3.3.5.
4- The separation measures are obtained (distance to ideal and negative-ideal solutions) for each alternative:
S i+ = ∑ nj =1 Dij+
(2.3-84) S i− = ∑ nj =1 Dij−
(2.3-85) where Dij+ = 1 − Sup( Min( µ v~ij , µ ~v + )) j
(2.3-86) Dij− = 1 − Sup( Min( µ ~vij , µ v~ − )) j
(2.3-87) 44
As Dij+ and Dij− are crisp values, Si+ and Si− will be crisp.
5- The relative closeness to the ideal solution is defined as: RCi =
S i− S i− + S i+
(2.3-88) 6- Alternatives are ranked in descending order of their relative closeness values.
2.3.3.5. Fuzzy Rank Ordering Methods
~ After finding U i sets, it is required to compare these sets to rank and prioritize the associated alternatives. There are various algorithms proposed to deal with ranking of such fuzzy sets:
2.3.3.5.1. Distribution-based Approaches ~ In this category of methods, we calculate an index based on distribution of U i sets. Alternatives are then ranked on the basis of these indices. The following methods have been proposed in this category:
2.3.3.5.1.1. Centroid Value ~ For U i = {(ui , µU~ (ui ))} , centroid value of membership distribution is defined as (Yager i
1980a): 45
For discrete membership functions: ∑ui µU~ (ui )ui i
Ci =
∑ui µU~ (ui ) i
(2.3-89) For continuous membership functions: ∫ µU~ (ui )ui dui i
Ci =
ui
∫ µU~ (ui )dui i
ui
(2.3-90) Note – As centroid value measures mean of a fuzzy distribution, ranking based on this method is regardless of the membership function spread.
2.3.3.5.1.2. Mean-Spread Values Lee and Li proposed that human intuition would favor a fuzzy number with a higher mean value and a lower spread (Lee and Li 1988). Therefore, mean and variance of a fuzzy number can be suitable indices to specify its rank among the others. These values ~ for a continuous fuzzy set U i = {(ui , µU~i (u i ))} are defined as the followings:
∫µ Mi =
~ Ui
(ui )ui dui
ui
∫µ
~ Ui
(ui )dui
ui
(2.3-91)
46
∫ (u Vi =
i
− M i ) 2 µU~i (ui )dui
ui
∫µ
~ Ui
(ui )dui
ui
(2.3-92) ~ ~ Then, the comparison of two fuzzy numbers U i and U j is as:
M i > M j → U~i > U~ j ~ ~ M i = M j and Vi < V j → U i > U j M = M and V = V → U~ = U~ j i j i j i ~ ~ otherwise → U i < U j (2.3-93) ~ Note – For a triangular fuzzy number U i = (ai , mi , bi ) , we have: Mi =
ai + mi + bi 3
(2.3-94) 2
Vi =
2
2
ai + mi + bi − ai mi − ai bi − mi bi 18
(2.3-95)
2.3.3.5.1.3. Preference Degree This method identifies a preference degree for each fuzzy set in comparison to the others. ~ ~ ~ The preference is given to normal fuzzy set U i = {(ui , µU~i (u i ))} when U i > U j ∀j . ~ Therefore, preference for U i can be identified as (Baas and Kwakernaak 1977):
47
~ Pi = U (IU j ) = Sup{Min( µU~ (u j ))} j j ui ≥u j
j u i ≥ u j , ∀j
(2.3-96) Then, fuzzy sets are ranked based on their preference degrees.
Note – This method in some cases may generate counter-intuitive results (Chen and Hwang 1992) by giving a higher rank to an alternative with higher spread (among those with equal mean values).
2.3.3.5.1.4. Fuzzy Reasoning Method This method is a different expression of preference degree method (Watson et al. 1979). ~ To find degree of optimality for U i = {(ui , µU~ (ui )) | i = 1, 2, ..., m} : i
1- Assume the following statements: ~ ~ X= “ U i is a fuzzy set and U j is a fuzzy set” ~ ~ Y=“ U i is preferred to U j ”
~ ~ 2- Then, preference of U i over U j is defined by a fuzzy reasoning process:
Pij = µ ( X → Y ) (2.3-97) where
48
~ ~ ~ ~ X → Y ≡ (U i ∧ U j ) → (U i ≥ U j )
(2.3-98) ~ ~ ~ ~ ≡ ~ (U i ∧ U j ) ∨ (U i ≥ U j )
(2.3-99) ~ ~ ~ ~ ≡ ~ [(U i ∧ U j ) ∧ (U i ≤ U j )]
(2.3-100)
~ ~ ≡ ~ (U i ∧ U j ) ui ≤u j
(2.3-101) Thus: Pij = 1 − Sup( Min( µU~ (ui ), µU~ (u j ))) i j ui ≤u j
(2.3-102) 3- Finally, by pair-wise comparisons of all sets we can obtain their order. To facilitate these comparisons, a preference matrix P = [ Pij ]m× m can be formed ( Pii = 1 ).
Note – As this method is a different expression of the preference degree method, getting a counter-intuitive result is possible.
2.3.3.5.1.5. Haming Distance This category of methods measures Haming distance between fuzzy sets to compare them. Haming distance between real-valued functions of f (x) and g (x) is defined as: 49
Continuous: H = ∫ | f ( x) − g ( x) | dx x
(2.3-103) Discrete: H = ∑ x | f ( x ) − g ( x ) | (2.3-104)
For fuzzy sets, on the other hand, we can compare them by computing their Haming distance from a maximum reference set. Then, sets can be ranked in descending order of those computed distances. The methodology is as follows (Yager 1980b, Kerre 1982, Nakamura 1986):
~ 1- For U i = {(u, µU~ (u )) | u ∈ [umin , u max ]} , fuzzy maximum is defined as: i
Ma~x = {(u , µ Ma~x (u ))} (2.3-105) where
µ Ma~x (u ) =
u − u min u max − u min
(2.3-106) 2- Then: H i = ∫ |µU~ (u ) − µ Ma~x (u ) | du i
(Continuous membership)
u
(2.3-107) 50
H i = ∑ | µU~ (u ) − µ Ma~x (u ) | u
i
(Discrete membership) (2.3-108)
~ 3- Rank U i sets based on their H i values.
Note – As the comparison factor in Haming distance is the measurement of areas between fuzzy sets and fuzzy maximum, and it does not consider mean value and spread of membership functions, we may encounter counter-intuitive results.
2.3.3.5.1.6. Left-Right Scores To avoid counter- intuitive results of Haming Distance method, a method called LeftRight Scores (LRS) has been proposed (Chen 1985). The LRS method utilizes two distribution parameters of a fuzzy number (left score and right score) to define its rank.
~ For fuzzy sets U i = {(u, µU~ (u )) | u ∈ [umin , u max ]} : i
1- Define fuzzy maximum and fuzzy minimum sets:
Ma~x = {(u, µ Ma~x (u )} (2.3-109) ~ M i n = {(u, µ M~i n (u )} (2.3-110) where 51
µ Ma~x (u ) = (
u − u min k ) u max − u min
k >0
(2.3-111)
µ M~i n (u ) = (
u − u max k ) u min − u max
k >0
(2.3-112) where k is a factor defined by decision-maker, the more the value of k grows the more the difference between maximum and minimum sets grows. ~ 2- Then right and left scores of fuzzy set U i are calculated as:
~ Ri = Sup(U i I Ma~x) = Sup( Min( µU~ (u ), µ Ma~x (u ))) u
u
i
(2.3-113) ~ ~ Li = Sup(U i I M i n) = Sup( Min( µU~ (u ), µ M~i n (u ))) u
u
i
(2.3-114) 3- Finally, alternatives are ranked based on their LR scores defined as:
LRS i = Ri − Li (2.3-115) Note – This method may encounter two types of problems. It has indiscrimination situation for different fuzzy sets with equal right and left scores. Also, this method ignores absolute location of fuzzy numbers. To resolve the first problem, Chen has proposed that when we get equal LR scores for different fuzzy sets, we can compare them based on a second factor like ‘mean” or “variance” (Chen 1985). To solve the second 52
problem, Chen and Hwang redefined fuzzy maximum and fuzzy minimum sets as follows (Chen and Hwang 1989):
~ u Max = o
0 ≤ u ≤ u max
otherwise
(2.3-116)
~ 1 − u Min = o
0 ≤ u ≤ u max otherwise
(2.3-117) 2.3.3.5.2. α-Cut-based Approaches
This category of methods compares fuzzy sets using α-cuts: 2.3.3.5.2.1. Upper Point Method ~ For fuzzy sets U i = {(ui , µU~ (ui ))} (Adamo 1980): i 1- Select an α-cut, for example α = α 0 ~ 2- Define the α-cut set U i (α 0 ) : ~ U i (α 0 ) = {(ui , µU~ (ui )) | µU~ (ui ) ≥ α 0} i i (2.3-118) ~ 3- Define upper point of U i (α 0 ) : ~ ~ M i (α 0 ) = U iU (α 0 ) = {Max(ui ) | ui ∈U i (α 0 )}
(2.3-119) 53
~ 4- For each α-cut, U i sets are ranked based on their M i (α 0 ) values.
Note – The problem in this method is that the ranking of fuzzy sets is dependent to the value of ‘α’. For different values of ‘α’ we may have different ranking orders. Therefore, our decision in ranking of alternatives is not consistent.
2.3.3.5.2.2. Dominance Index To avoid the inconsistency problem of using α-cuts in upper point method, the Dominance Index method has been proposed (Mabuchi 1988):
~ ~ ~ 1- To compare each pair of (U i ,U j ) , a difference set Dij is computed:
~ Dij = {( d ij , µ D~ (d ij ))} ij
(2.3-120) where d ij = ui − u j (2.3-121)
µ D~ (dij ) = Sup( Min( µU~ (ui ), µU~ (u j ))) ij
d ij
i
j
(2.3-122)
~ ~ 2- Dij (α ) is the ‘α’ level set of fuzzy set Dij :
54
~ Dij (α ) = [d ijL (α ), d ijU (α )] = [d L , dU ] (2.3-123) 3- Define L+ and L− , which are lengths of positive and negative regions of the abovementioned interval, respectively:
0 L = dU d − d L U
d L , dU ≤ 0
+
d L ≤ 0, dU ≥ 0 d L , dU ≥ 0 (2.3-124)
0 L = − d L d − d L U
d L , dU ≥ 0
−
d L ≤ 0, dU ≥ 0 d L , dU ≤ 0 (2.3-125)
~ ~ 4- Degree of dominance of U i over U j at ‘α’ level is defined as: − 1 0 Pij (α ) = +1 L+ − L− L+ + L−
d L = dU < 0 d L = dU = 0 d L = dU > 0 otherwise
(2.3-126) ~ ~ ~ 5- Therefore, degree of dominance of U i over U j is defined as fuzzy set Pij : ~ Pij = {( ρ (α ), µ P~ ( ρ (α ))) | ρ (α ) = Pij (α ), µ P~ ( ρ (α )) = α } ij
ij
(2.3-127) 55
~ ~ 6- To have a single index for comparison of fuzzy sets U i and U j , compute the following index:
Pij = ∫ αρ (α )dα α
(2.3-128) 7- As the above integral may be cumbersome to compute, its approximation has been proposed as:
Pij ≅
1
k n ([∑ nk = 0 kρ (α = α max )] − ρ (α max )) n 2 n2 (2.3-129)
where ‘n’ is the number of sub-intervals defined by α-cuts over [0,1] and ‘αmax’ is the maximum value of α.
8- Then, ranking of alternatives is based on the following pair-wise comparison:
~ ~ U i < U j ~ ~ U i = U j ~ ~ U i > U j
Pij < 0 Pij = 0 Pij > 0 (2.3-130)
56
~ Note – Defining a fuzzy difference set Dij in this method reduces the comparison of
~ ~ ~ fuzzy sets U i and U j to a comparison of Dij with zero. The Dominance Index method provides a consistent way for using α-cuts without any additional assumptions. However, the larger the ‘n’ value, the better the approximation of (2.3-129).
2.3.3.5.2.3. Height of Fuzzy Set Method ~ ~ For fuzzy sets U i = {(ui , µU~ (ui ))} , height value or hgt (U i ) is defined as: i ~ hgt (U i ) = Sup ( µU~ (ui )) i ui
(2.3-131) On the basis of this value, two different indices have been proposed for rank ordering of ~ U i sets: •
Choobineh and Li index (Choobineh and Li 1993):
Let
a
and
d
be
two
numbers
satisfying
~ a ≤ inf{u i | ui ∈ Uim=1 S (U i )} and
~ ~ d ≥ Sup{ui | ui ∈ U im=1 S (U i )} . Then, U i sets are ranked in descending order of the following index:
~ ~ 1 1 ~ ~ ~ ~ CL (U i ) = ( hgt (U i ) − ( ∫ 0hgt (U i ) ( d − U i U (α ) ) dα − ∫ 0hgt (U i ) (U i L (α ) − a ) dα )) 2 d −a
(2.3-132) •
Fortemps and Roubens index (Fortemps and Roubens 1996):
~ Convex fuzzy sets U i are ranked in descending order of the following index: 57
~ FR(U i ) =
~ ~ 1 ~U L hgt (U i ) (U ~ ∫0 i (α ) + U i (α )) dα 2hgt (U i )
(2.3-133)
2.3.3.5.2.4. Degree of Optimism Method ~ Suppose that li and ri are left and right spreads of U i normal fuzzy sets, which are monotonically increasing and decreasing continuous functions, respectively. The ordering ~ index of U i sets is then defined as (Liou and Wang 1992):
~ 1 1 LWλ (U i ) = λ ∫0 ri−1 (α )dα + (1 − λ ) ∫0 li−1 (α )dα (2.3-134) where λ ∈ [0,1] is decision-maker degree of optimism, which the higher the value of λ is, the more optimistic the decision-maker is. In this sense, λ = 1 reflects a completely optimistic decision-maker (risk seeking) while λ = 0 represents a completely pessimistic decision-maker (risk averse).
58
2.4. Energy-Environmental Policy Analysis and Assessment: Areas and Foundations
This section provides a bridge between the theories of fuzzy-based multiple-attribute decision-making reviewed so far in this chapter and policy analysis for energy and environment. Several policy areas of concern will be reviewed with discussion on background and foundations of their associated assessment process. In this way, necessity and importance of developed methodologies, presented in Chapter 3, will be revealed.
2.4.1. Groundwater Remediation Policy Area: Compatibility Assessment Groundwater remediation (GWR) programs comprise a wide range of efforts implemented to protect, or treat groundwater resources against hazardous compounds called “contamination”. There are many sources for contamination threatening groundwater quality. The most important sources in terms of the volume of discharged pollution, and the amount of possible environmental-health risks are landfill sites, industrial facilities, petroleum and gas processing plants, oil spills and leaks, mining wastes, and pesticides in agricultural zones. The importance of groundwater remediation can be seen from different points of view (Nace 1971 and Lehr et al. 2002a): (i) a large portion of the water supply for drinking, aquatic, agriculture and other purposes comes from subsurface resources, (ii) the transport time for groundwater flows is generally long (even up to hundred or thousand years) and therefore the released pollution will remain for a long time with continuous 59
environmental impacts and, (iii) groundwater has many connections to other natural resources, thus any disorder in its quality characteristics may impose harmful impacts on those resources. There are many alternative remediation strategies and technologies to apply in the area of groundwater management. To choose the best possible plan for each contaminated site, environmental decision-makers must evaluate and prioritize remediation policies and technologies, based on their efficiencies, expenditures, environmental risks and compatibilities. To facilitate this decision-making process, expert systems and decision support tools can be employed, especially when the decision environment contains uncertain parameters (Simonovic 2000). The main sources for uncertainty in GWR decision-making in comparison with surface water, arise from less accurate, less deterministic information about subsurface layers, groundwater flow, and interaction between contamination, soil, water and employed policies. This especially applies for dense non-aqueous phase liquid, (DNAPL) (Mercer and Cohen 1990, Huling and Weaver 1991 and Cohen and Mercer 1993) and light non-aqueous phase liquid (LNAPL) (Newell et al. 1995) compounds whose transport and fate are often complex. There are two major types of methodologies to deal with the uncertainties; a stochastic approach is applied when the uncertainty originates from the randomness of parameters, and a fuzzy logic approach is applied when we encounter vague parametric values based on imprecise, or incomplete data and subjective opinions or judgments. Development of decision support architectures for water resources planning, and management using fuzzy set theory has been widely studied for different types of decision-making problems (Kung et al. 1992, Lee et al. 1994, Ravi and Reddy 1999, Yin
60
et al. 1999, Chang et al. 2001 and Prodanovic and Simonovic 2002). However, this literature has not provided a structure for multi-level decisions, and most importantly has not considered the interactions between decision criteria while becoming aggregated. These methodologies use a single aggregation operator, which is not always the most appropriate way to reach a decision. Moreover, none of these contributions target the area of groundwater remediation from a compatibility assessment point of view. The idea of a GWR compatibility assessment was first employed by Knox et al. in 1986 to investigate the compatibility of different remediation technologies with generic pollution classes and soil types. In 1995, the US Environmental Protection Agency (EPA) published an issue paper (McCaulou et al. 1995) to provide a comprehensive literature review regarding the compatibility of non-aqueous phase liquids (NAPLs) with a wide variety of materials used at hazardous waste sites. This included a reference table of compatibility data in form of linguistic attributes for 207 chemicals, and 28 commonly used well construction and sampling equipments. Considering the above facts, in Chapter 3, a novel fuzzy logic based multi-level, multiple-attribute decision aid system, is introduced to assess the compatibilities of the groundwater remediation technologies, with the characteristics of the contaminated zone including contaminants, their distribution and soil properties (Nasiri et al. 2007). The proposed model is then implemented in a real-world case study in Chapter 4.
2.4.2. Transportation Policy Area: Environmental Optimality Assessment Transportation activities are primary contributors to environmental pollution. Mobile sources are among the largest sources of greenhouse gases (GHG) and air pollutants such 61
as NOx and SO2. In Europe, transportation is the source of nearly one-third of energy related greenhouse gas emissions (EC 2001). Between 1990 and 2001 greenhouse gas emissions in the European Union (EU) decreased in most sectors (industry, energy supply, agriculture, waste management); however, emissions from transportation sector showed an increase of 20% in the same period (EEA 2003). In light of the above facts and considering the Kyoto protocol, reduction of GHG to 8% below 1990 levels by 2010, transportation decisions are necessary to be made in an environmentally sensitive way. They should address the aspirations of the society served by these plans from a local point of view, as well as global environmental and energy consumption concerns. Having these complex and conflicting goals in design and operation of transportation systems makes it necessary to apply more advanced models for transportation planning. A typical transportation planning (TP) model is developed to minimize, or effectively reduce the overall transportation cost of a distribution system from manufacturer to customer (Hillier and Lieberman 2002, Goulias 2002). This cost minimization problem is generally subject to supply and demand constraints as well as availability of routes. In case of having decision objectives rather than cost, the model will be converted to a multiple-objective decision-making (MODM) problem (Ringuest 1992, Lai and Hwang 1994, Michnik and Trzaskalik 2002). In this way, objectives such as delivery time, shortest path and other technical/technological goals could be considered in the model (Climaco et al. 1993, Current and Marish 1993, Rakas et al. 2004). Although design of such a model seems not much difficult, in actual transportation problems interactions and/or conflicts among the objectives prevents the model to reach a unique optimal solution for all objectives. Therefore, intensive investigations have been
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made to provide methodologies for obtaining the most preferred solution in terms of decision-maker preferences (Li and Lai 2000). The two of the most common methodologies for dealing with MODM problems are compromise programming and aspiration approach. In compromise programming (Merino et al. 2003, Prodanovic and Simonovic 2002), each solution is evaluated in terms of its distance from the optimal (ideal) solutions of objectives. A feasible solution that provides the shortest overall distance to the ideal solutions by considering all objectives would be considered as the most preferred solution. To calculate such an overall distance, which is the weighted confluence of all distances, the decision-maker should provide weighting values to specify the importance of each objective (distance). In aspiration approach (Abel and Korhonen 1996, Buchanan and Gardiner 2003), the MODM problem will be converted to a single objective model, by optimizing one objective and transferring the others to problem constraints by considering some aspiration levels for those objectives. In comparison, the aspiration approach provides a less complex and more practical methodology, but the solutions are not as preferred as those provided by compromising approach. On the basis of the above review, in Chapter 3, a combination of the above approaches is considered to provide the optimal solution for a transportation planning problem with cost and environmental objectives. The proposed methodology (Nasiri et al. 2006c) has some innovative features. Firstly, it provides an opportunity to combine the simplicity and applicability of aspiration approach, with the optimality of the compromise methods. In addition, to assess the optimality of solutions, instead of finding an overall
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distance, the concept of fuzzy membership function has been used to interpret the optimality of solutions. By this interpretation, the assessment indices can be simply obtained using fuzzy aggregation operators (Yager and Kacprzyk 1997). The associated case study in this dissertation, is also placed among the very few studies (Bielli et al. 1998, Nagurney 2000, Tsamboulas and Mikroudis 2000, Himanen et al. 2005, Yedla et al. 2005) carried out to consider the environmental consequences of transportation activities in a real-world transportation planning problem, presented in Chapter 4.
2.4.3. Waste Recycling Policy Area: Performance Assessment Waste management activities consist of two major different levels of activities in terms of targets and technologies involved, Waste Minimization and Waste Disposal. Waste minimization refers to actions that target the reduction in the quantity of waste managed through disposal. This level of activities includes waste prevention, recycling and waste combustion with energy recovery (Stutz, 2002). Waste disposal takes place when there is not a feasible way to minimize the waste. The disposal is accomplished via burning (without energy recovery) and land-filling (Lehr et al., 2002b). Disposal of wastes has been the main waste management strategy over the past decades. Today, this legacy of the past is a major environmental problem. Waste material may migrate into the surface water or groundwater resources, where it can be ingested and harm the human body and other living organisms. It may enter the food chain via uptake by plants and consequently by humans, with possible long lasting impacts on human health. The gradual decomposition of waste materials generates organic toxic gases, methane and carbon dioxide, which are categorized as air pollutants. Moreover, 64
from a land use perspective, especially in urban areas, space for landfill is either limited or unavailable. On the other hand, open burning and inefficient combustion of waste materials produce air pollution and toxic residues. In addition, the clean-up and remediation technologies applied to eliminate or to reduce the environmental consequences of waste disposal are very expensive and slow to implement due to technological complexities involved (Rubin, 2001). Considering the above-mentioned challenges, in recent years, the waste minimization technologies have been used in many waste management practices (Lehr et al., 2002b). Besides the implementation of waste prevention strategies in product design, manufacturing and packaging and improvement of product reuse opportunities, waste recycling technologies have been applied to recover a wide range of materials from the waste flow. For example, in the United States, recycling rate for municipal solid waste (including paper, plastics, glass, metals, foods, woods, etc.) has reached about 22 percent, and some states have designed a target diversion rate of 50 percent (Kreith and Tchobanoglous, 2002). Despite the many advantages obtainable from developing an effective recycling process, experience has shown that there are several obstacles which hinder the development of effective recycling markets. From a consumer point of view, there is not enough awareness about the recycled products and enough confidence in the quality of recycled material. From a technological point of view, recovery of certain types of recyclable materials such as plastics and oil is difficult and costly, and more technological improvements are still required. Finally, from an environmental perspective, recycling activities (collection, processing and marketing) have certain types
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of cross-media environmental impacts. Therefore, to ensure that the recycling programs, in terms of design and operation, are economically and environmentally responsible, and to keep track of the environmental achievements obtained through these programs, environmental performance measurement indices have been developed for waste recycling programs. An environmental performance index (EPI) is a score value derived from certain economical and environmental parameters and indicators, which represents the state of a program in comparison with the others or a predetermined target or standard. It is similar to the UV (Ultra Violet) index we check before spending the day in the sun. This index takes complex scientific information and synthesizes it in a way that makes it easily understandable. It can help to translate a wide variety of environmental indicators into a simple system that can be easily communicated. This index provides a decision-making tool for the governments in design, implementation and control of environmental policies. The last but not the least, it can provide easy-to-understand information to citizens in order to help them comply with regulations, (like no-burn days and air quality index) or make personal lifestyle choices that will benefit the environment. Measurement of environmental performance using an EPI has been studied in various domains (Tyteca, 1996 and 1997; Courcelle et al., 1998; Ethridge, 1998; Jung et al., 2001; Lippke et al., 2004; Färe et al., 2004). In practice, at corporate level, Nortel was the first company in North America to develop an EPI in 1993 to study the environmental impact of the company operations (Poltorzycki, 1996). At national level, the U.S. Environmental Protection Agency (EPA) launched the national environmental performance track in 2000, a voluntary partnership program to recognize and reward the
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annual improvements in a set of core performance indices (EPA, 2003). Later, World Economic Forum organization introduced the first serious attempt to measure environmental performance in a global scale via one summary index by compounding 22 environmental indicators (Easty and Cornelius, 2002). Although using the environmental performance indices has many beneficial aspects, the uncertainties involved in index making process may result in misleading outputs. To define an EPI, several parameters and indicators are required to be aggregated in relation to their relative weights. Therefore, this index making process relies on the experts’ judgments in selecting the appropriate parameters, and in assigning the appropriate aggregation weights. The situation becomes more complex when we encounter several performance measurement scenarios (such as corporate level scenarios, regional scenarios, etc.). Moreover, there are uncertainties involved in measurement of performance parameters because of incomplete and imprecise data or measurement errors. Fuzzy set theory (Zadeh, 1965) as an advanced method capable of accounting for imprecise information offers a way to deal with these uncertainties. In performance assessment studies using fuzzy set theory, emphasis has been placed on two directions of fuzzy rule-based modeling (Plantamura et al., 2003; Sadiq et al., 2004) and fuzzy data envelopment analysis (Triantis and Girod, 1998; Tsou et al., 2004). In fuzzy rule-based modeling (FRM), a set of fuzzy logic if–then rules are used to build a model for the prediction of environmental performance in form of linguistic assessments. Fuzzy data envelopment analysis (FDEA) is a nonparametric estimation methodology based on linear programming, which measures the relative performance of a collection of decision-
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making units (alternatives) by doing an input-output analysis. Although, both FRM and FDEA methodologies are capable of dealing with the uncertainties embedded in the environmental performance assessment problems, there have been some complexities, rather than fuzziness of data, involved in both methodologies. Design of a fuzzy logicbased rule becomes complex if the number of decision parameters increases. Reaching a decision becomes hard or even impossible when we encounter a large number of rules, especially, when there are overlaps between them. In FDEA, on the other hand, a performance assessment study is computationally intensive. This method works based on the repeated application of linear programming. To solve an LP model, this number would be multiplied by the number of iterations. To avoid the above-mentioned complexities, in Chapter 3, an innovative fuzzy multiple-attribute analysis (FMAA) approach is proposed for the environmental performance assessment of waste recycling programs (Nasiri and Huang 2005). This methodology calculates an environmental performance index, which has the previously mentioned benefits of an index-based performance assessment system with the capability of using fuzzy information. To show the applicability of this methodology, a real-world case study is illustrated in Chapter 4, which investigates the performance of waste recycling programs in Canada.
2.4.4. Surface Water Policy Area: Water Quality Assessment To prevent, control or treat environmental problems related to quality of water, a wide range of technical, economical and social policies classified as water quality management have been proposed at different levels, from the general public to governments, especially 68
in the final decades of the 20th century (Rubin 2001). During this time, water quality management has been considered as one of the most challenging areas of environmental policy analysis. There are several reasons behind this criticality. First of all, priorities and preferences for water quality management always vary from community to community and from country to country. For instance, a nation struggling to provide its citizens with the basic necessities of life is unlikely to be as worried about effects of water quality on wilderness preservation as a wealthy nation. Even in wealthy societies, different ranks have been assigned to the policies in people’s lives, and in the government agenda from time to time and across many regions. Water quality management policies are broad and complex issues, that involve not only technical efforts in the realm of engineering, but also a host of different professions and interests beyond that, such as city and regional planners, government regulatory agencies, and other groups responsible for review and approval of policies. This complexity makes the task of arriving at a final decision a time consuming process, and on occasion impossible. Within boundaries of these complexities, there are many alternative strategies and technologies available to be applied for water quality management (Lehr et al. 2002c), which must be evaluated and be compared by environmental decision-makers for implementation of an optimal choice. To provide a comprehensive but easy to understand methodology for assessment and evaluation of water quality policies, by considering previously mentioned complexities and boundaries, concept of water quality index (WQI) was introduced by Horton in 1965. A water quality index is a composition of parameters affecting the quality of water to summarize the water quality in a single score. The use of this type of index to grade water
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quality has been a controversial issue among water quality scientists, as a single number may not tell the whole story of water quality. However, using a WQI has certain benefits, which made it quite popular in water quality management practices. It takes complex scientific information, and synthesizes it in a way that makes it easily understandable. This index can help translate a wide variety of environmental indicators into a single integrated index (compared to process-based methods such as Total Maximum Daily Load - TMDL (EPA 2002), which is a portfolio of different pollution indicators) that is easily communicated. In practice, different versions of such an index have been developed to address the quality of water bodies for environmental monitoring, regulation and planning (RMP 2002, Environment Canada et al. 2005, OWEB 2005). In the United States, the National Sanitation Foundation is currently using a water quality index developed in 1970. To develop such an index, approximately 142 water quality scientists were surveyed to provide their knowledge about the functioning of 35 water quality parameters. They were asked about the parameters that should be included in WQI. Dissolved oxygen, fecal coliform, biochemical oxygen demand, PH, nitrates, total phosphate, temperature change, turbidity and total solids were selected for index-making. A weighted mean is then used to combine the associated values of parameters to get an overall index, ranging from 0 to 100. According to this calculated index, the quality of water is then represented by one of these categories: excellent, good, fair, poor or very poor (Brown et al. 1970 and Mitchell and Stapp 2000). Although, for four decades, this process has been widely accepted as the way to measure WQI (Prati et al. 1971, Landwehr et al. 1974, Inhaber 1975, House and Ellis 1987, Dojlido et al. 1994, Heinonen and Herve 1994, Suvarna and Somashekar
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1997, Yazdandoost and Katdare 2000, Cude 2001 and Maqsood et al. 2003), there are a number of uncertainties that have not been addressed in this traditional way of index making. This may result in misleading outputs of WQI. First, there are uncertainties involved in measurement of water quality parameters, which are not considered in traditional WQI, especially uncertainties caused by data collection errors. In addition, in measurement of water quality parameters one may encounter uncertain spatial distributions over the water basin. Definitions for quality of water may differ as a result of different possible types of water-uses such as drinking, agriculture, etc. This comes from the fact that for each water quality parameter, there would be different criticalities (weights) and acceptable levels (standards) within different water-uses (Table 2.4.1). For instance, the concentration of coliform bacteria is a valuable indicator of potential microbiological contaminants (for limitations of using such an indicator see (WHO 2001)), which are an important quality factor for drinking purposes. But this is not an important parameter to be monitored when we are assessing the water quality for an agricultural purpose. Therefore, a fixed crisp weighting system like traditional WQI is not able to correctly represent the water quality in a multiplewater-use practice. As mentioned before, finding a WQI for a body of water, very much relies on expert judgments in selection of appropriate water quality parameters and in assignment of their associated weights in relation to water-uses. Therefore, there is another type of uncertainty raised during index making process, caused by personal preferences and linguistic judgments of experts and decision-makers. The traditional WQI is not equipped to address this imprecision. Moreover, in terms of output, there might be overlaps
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between linguistic categories defined to interpret the quality of water. The larger the overlaps, the more difficulties generated to correctly express the quality of water by WQI. To deal with the uncertainties that arise during the water quality index making process, fuzzy set theory (Zadeh 1965) was introduced to the field as an advanced method capable of accounting for imprecise information (Sii et al. 1993). In this application, emphasis has been placed on two directions of fuzzy clustering analysis, (FCA) and fuzzy synthesis analysis (FSA). In fuzzy clustering analysis (Zheng and Ying 1984, Ying 1986, Kung and Ying 1991 and Kung et al. 1992), monitored water quality parameters are compared with standard levels to structure a fuzzy value-standard relation, then based on this matrix of relations, a cluster of water quality parameters is built to address the overall water quality index. In a fuzzy synthesis analysis (Lu et al. 1999 and Chang et al. 2001), we consider tolerance levels for standard values of water quality parameters to describe them in relation to differing water-uses. This provides an opportunity to interpret the standards, and consequently their associated quality categories, via fuzzy numbers. By comparing the outcome of monitoring and sampling works with the pre-determined fuzzy standards, a quality category is assigned to each water quality parameter. Consequently, an overall water quality index is calculated by weighted average of defuzzified parameter values. The FCA and FSA techniques are capable of dealing with all types of previously mentioned uncertainties, except in situations where there is a multiple-water-use framework. Both techniques are able to adjust the water quality index with each wateruse, but are not capable of combining these different uses together. In a fuzzy clustering
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analysis, the method used to structure the cluster is not consistent and is different for each individual case study (Kung et al. 1992). Therefore, there is a need for a more comprehensive approach in water quality index making, to utilize fuzzy set theory in a consistent way and to target a broader range of uncertainties. In Chapter 3, a new fuzzy approach is presented to deal with the uncertainties surrounding the water quality index making, including the problem of having multiplewater-uses (Nasiri et al. 2006a). This approach is formed by collaboration of fuzzy set theory, and theory of multiple-attribute decision-making (Yoon and Hwang 1995). The reason behind using a multiple-attribute analysis to calculate the water quality index is because a water quality index, as previously mentioned, is formed by composition of several water quality attributes (multiple water quality parameters) whereas these attributes are examined against a number of standards to provide the quality attainments. This multiple-attribute decision-making (MADM) framework also enables us to design a structure for modeling the index making process within boundaries of multiple-wateruses. In this way, by providing a multiple-layer analysis structure, we are also able to consider the spatial distributions of water quality attributes. The developed methodology is then followed by a case study in Chapter 4 to illustrate its applicability.
2.4.5. Electricity Generation Policy Area: Capacity Planning EnergyEnvironmental Assessment There is growing consensus worldwide, that climate change is becoming one of the most challenging problems of the 21st century. Climate scenarios predict that the average global surface temperature is to increase by 1.4-5.8oC over this century (NRCAN 2004). 73
In Canada, this climatic change is likely to be felt with a higher magnitude. Warmer and longer summers along with milder winters will cause decrease in sea/land-ice cover, changes in biodiversity, increase in severe weather conditions (i.e. more extreme precipitation and flooding in some areas and more droughts, water shortage and wild fire in the other parts). These environmental shifts are expected to impose social, economic and health consequences and to have great influence on the sustainability of natural resources (NRCAN 2004). To effectively deal with climate change, two complementary courses of actions, prevention and adaptation, have been recommended by the environmental scientists over the past decades to be practiced. Preventive strategies, such as Kyoto protocol, consist of measures introduced to reduce the magnitude of climatic change by aiming at change causes. Adaptation strategies, such as modification of urban infrastructures for flooding control, are on the other hand strategies that target reduction of climatic change risks, and adverse impacts by encouraging the societies to change in accordance with, and in response to climate change. In spite of the broad consensus about the occurrence of global climatic change, there has been a huge debate and controversy in the international scientific community over the major cause of such a change. Some, especially in the United States, believe that climate change is direct consequence of heat discharge to the atmosphere, which has been dramatically increased over the past decades due to growing energy consumption. The others, on the other hand, are pointing to increase in human-induced greenhouse gases (such as CO2, CH4, N2O and halocarbons) over the past century, arguing that these gases
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that trap solar energy and disrupt the heat balance of the planet, have been the major cause of climate change. This controversy about the principal cause of climate change has been a key decision factor worldwide, in choosing the most appropriate prevention strategies. In this sense, electricity generation sector, which is a major producer of greenhouse gases (GHG) due to fossil fuel combustion, and also is a major consumer of energy due to internal needs of generation facilities and life cycle energy requirements (mining, refinery, transportation, etc.), has followed two different directions. In Canada, there has been a growing trend toward more energy efficient power generation (NRCAN 2005a). Also, there has been a significant investment in development of technologies that utilize renewable energy sources, or less GHG-intensive fuels such as nuclear power and natural gas (Tampier 2002, CBCanada 2003). Moreover, some power plants have already started to build up and use sequestration facilities to capture carbon dioxide produced during electricity generation (NRCAN 2006). In addition to the above technological improvements in design of environmentally responsible power generation systems, large-scale (regional, provincial or national) integrated planning models can be used to optimize electricity generation capacities in order to provide a minimum energy and/or GHG intensity, while having a least-cost generation plan. In doing so, a number of studies have proposed use of multiple-objective programming approaches to address the integrated electricity generation planning with conflicting objectives such as cost, environmental concerns and energy efficiency measures (Martins et al. 1996, Mavrotas et al. 1999, Hsu and Chou 2000, Antunes et al. 2004).
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In general, there are two major approaches for dealing with large-scale multiple-objective programming problems: compromise programming and aspiration approach. In compromise programming (Merino et al. 2003, Prodanovic and Simonovic 2002), solutions of the problem are evaluated in terms of their distance to the ideal value of objectives. A feasible solution that provides the shortest overall distance to the ideal values is considered as the most optimal one. To calculate such an overall distance, which is the confluence of all distances, the decision-maker may provide weighting factors to specify the importance of each objective. In aspiration approach (Abel and Korhonen 1996, Buchanan and Gardiner 2003), a multiple-objective programming model will be converted to a single objective model, by optimizing one objective and converting the others to model constraints by considering some aspiration levels for them. In comparison, the aspiration approach provides a less complex and more practical methodology especially for large-scale problems, but the solutions may not be as preferred as those provided by compromising approach. In the light of the above facts, in Chapter 3, this dissertation will propose a multipleobjective optimization model for large-scale planning of electricity generation (Nasiri and Huang 2006b). To reach an optimal solution, a combination of aspiration and compromise approaches is used. First, by applying aspiration approach energy and/or GHG objectives are transferred to model constraints considering their associated target values. Then, on the basis of this transformed model, a set of optimal solutions is obtained. Finally, a complementary compromise programming optimality assessment process using fuzzy set theory concepts is developed, which determines the best compromise scenario.
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The proposed capacity planning methodology combines major multiple-objective programming approaches to take the advantages from both methods. Moreover, it provides a more comprehensive approach compared to studies that use fuzzy set theory only to treat the uncertainties and imprecision embedded in defining the parameters of model (Borges and Antunes 2003, Sadeghi and Hosseini 2006). This dissertation employs fuzzy concepts not only in constructing the optimization model, but also in designing a process for post-optimization assessment of solutions. The constructed methodology will then be utilized in Chapter 4 to define an optimal medium-term integrated capacityplanning scenario for Canadian electricity generation sector.
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Chapter 3 - METHODOLOGY
3.1. Groundwater Remediation Policy Area: Compatibility Assessment Consider a case in which a number of GWR plans are under investigation to determine those that have a higher compatibility with the characteristics of a contaminated zone. The term ‘compatibility’ is identified by a number of factors such as plan durability and chance of completion, which together will make a remediation plan compatible or not. These compatibility factors are affected by several categories of parameters, called effective parameters, which are the characteristics of the effected site, such as soil type and contamination type. Each category of effective parameters may contain several elements.
For instance, in contamination category, different contaminants might be
involved in the remediation site such as petroleum hydrocarbons, heavy metals, etc. Using an MADM approach enables us to compose the interactions between the elements in each category of effective parameters (for a contaminated site) and the compatibility factors of a technology, to define the technology compatibility index. This will lead to the ranking of remediation technologies in terms of compatibility. In many real cases, the contaminated site may also involve regions with different properties. Again an MADM outline can be used to compose the regional indices obtained for compatibility, to obtain an overall compatibility index. In doing so, a methodological decision support system (DSS) is proposed for the compatibility assessment of groundwater remediation technologies with several compatibility factors, and in a remediation site with different regional characteristics. The applicability of this DSS is then illustrated via a case study. 78
Before going through the methodology, let us assume the following definitions for the compatibility assessment problem: -
‘ i = 1, 2, ... , M ’ shows the number of remediation plans under investigation
-
‘ j = 1, 2, ..., G ’ shows the number of compatibility factors to be studied
-
‘ k = 1, 2, ..., K ’ denotes the number of categories for effective parameters
-
‘ n k = 1, 2, ... , N k ’ denotes the number of effective parameters in category ‘k’
-
‘ r = 1, 2, ... , R ’ represents the number of regions at the contaminated site
By considering the above notations, a decision support system for compatibility assessment of groundwater remediation policies is proposed. This decision aid is summarized in Figure 3.1.1 with the following steps:
3.1.1. Effect Sets For a compatibility assessment, first it is required to define the impact of each effective parameter of the remediation site such as soil type, or contaminants on the compatibility factors of the remediation technologies such as durability or set time. The uncertainties involved in defining those impacts lead us to rely on experts’ judgments and assessments. In this way, the impacts are assessed in form of linguistic attributes expressed via fuzzy sets (Zadeh 1975a, 1975b and 1975c). The impact assessment using linguistic terms and fuzzy sets has been recently considered as a significant field of study with numerous realworld applications (Silvert 2000, Marusich and Wilkinson 2001, Martín Ramos et al. 2003, Sadiq et al. 2005, and Shepard 2005). In this compatibility assessment, impact of
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effective parameters (such as contaminant, soil type, etc.) on the value of compatibility factors (i.e. set-time, durability, etc.) are assessed via linguistic terms of VI, I, D and VD (abbreviated form of ‘Very Increasing’, ‘Increasing’, ‘Decreasing’ and ‘Very Decreasing’). These linguistics terms are expressed by fuzzy sets:
~ ij ij ij E ={( e , µ ~ ij ( e ))} nk nk E nk nk
(3.1-1) where e nij ∈ [ −100,+∞) is a real number called effect number, which shows that how much k
change (decrease or increase in percentage) an effective parameter ‘ nk ’ (of category ‘k’) can possibly impose on a compatibility factor ‘j’ (associated with the remediation ~
technology ‘i’), and E nij , called ‘Effect Set’, is a fuzzy set, which depicts the possibility k of these changes via a membership function µ ~ ij (e nij ) . E k n k
3.1.2. Joint Effect Sets To obtain the impact of a remediation region (of the site) on a compatibility factor of a technology, it is required to consider and compose the impacts of all effective parameters associated with that region. The first step of this combination is to investigate the importance (relative weight) of each effective parameter. Relative importance of an effective parameter shows its contribution to the overall impact. For instance, if in a region there are two active chemicals as the possible contaminants with similar impacts on a remediation technology but with different concentrations, the concentrations will act 80
as a weight for those effective parameters (i.e. contaminant). These weights are also determined by experts’ judgments via the linguistic attributes of ‘L’ for low, ‘M’ for medium or ‘H’ for high, and are represented by fuzzy sets for each effective parameter in different categories:
~ Wk , n = {( w, µW~ ( w)) | 0 ≤ w ≤ 1} k k , nk
(3.1-2) As each region is discriminated by its associated contaminant(s) and soil type(s), it can be illustrated by the following set:
r ={( k , n k )}
(3.1-3) For a region, the confluence of the impacts on a technology compatibility factor can be defined via a weighted sum of impacts of all effective parameters of that region. This weighted summation is an arithmetic aggregation, which is a function of fuzzy quantities (Zimmermann 1990, and Mesiar and Fuller 1997). These types of aggregators are obtained based on the extension principle (2.3-16). In this way, by considering the rational behind using the weights as exponents (see below (2.3-69) and (2.3-70)), the weighted sum of impacts, called joint effect set, is defined as:
* Nk K ~ ~ ij wn k JEij ( r ) = ∑ ∑ [ E ] = {(eij , µ JE~ ( r ) (eij ))} nk ij k =1 n k =1
(3.1-4) 81
based on the extension principle (2.3-16) and definition of exponent (power) of a fuzzy set (2.3-13), we have: w*n ij ~ (e ) = Sup ( Min ( Min ([ µ ~ ij (e )] k ))) µ JEij ( r ) ij nk E nk k nk
(3.1-5) and as the percentage of change (joint impacts) should be in [ −100,+∞ ) interval, from (2.3-16):
Nk K ij eij = Max ( −100, ∑ ∑ e nk k =1 n k =1
)
(3.1-6) where wn*k is the representative (mean) value (2.3-90) of fuzzy set W~n
k
The remediation technologies can be regionally ranked in terms of compatibility factors by comparing their associated joint (fuzzy) effect sets. There are a number of methods in literature for comparison of fuzzy sets (see section 2.3.3.5). A simple and easy to understand approach is to compare fuzzy sets based on their mean values, as defined in (2.3-90). Therefore, JE ij* ( r ) can be computed, as the ranking index, which is the mean value of the joint effect set JE~ ij ( r ) . If a compatibility factor is to be maximized to reach a better remediation (such as durability), a higher JE ij* ( r ) value gives a higher compatibility rank, and if a compatibility factor is to be minimized to reach a better remediation (such as set time), a lower JE ij* ( r ) value will show a better rank.
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3.1.3. Regional Compatibility The importance of a factor is the order of magnitude for the influence of that factor on the plan’s success in comparison with the other compatibility factors (relative weight). For example, set time could be a very vital factor in the implementation of some remediation plans such as slurry walls or grouting (Lehr et al. 2002a), but meanwhile it is not as important for the success of the other kinds of applicable technologies. Therefore, similar to section 3.1.2, importance of compatibility factors can be defined in form of linguistic ~ attributes of ‘L’, ‘M’ or ‘H’ interpreted as a fuzzy set W j with a mean value of w*j .
By considering the joint impact of effective parameters of a region on all associated compatibility factors of a technology (joint effect sets) as a compatibility score, the compatibility of that technology with the region (regional compatibility) can be assessed. In doing so, the relative importance of compatibility factors should be considered. In this sense:
~ ~ ~ ~ ~ ,w ~ ~ COM i ( r ) = { JEi1 ( r ), JEi 2 ( r ), ... JEiG ( r )} in relation with {w 1 2 , ... wG }
(3.1-7) From the rational behind using the weights as exponents (see below (2.3-69) and (2.3-70), we can say that:
~ ~ ~ ~ v v v COM i ( r ) = {[ JEi1 ( r )] 1 , [ JEi 2 ( r )] 2 , ... [ JEiG ( r )] G }
(3.1-8) 83
where v j = 1 w*j
(3.1-9) v ~ ~ ~ As in this way: {[ JE~ij ( r )] j | j = 1, 2, ... , G} ⊆ COM~ i ( r ) , and also A ⊆ A ∪ B (see (2.3-2) and
(2.3-10)), the regional compatibility can be mathematically represented by (weighted union of scores):
G v ~ ~ ~ COM i ( r ) = U ( g j .[ JE ij ( r )] j ) = {( ei , µ COM (ei ))} i (r ) j =1
(3.1-10) where ‘ g j ’ is a scalar applied to unify the compatibility factors for the above union aggregation, g j = +1 if the compatibility factor ‘j’ is to be maximized to reach a better remediation (such as ‘Durability’ factor) and g j = −1 if the compatibility factor ‘j’ is to be minimized to reach a better remediation (such as ‘Set time’). Therefore:
ei ={g j ⋅eij }
(3.1-11) and µ
~ (e ) = Max ([ µ ~ (e )] COM i ( r ) i JEi , j ( r ) ij j
vj
)
(3.1-12)
84
The reason for using importance of compatibility factors in an inverted form as exponents to express their weights, is that all factors having higher importance will have lower v j exponents which will result in higher values in ‘under Max’ part of (3.1-12) (memberships range from 0 to 1). This has the effect of making the membership function of (3.1-12) being more determined by the more important attributes (Zimmermann 1990). Similar to section 3.1.2, the remediation technologies can be regionally ranked in terms of compatibility by comparing their regional compatibilities (3.1-10). To compare these fuzzy sets, similar to section 3.1.2, from the alternative methods proposed in the literature for comparison of fuzzy sets (Section 2.3.3.5), for the sake of simplicity, the mean value indicator (2.3-90) is chosen denoted by COMi*(r ) .
3.1.4. Overall Compatibility The objective of the technology prioritization study might be beyond a regional investigation and is to get an overall ranking for the remediation technologies, which considers all regions of the remediation site and their properties. In this way, it is required to compose the regional indices, obtained in section 3.1.3, with respect to the region criticalities (weights). The weight of a region depends on several criteria such as type of water-use in the region (drinking, aquatic, agricultural, etc.), population of the area, volume of groundwater resources in that region, etc. Similar to section 3.1.2, region weights are defined in form of linguistic attributes of ‘L’, ‘M’ and ‘H’ expressed by a ~ fuzzy set Wr with a mean value of w*r .
85
Based on the logic presented in section 3.1.3, the union of the regional compatibility sets, defined for a remediation technology, in relation to the region weights gives us the overall compatibility set associated with that technology:
R v ~ ~ ~ (ei ))} COM i = U [COM i ( r )] r = {(ei , µ COM i r =1
(3.1-13) where v r weights and membership function µ COM~ (ei ) are defined in a way similar to i ((3.1-9) and (3.1-12)).
~
Then by using the mean values of the above sets, denoted by COM i* , an overall compatibility rank can be assigned to the remediation technologies.
In Chapter 4, a real-world case study will be provided to illustrate the applicability of the proposed decision aid model.
86
3.2. Transportation
Policy
Area:
Environmental
Optimality
Assessment Let us consider a transportation network of ‘n’ nodes. Assume that the purpose of this network is to transport R different types of products produced in node ‘1’, which is a production plant, by M different types of transportation means (modalities) to a customer company, which is represented by node ‘n’. In this process the main objective is to minimize the total cost of transportation, while minimizing the environmental impacts of transportation activities, interpreted via e = 1, 2, ... E different environmental objectives. In this sense, considering the literature reviewed in section 2.4.2, the problem is formulated as follows: n −1 n
Minimize
R
M
f C = ∑ ∑ ∑ ∑ α ijrm X ijrm
i≠ j
i =1 j = 2 r =1 m =1
(3.2-1) n −1 n
Minimize
R
M
f1 = ∑ ∑ ∑ ∑ ∆ ijrm ( 1 ) X ijrm
i≠ j
i =1 j = 2 r =1 m =1
(3.2-2) n −1 n
Minimize
R M
f 2 = ∑ ∑ ∑ ∑ ∆ ijrm ( 2 ) X ijrm
i≠ j
i =1 j = 2 r =1 m =1
(3.2-3)
M n −1 n
Minimize
R M
f E = ∑ ∑ ∑ ∑ ∆ ijrm(E) X ijrm
i≠ j
i =1 j =2 r =1 m =1
(3.2-4) 87
Subject to:
n
n −1 M
M
X rm − ∑ ∑ β rm X rm = 0 ∑ ∑ β rm kj kj ik ik
j = 2 m =1
k = 2, 3, ... , n − 1
( i, j ≠ k )
i =1 m =1
(3.2-5) n
M
∑ ∑ β 1rmj X 1rmj = Dr
r = 1, 2, ... , R
j = 2 m =1
(3.2-6) n −1 M
rm X rm = D ∑ ∑ β in r in
r = 1, 2, ... , R
i =1 m =1
(3.2-7) R
m ∑ X ijrm ≤ Π ij
m = 1, 2, ... , M
r =1
(3.2-8) rm X ij ≥ 0
i , j = 1, 2, ... , n
( i ≠ j) (3.2-9)
where X ijrm =
Amount of product ‘r’ (ton) which is transported by transportation mean type ‘m’
from node ‘i’ to node ‘j’ ( i ≠ j )
β ij = 0 rm
1
if i → j is a feasible route for mean ' m ' to transport product ' r ' otherwise
(3.2-10)
88
rm
α ij
C rm = ij 0
if i → j is a feasible route for mean ' m ' to transport product ' r ' otherwise
(3.2-11) rm = C ij
Transportation cost for transferring (each ton of) product ‘r’ by transportation
mean ‘m’ from node ‘i’ to node ‘j’ ( i ≠ j ) E rm (e) rm ∆ ij (e) = ij 0
if i → j is a feasible route for mean ' m ' to transport product ' r ' otherwise
(3.2-12) rm E ij (e)
= Amount of environmental impact in terms of objective ‘e’ (emission or energy
consumption) resulted from transferring each ton of product ‘r’ by transportation mean ‘m’ from node ‘i’ to node ‘j’ ( i ≠ j ) m Π ij = Capacity of transportation mean type ‘m’ (ton) for transportation from node ‘i’ to
node ‘j’ ( i ≠ j ) Dr
= Amount of product type ‘r’ (ton) requested by the customer company (demand)
3.2.1. Upper and Lower Solutions The lower set of solutions is obtained by minimizing each individual objective function with respect to problem constraints. This set is represented by f L = { f CL , f1L , f 2L , ... , f EL } , in which for instance f CL represents the minimum value of the objective function f C with respect to model constraints ((3.2-5) to (3.2-9)).
89
Each lower objective value is associated with a set of values for other objectives. The upper set of solutions consists of the maximum possible values of objectives obtained from these associated sets. This set is denoted by f U = { f CU , f1U , f 2U , ... , f EU } . It can be concluded that: L
f =[f , f
U
]
L
U
L
U
L
U
L
U
= {[ f C , f C ], [ f1 , f1 ], [ f 2 , f 2 ], ... , [ f E , f C ]}
(3.2-13)
3.2.2. Target Solutions Assume that there are P alternative policy packages to address the environmental targets in transportation planning. A policy package ‘p’ is a set of target values for E different environmental objectives defined as: L p = {l ep | e = 1, 2, ... , E}
(3.2-14) where l ep is the target (maximum allowable) value of objective ‘e’ in policy package ‘p’.
In this sense, t ep which is a feasible target value of objective ‘e’ based on policy package ‘p’ is defined as: fU e t ep = l ep L U λ p f e + (1− λ p ) f e
l ep ≥ f eU f eL ≤ l ep < f eU l ep < f eL or if there is no policy target
(3.2-15)
90
where λ p ∈[0, 1] represents the potential optimality degree of objectives which cannot feasibly attain the target of a policy package ‘p’ or there is not a target for them in ‘p’. In this way, having a policy package ‘p’, the transportation planning problem can be rewritten as a single objective optimization problem with the following objective function:
Maximize
λp
(3.2-16) By replacing equations (3.2-1) to (3.2-4) with the following constraints:
n −1 n R M f C = ∑ ∑ ∑ ∑ α ijrm X ijrm ≤ λ p f CL + (1 − λ p ) f CU i =1 j = 2 r =1 m =1
i≠ j
(3.2-17) n −1 n
R M
f1 = ∑ ∑ ∑ ∑ ∆ ijrm ( 1 ) X ijrm ≤ t1 p
i≠ j
i =1 j = 2 r =1 m =1
(3.2-18) n −1 n
R
M
f 2 = ∑ ∑ ∑ ∑ ∆ ijrm ( 2 ) X ijrm ≤ t 2 p
i≠ j
i =1 j = 2 r =1 m =1
(3.2-19)
M n −1 n
R M
f E = ∑ ∑ ∑ ∑ ∆ ijrm(E) X ijrm≤ tEp
i≠ j
i =1 j =2 r =1 m =1
(3.2-20)
91
Subject to:
0 ≤ λp ≤ 1
(3.2-21) and subject to constraints (3.2.5) to (3.2.9).
Let f ( p ) = { f C( p ) , f1( p ) , f 2( p ) , ... , f E( p ) } denotes the optimal set of objective values obtained based on the above model (policy p-based model).
3.2.3. Optimality Assessment In the presence of different policy packages, there is a need for a post optimization method to assess the optimality of policy-based solutions. In doing so, an optimality set is defined for each solution, which gives a membership degree to each objective. This membership measures to what extent the objective is optimized, which can be obtained by comparing the lower and upper solutions with the obtained solutions. Thus, by comparing these optimality sets one may find the most optimal policy-based solution, which gives us the preferred policy for implementation. To determine this preferred policy, an approach based on fuzzy set theory is applied. In case of having a multiple-criteria decision-making with unequal weights for criteria, based on Bellman and Zadeh definition for decision ((2.3-23) and (2.3-24)), the decision can be defined as (Yager and Kacprzyk 1997):
92
~ D ={( xi , µ ~ ( x i ))} D
(3.2-22) where ~ ( xi ) = Agg f ( µ ~ ( x ), w ), f ( µ ~ ( x ), w ), ..., f ( µ ~ ( x ), w ) µD i 1 i 2 i n A A A 1
2
n
(3.2-23) where for j = 1, 2, ... n ,
~ ~ ( x i )) | i = 1, 2, ... , m} A j = {( x i , µ A j
is a fuzzy set that its
memberships represent the attainments of the decision attribute ‘j’ by different decision alternatives ( i = 1, 2, ... , m ), w
j
represents the weight (importance degree) of this
attribute, ‘ Agg ’ indicates any aggregation operator, and f is a function that satisfies the following properties: •
If a > b then f ( a , w) > f (b, w)
•
f ( a, w) is
•
f ( a , 0) = ID
monotone in w and f ( a, 1) = a
where ID (identity element) is such that if it is added to the aggregation it does not change the aggregated value. In such a decision-making problem, selection of the aggregator ‘ Agg ’ and function f
are key issues. The selection of the aggregator is based on the decision-maker degree of
optimism. ‘ Agg ’ could be a fuzzy intersection (2.3-9) for a highly pessimistic decisionmaker or a fuzzy union (2.3-10) in a highly optimistic case (Slowinsky, 1998). The associated function f should be defined according to ‘ Agg ’ operator in such a way that
93
enables us to reduce the effect of the attributes with low importance, and to increase the effect of the attributes with high importance (Yager and Kacprzyk, 1997). In this sense, the optimality set for a policy p-based solution is defined as the following fuzzy set:
~* ( p) ( p) ( p) ( p) ( p) ( p) ( p) ( p) f p = {( f C , µ ~ * ( f C ), ( f1 , µ ~ * ( f1 ), ( f 2 , µ ~ * ( f 2 ), ..., ( f E , µ ~ * ( f E )} f f f f
(3.2-24) where ( p)
µ ~* ( fC ) = λ p fp
(3.2-25) and ( p)
µ ~* ( fe fp
)=
f eU − f e( p ) f eU − f eL
e = 1, 2, ... , E
(3.2-26) By applying (3.2-23) to the above set, an assessment index is defined for measuring the optimality of each policy-based solution. In this way, the most pessimistic aggregation is represented by the following equation, where f ( a , w) = a w : w w w w I pL = [ µ ~ * ( f C( p ) )] c ∧ [ µ ~ * ( f1( p ) )] 1 ∧ [ µ ~ * ( f 2( p ) )] 2 ∧ ... [ µ ~ * ( f E( p ) ] E f
f
f
f
(3.2-27) where {wc , w1 , w2 , ... , w E } are the importance degree (weight) of objectives from decision-maker point of view.
94
The rationale behind using the weights as exponents to express the importance of objectives in (3.2-27) aggregation (intersection) is the fact that all objectives having higher importance will have higher exponents, which will result in lower weighted values for them (memberships range from 0 to 1). This has the effect of making the index being more determined (via minimization) by the more important objectives.
Similarly, the most optimistic aggregation has the following property (note that f ( a , w) = a
1/ w
and
1 0+
= +∞ ):
1 / wc
I Up = [ µ ~ * ( f C( p ) )] f
( p)
∨ [ µ ~ * ( f1 f
1 / w1 ( p ) 1 / w2 ( p) 1 / wE )] ∨ [ µ ~ * ( f 2 )] ∨ ... [ µ ~ * ( f E ] f f
(3.2-28) The reason behind using the inverse weights as exponents to express the importance of objectives in (3.2-28) aggregation (union) is the fact that all objectives having higher importance will have lower exponents, which will result in higher weighted values for them (memberships range from 0 to 1). This has the effect of making the index being more determined (via maximization) by the more important objectives. From the above aggregations, the optimality index of the transportation plan obtained based on policy package ‘p’ is then determined as:
I p = α I Up + (1 − α ) I pL
(3.2-29)
95
where
α ∈ [0, 1]
is the optimism index reflecting the risk loving (optimism) degree of the
decision-maker. The higher α is, the more optimistic the decision-maker is.
The proposed transportation planning optimality assessment decision aid model could be summarized as Figure 3.2.1. In Chapter 4, a real-world case study will be provided to illustrate the applicability of the proposed decision aid model.
96
3.3. Waste Recycling Policy Area: Performance Assessment As discussed in Chapter 2, in this dissertation, a methodology is proposed to compute an index for the environmental performance assessment of the waste recycling programs based on fuzzy information. The Environmental Performance Index (EPI) is interpreted in two directions of efficiency and effectiveness. An efficiency index represents the environmental benefits obtained per each dollar spending. It compares the environmental benefits of a program (as the program outputs) with the program expenditures (as the program inputs) to assess the performance. On the other hand, an effectiveness index compares the environmental benefits of a program (program output) with the volume of waste to be managed through the program (as the inputs). Therefore, an efficiency index targets the economical aspects of the environmental performance, while the effectiveness index investigates the technical/environmental aspects. By considering these two directions, a procedure is proposed with the following steps for the environmental performance assessment of the waste recycling programs (Figure 3.3.1):
3.3.1. Environmental Benefits The environmental performance of a recycling program is assessed based on the comparison of the environmental benefits earned via the program and the program inputs (expenditures/technologies). Therefore, the first step in this assessment process is to define the environmental benefits. As proposed by recent studies (Beck, 2002; Morawski and Felder, 2002a and 2002b; Morris et al., 2005), in this dissertation, the following
97
benefits are considered as the major environmental expectations from a recycling program: •
Energy saving
•
Avoiding greenhouse gases (GHG)
•
Avoiding smog (SOx + NOx)
•
Avoiding litter
•
Avoiding landfill space
These environmental objectives of a recycling program have unequal importance (criticality) from a local or a global perspective. For instance, avoiding GHG emission is an important environmental objective from a global point of view, while the avoidance of litter is not. Instead, it could be a critical issue in decision-making for recycling programs of a city. Moreover, an efficiency study generally targets the local environmental concerns, as the cost factor is not a global issue. A survey conducted among the environmental experts in province of Saskatchewan in Canada suggests Table 3.3.1 as a reference outline for these criticalities, presented by linguistic terms of High, Medium and Low, for the local and the global assessments. Based on the opinion of survey participants about the numerical representation of linguistic terms, these terms are illustrated by triangular fuzzy quantities (Zadeh, 1975a, 1975b and 1975c) as shown in Figure 3.3.2.
3.3.2. Performance Indicators Based on the above five categories of environmental benefits, a set of five efficiency, and five effectiveness indicators are definable. Table 3.3.2 describes these indicators in 98
details. Assume that there are i = 1, 2, ..., m alternative recycling programs to be compared in terms of environmental performance. In this way, if ~
µ ET j ( x i ) represent
the attainments of the j
th
~
µ EF j ( x i ) and
efficiency and effectiveness indicators
( j = 1,2, ..., 5 ) by recycling program ‘i’, respectively, then these indicators will form the following fuzzy sets: ~ ~ EF j = {( xi , µ EF j ( x i )) | i = 1,2, ... , m}
(3.3-1) ~ ~ ET j = {( xi , µ ET j ( x i )) | i = 1, 2, ..., m}
(3.3-2) where the membership functions are in normalized form (see section 2.3.1) to express the above mentioned attainments.
3.3.3. Performance Scenarios and Indices Local and global performance indices are obtained by the aggregation of the performance indicators in relation to their criticalities in local, and global scenarios (Table 3.3.1). In this sense, the efficiency, and the local and the global effectiveness indices are determined on the basis of definition (3.2-23) for weighted aggregation as the follows ( i = 1, 2, ... , m ): µ
~ EF
( x i ) = Agg f ( µ ~ ( x i ),W1L ), f ( µ ~ EF1
EF 2
( x i ),W 2L ), ..., f ( µ ~ ( x i ),W5L ) EF 5
(3.3-3) 99
µ
~ ETL
( x i ) = Agg f ( µ ~ ( x i ),W1L ), f ( µ ~ ( x i ),W 2L ), ... , f ( µ ~ ( x i ),W5L ) ET 1
ET 2
ET 5
(3.3-4)
µ
~ ETG
( xi ) = Agg f ( µ ~ ( x i ),W1G ), f ( µ ~ ( xi ),W 2G ), ... , f ( µ ~ ( x i ),W5G ) ET 1
ET 2
ET 5
(3.3-5)
where W jL and W jG represent the local and the global weights (criticalities) of the j
th
indicator, respectively. To compute the above fuzzy indices, the following definitions are considered:
Definition – For real numbers a1 , a 2 , ... , a n ∈ (0,1] , an aggregation operator ‘ o ’ is defined
as: o a1 , a 2 , ... , a n = Avg ( a1 , a 2
... , a n ) = ( a1 + a 2 + ... + a n ) / n (3.3-6)
Definition – For real numbers a ∈ (0, 1] and W ∈ ℜ + , a function f , is defined as: f ( a, W ) = a
1/ W
(3.3-7) where this function satisfies the properties of (3.2-23) (note that
100
1 0.00+
= +∞ )
Based on the above definitions, for i = 1,2, ... , m , the efficiency, the local and the global effectiveness indices ((3.3-3), (3.3-4) and (3.3-5)) are identified as:
~
µ EF ( x i ) =
1 ~ ~ ~ ~ ~ 1 /M 1 /L 1/ M 1 /M 1 /H ( µ EF1 ( x i ) + µ EF2 ( x i ) + µ EF3 ( x i ) + µ EF 4 ( x i ) + µ EF5 ( x i ) ) 5
(3.3-8)
~
1
~
µ ET ( x i ) = ( µ ET ( x i ) L 1 5
1 /M
~
+ µ ET2 ( x i )
1 /L
~
+ µ ET 3 ( x i )
1/ M
~
+ µ ET4 ( x i )
1 /M
~
+ µ ET5 ( x i )
1 /H
)
(3.3-9)
~
1
~
µ ET ( x i ) = ( µ ET ( xi ) G 1 5
1 /M
~
+ µ ET2 ( x i )
1 /H
~
+ µ ET 3 ( xi )
1/ M
~
+ µ ET4 ( x i )
1 /L
~
+ µ ET5 ( x i )
1 /L
)
(3.3-10)
The reason behind using the inverted weights as exponents to express the importance of indicators, is to recognize even very little differences in the attainments. In this way, all indicators having little importance will have larger exponents, which will result in lower (under aggregation) values in (3.3-3), (3.3-4) and (3.3-5) (membership functions range from 0 to 1). The rational behind using an averaging operator to aggregate the indicators, is the fact that this operator is a compromise between the pessimistic intersection operator (2.3-9) and the optimistic union operator (2.3-10), which are the extreme cases for ‘ o ’ operator in (3.3-6) (Slowinsky, 1998).
101
As (3.3-7) is defined for a real numbers W, to compute the above indices it is required to break down the fuzzy quantities associated with the linguistic terms (Figure 3.3.2) to their α-cuts (see (2.3-4), which are real numbers (Table 3.3.3). For instance, by considering Table 3.3.3 for α = 0.00, the efficiency index (3.3-8) is defined as:
Lower bound: ~
µ EF ( x i )
1 5
~ ~ ~ ~ 5.0 5.0 5.0 2.5 = ( µ EF ( x ) + µ EF3 ( x i ) + µ EF 4 ( x i ) + µ EF5 ( x i ) ) 1 i
(3.3-11) Upper bound: ~
µ EF ( x i ) =
1 ~ ~ ~ ~ ~ 1.25 1.67 1.25 1.25 ( µ EF1 ( x i ) + µ EF2 ( x i ) + µ EF3 ( x i ) + µ EF 4 ( x i ) + µ EF5 ( xi )) 5
(3.3-12) (Note that
1 0.00+
= +∞ and memberships range from 0 to1)
Thus, for each α-cut two values could be found for the efficiency, the local effectiveness ~ ~ ~ and the global effectiveness indices ( µ EF ( x i ) , µ ETL ( xi ) and µ ETG ( x i ) ). Therefore, each of
these performance indices will form a triangular fuzzy set.
3.3.4. Performance Assessment Performance ranking of the alternatives is based on the comparison of their associated fuzzy performance indices. In this way, a method is needed for the comparison of fuzzy quantities (see section 2.3.3.5). As the fuzzy indices are defined via their α-cuts, for the 102
sake of simplicity, it is better to build on an ordering method that works with the α-cuts. There are three classes of ordering methods in fuzzy literature (excluding the linguistic approaches). In the first class, an ordering index is designed to transform the fuzzy quantities into the real numbers. Then, fuzzy quantities are ranked according to the corresponding real numbers (Liou and Wang, 1992; Wang and Kerre 2001). In the second class, a reference set (or more) is designed and then fuzzy quantities are compared with it to be ranked (Kim and Park, 1990; Wang and Kerre, 2001). In the third class, a fuzzy relation is designed to provide pair-wise comparisons of decision alternatives (Yuan, 1991; Wang and Kerre, 2001). In this latter class, there are methodologies that use α-cuts of fuzzy sets to build the pair-wise comparison relations. For instance, suppose the following fuzzy relation (Yuan, ~
~
1991; Wang and Kerre, 2001), which compares two fuzzy sets of Ai and Ak by comparing the upper bound of α-cuts of one quantity, with the lower bound of α-cuts of the other and vice versa:
Pik = ∑ ULik (α ) + ∑ LU ik (α ) α
α
(ULik > 0 )
( LU ik > 0 )
(3.3-13) where: ULik (α ) = U i (α ) − L k (α )
(3.3-14) LU ik (α ) = Li (α ) − U k (α )
(3.3-15) 103
where
U i (α )
and
Li (α )
~
are the upper and lower bounds of α-cut of fuzzy set Ai ,
respectively.
~
~
In this way, fuzzy set Ai is preferred to (is greater than) fuzzy set Ak if Pik > Pki .
For better understanding of the above procedural steps of the proposed decision aid, a real-world case study for performance assessment of provincial recycling programs in Canada will be illustrated in details in Chapter 4.
104
3.4. Surface Water Policy Area: Water Quality Assessment In this dissertation, a methodology is proposed to compute water quality index for rivers, to evaluate and prioritize water quality management policies. The goal is to choose the optimal plan in terms of improvement opportunities for water quality. In doing so, a procedure, as shown in Figure 3.4.1, is proposed with the following steps:
3.4.1. Quality Factors A water quality factor is defined as any possible contamination that could prevent water from supporting its different beneficial uses such as drinking, agriculture, aquatic life and so forth. Pathogens, organic wastes, toxic metals and chemical compounds and even nutrient materials such as nitrogen and phosphorous are considered as major harmful pollutions. Thus, for a body of water, contaminants can be from a various classes. Therefore, the first step in each quality index making process is to determine, based on the types of water-use, which contaminant is a quality factor and which is not. For example, in an agriculture water-use circumstance, pathogens are not a qualitythreatening factor, but some toxic chemicals may impose serious harms on crops and must be considered as a quality factor.
3.4.2. Quality Indicators and Quality Attributes Status of each quality factor in a water body is interpreted via its associated indicator(s). For instance, concentration of coliform bacteria is a valuable indicator of potential microbiological contaminants (for limitations of using such an indicator see WHO 2001), 105
which are an important quality factor for drinking purposes. Thus, the next step in the index making process is to define and estimate the indicators that are associated with previously determined quality factors. These estimations should be accompanied by expert-based assessments to give the decision-makers, especially non-experts, a meaningful decision factor. For example, a concentration of 3.0 mg/L measured for Dissolved Oxygen (DO) may have no meaning for municipal lawmakers unless they could find out whether it is a high or low level of DO. The judgments about quality attributes, are derived from water quality experts (or expert systems) those that consider both characteristics of different water bodies (river, lake, groundwater, etc.) and standards for different water-uses (drinking, agriculture, etc.) in the assessment of quality measurements. In such a direction, two types of efforts should be realized, collection of experts’ judgments (or designing a system to assign these judgments based on experts’ beliefs) and processing these assessments in order to reach an appropriate decision. In case of data collection, experts’ beliefs are gathered via a Delphi process (Linstone and Turoff 1975 and Porter et al. 1991). In the proposed decision support expert system, quality attributes are assigned based on the measured level (concentration) of quality factors (pollutants), which are described by five linguistic terms: “Extra Clean”, “Very Clean”, “Medium Clean”, “Fairly Clean” and “Contaminated”. Then, a fuzzy logic-based process, with the following steps, manages these linguistic attributes to a decision point.
106
3.4.3. Scoring Process To work with the above-mentioned linguistic quality attributes, a system of scoring is considered, where a lower score number represents a better quality of water. This system has maximum and minimum possible scores Smin and Smax. Therefore, each linguistic term, as an uncertain value, can be illustrated by a fuzzy set (Zadeh 1975) distributed within [Smin,Smax] interval. For a river system with i = 1, 2, ... , I different water quality factors and r = 1, 2, ... , R different reaches, the score of the quality factor ‘i’ in reach ‘r’ is ~
expressed by fuzzy set S i, r :
s
s
s
s
~ Si, r = {( , µ S~ ( )) | ∈ ℜ, S min ≤ i, r i =1, 2, ... , I r =1, 2, ... , R
≤ S max }
(3.4-1)
3.4.4. Importance of Factors To obtain a water quality index for each reach, it is required to aggregate all quality factors associated with that reach, with respect to their importance. In each reach of river, there may be different water-uses and demands. Therefore, importance of a water quality factor could be a different value in each reach of a river. For instance, dissolved oxygen could be a vital quality factor in a reach with demand for aquatic life protection, but a less important factor in agricultural zones. Similar to the procedure done to obtain quality attributes, importance of quality factors in each reach can be derived from water quality stakeholders and experts in each reach via a Delphi process, to build a weighting system for quality factors based on types 107
of water-uses and demands. In the proposed decision support expert system, importance degrees for quality factors are categorized into four linguistic classes: ‘Very Important’, ‘Important’, ‘Less Important’ and ‘Unimportant’, which can be interpreted, as uncertain values, by fuzzy sets. Therefore, for reach ‘r’, the importance degree (weight) for quality ~
factor ‘i’ is defined by a fuzzy set Wi, r :
~ Wi , r = {( w , µW~ ( w ) | w ∈ ℜ,0 ≤ w ≤ 1} i, r i =1, 2, ... , I r =1, 2, ... , R
(3.4-2) As in the index-making procedure, a weighted confluence of quality attributes should be considered (see section 2.3.3.3), for the sake of simplicity weights are considered crisp (as proposed by Yager in 1978); representative values need to be found for the importance sets to be used as the model weights. Based on the Centroid Method (see section 2.3.3.5.1.1) which calculates the centroid or centre of gravity of the area under the membership distribution, a representative value w * i,r can be obtained for each fuzzy set ~ Wi, r : 1 ∫ w = 0 w.µW~ ( w ) dw i, r * w i,r = 1 ∫ w = 0 µW~ ( w ) dw i, r
(3.4-3)
108
3.4.5. Reach Quality Index For each river reach, a water quality index can be computed by aggregating the water quality attributes with respect to their weights via a max-min multiple-attribute decision~
making model (section 2.3.3.3). In doing so, for reach ‘r’, a quality index set QI r is defined as the following:
* ~ ~ w* = (s) = QI r S i, r w i,r = {(s, µ QI~ (s)) | µ Q~ Min (( µ S~ (s)) i,r )} I I r r i, r i =1, 2, ..., I r =1, 2, ..., R i =1, 2, ..., I
(3.4-4) In this way the element with the maximum membership value shows the decision point in terms of reach quality index QIr * :
* QIr = {s | µ QI~ (s ) = Max ( µ Q~ I r )} r
(3.4-5)
3.4.6. Prioritizing per Reach A major objective for reach water quality index, is to allow the evaluation of the benefits associated with different alternative water quality management plans in river reaches. This index is well suited to the objective, because it is correlated directly with the water quality classifications, and provides a direct measure of the relative water quality of a reach within the boundaries of water-uses. Thus, a water quality management plan that can potentially decrease a reach quality index, represents an improvement in water
109
quality of that reach, which can be a base for comparing the alternative plans to prioritize them per reach. Therefore, for each plan the effectiveness indicator can be defined in reach ‘r’ as:
+ QIr
*
* * = 1−(QIr QIr (0)) 0
* * QIr < QIr (0)
Otherwise
(3.4-6) where QIr * and QIr * (0) are the estimations for the water quality index of reach ‘r’ (3.4-5) after and before a water quality management plan, respectively.
3.4.7. Reach Importance Degree In terms of water quality, a reach is defined via its associated quality factors. Therefore, it could be assumed that the criticality of a reach, (in terms of water quality) is derived from the criticalities of water quality factors in that reach (which are defined in relation to the reach water-uses). In this sense, the reach importance will be a set, which consists of importance degrees of its quality factors:
~ ~ ~ ~ Wr = {W1, r , W2, r , ... , W I , r }
(3.4-7) Therefore, in a fuzzy sense:
~ Wr =
~ U Wi , r = {( w , µW~ ( w )) | µW~ ( w ) = Max ( µW~ ( w ))} r r i, r i i =1, 2, ..., I
110
(3.4-8) In this way, the fuzzy distribution for a reach importance contains (see (2.3-2)) all fuzzy distributions for criticalities of water quality factors in that reach ( W~i , r ⊆ W~ r ). Therefore, for each reach, there is a weight set that covers and depicts all possible importance degrees and criticalities involved in that reach. Consequently, by this fuzzy union, a reach that has a larger number of quality factors with higher criticalities will have a higher importance.
3.4.8. River Quality Index The river quality index is defined by aggregating the water quality indices for all reaches of the river computed in (3.4-5), with respect to the reach weights computed via (3.4-8). As the reach indices are crisp values, an arithmetic aggregation is used (such as simple additive, weighted product, etc.) instead of set theoretic operators (such as intersection, union, etc.). Using a simple additive weighting (SAW) method (section 2.3.3.1), and considering the representative values of reach importance sets ( w r * ), we have:
QI* = ∑ R w r * .QIr * ∑ R w r * r =1 r =1
(3.4-9)
111
3.4.9. Prioritizing per River Quality The plan effectiveness indicator for the river, similar to reach indicator, shows the amount of improvement in quality of river water that can be achieved via a water quality management plan. Thus, it is defined as:
* * + QI * = 10− (QI QI (0))
QI* < QI * (0)
Otherwise
(3.4-10) where QI* and QI* (0) are the estimations for the water quality index of the river (3.4-9) after, and before a water quality management plan, respectively.
Finally, by comparing these plan effectiveness indicators, a rank can be obtained for each water quality plan, as a higher value of indicator means a higher rank and priority in terms of effectiveness. In Chapter 4, a real-world case study will be provided to illustrate the applicability of the proposed decision aid model.
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3.5. Electricity Generation Policy Area: Capacity Planning EnergyEnvironmental Assessment A planning framework is considered consists of i = 1, 2, ..., m electricity generation regions, which produce electricity using j = 1, 2, ..., n different sources (i.e. hydro, natural gas, etc.). The goal is to optimize this network of electricity producers such that for a given year ‘t’ the electricity is generated with the least overall cost while minimizing total GHG emission, and overall internal energy consumption. On the basis of the literature reviewed in section 2.4.5, the following multiple-objective linear programming model is proposed to represent this optimization problem (with m (3n + m) variables and 2 m( 2 n + 1) constraints (except none-zero constraints)):
Minimize TC
(t )
(t ) = ∑ im=1 ∑ n (C j S ij (t ) + CC j ∆Rij T j ) j =1
(3.5-1) Minimize EM
(t )
= ∑ im=1 ∑ n EM j S ij (t ) j =1
(3.5-2) Minimize EN
(t )
= ∑ im=1 ∑ n EN j S ij (t ) j =1
(3.5-3) Subject to: ∑ nj =1 ( X ij (t 0 ) − Rij (t ) − ∆Rij (t ) + ∆X ij (t ) ) ≥ d i (t ) (1+ α i )
i = 1, 2, ..., m
( k ≠i ) (3.5-4)
113
∑ nj =1 S ij (t ) + ∑ m ( E ki (t) − Eik (t) ) = Di (t ) k =1
i = 1, 2, ..., m
( k ≠i ) (3.5-5)
∆X ij
(t )
i = 1, 2, ..., m ; j = 1, 2, ..., n
≤ Pij
(3.5-6)
∆Rij
(t )
(t ) (t ) ≤ X ij 0 − Rij
i = 1, 2, ..., m ; j = 1, 2, ..., n
(3.5-7)
S ij
(t )
(t ) (t ) (t ) (t ) ≤ ( X ij 0 − Rij − ∆Rij + ∆X ij ) T j
i = 1, 2, ..., m ; j = 1, 2, ..., n
(3.5-8)
S ij
(t )
(t ) (t ) (t ) (t ) ≥ L j ( X ij 0 − Rij − ∆Rij + ∆X ij ) TPi
i = 1, 2, ..., m ; j = 1, 2, ..., n
(3.5-9)
∆X ij
(t )
, ∆Rij
(t )
, S ij
(t)
, Eik
(t)
≥0
i = 1, 2, ..., m ; j = 1, 2, ..., n ; k = 1, 2, ..., m ( k ≠ i )
(3.5-10)
On the basis of the following denotations:
114
TC
(t )
: Total cost of electricity generation for year ‘t’ ($1000)
(t )
EM
: Total GHG emissions resulted from electricity generation in year ‘t’ (KT: Kilo-
Ton) EN
(t )
∆X ij
: Total energy consumption for electricity generation in year ‘t’ (TJ: Terra Joule)
(t )
: Development of electricity generation capacity (GW: Giga Watt) for source ‘j’ in
region ‘i’ for year ‘t’ compared to ‘t0’ (variable) S ij
(t)
: Electricity generation (GWh: Giga Watt Hour) from source ‘j’ in region ‘i’ for year
‘t’ (variable) E ik (t ) :
Total electricity (GWh) exported from region ‘i’ for consumption in region ‘k’ in
year ‘t’ (variable) X ij (t 0 ) : Cj:
Current electricity generation capacity (GW) from source ‘j’ in region ‘i’
Levelized
cost
of
electricity
generation
(including
capital
operation/maintenance and fuel) by source ‘j’ ($1000/GWh) CC j : Tj:
Levelized capital cost of electricity generation by source ‘j’ ($1000/GWh)
Average annual service time (hr) of electricity generation source ‘j’
where T j =365×24×L j =8760 L j Lj:
Loading factor for electricity generation source ‘j’
TPi :
Average annual peak-load time (hr) in region ‘i’
EM j :
GHG (emission) intensity for electricity generation source ‘j’ (KT/GWh)
EN j :
Energy (consumption) intensity for electricity generation source ‘j’ (TJ/GWh) 115
cost,
Rij (t ) :
Scheduled retirement of electricity generation capacity (GW) for source ‘j’ in
region ‘i’ until year ‘t’ (estimated based on service life of electricity generation stations) ∆Rij
(t )
: Additional retirement of electricity generation capacity (GW) for source ‘j’ in
region ‘i’ until year ‘t’ (variable) d i (t ) :
Forecasted peak demand (GW) in region ‘i’ for year ‘t’
α i : Region ‘i’ reserve capacity (% of peak demand) Di (t ) : Pij :
Forecasted electricity demand (GWh) in region ‘i’ for year ‘t’
Potential (maximum) development capacity (GW) for electricity generation from
source ‘j’ in region ‘i’
Note that CC j ∆Rij (t )T j term in total cost objective function, represents the lost opportunity cost (lost investment) due to additional retirement (before end of service life) of electricity generation source ‘j’ in region ‘i’. Also, it should be mentioned that (3.5-8) and (3.5-9) define upper and lower limits of production, as a capacity could be a baseload and generate electricity during the year (based on its loading factor) or at least generate electricity during peak times.
3.5.1. Optimization Using an aspiration approach, a fuzzy single-objective optimization model is constructed by transferring the GHG and energy objectives to model constraints, based on a set of target values. Given a GHG emission target of TGHG : 116
EM
(t )
= ∑ im=1 ∑ n EM j S ij (t ) ≤ TGHG j =1
(3.5-11) And an energy target of TEnergy : EN
(t )
= ∑ im=1 ∑ n EN j S ij (t ) ≤ TEnergy j =1
(3.5-12) The parameters of the above-proposed model could possibly change within a range of values due to uncertainties involved in their estimations even for a medium-term planning. This is because there are various technologies available, or used to generate electricity from each source. Moreover, these technologies could have different efficiency and performance rates due to their age and variations in operating conditions (temperature, humidity, altitude, availability of maintenance, etc.). If volatility of energy market in terms of technology/fuel costs and possible variations in environmental policies are also considered, definition of model parameters by interval values becomes a necessity. In this sense, using the concept of decision-maker degree of optimism (section 2.3.3.5.2.4), technology/policy related parameters of model could be represented by a set of pessimistic and optimistic approximations as follows:
C j = [C j , C j ] = {λ C j + (1 − λ )C j | 0 ≤ λ ≤ 1}
(3.5-13) CC j = [CC j , CC j ] = {λ CC j + (1 − λ ) CC j | 0 ≤ λ ≤ 1}
(3.5-14) 117
L j = [ L j , L j ] = {λ L j + (1 − λ ) L j | 0 ≤ λ ≤ 1}
(3.5-15) TPi = [TPi , TPi ] = {λ TPi + (1 − λ ) TPi | 0 ≤ λ ≤ 1}
(3.5-16) EM j = [ EM j , EM j ] = {λ EM j + (1 − λ ) EM j | 0 ≤ λ ≤ 1}
(3.5-17) EN j = [ EN j , EN j ] = {λ EN j + (1 − λ ) EN j | 0 ≤ λ ≤ 1} (3.5-18) Rij = [ Rij , Rij ] = {λ Rij + (1 − λ ) Rij | 0 ≤ λ ≤ 1}
(3.5-19) TGHG = [TGHG , TGHG ] = {λ TGHG + (1 − λ )TGHG | 0 ≤ λ ≤ 1}
(3.5-20) TEnergy = [TEnergy , TEnergy ] = {λ TEnergy + (1 − λ )TEnergy | 0 ≤ λ ≤ 1}
(3.5-21) where λ ∈ [0, 1] is decision-maker degree of optimism, which the higher λ is, the more optimistic the decision-maker is.
3.5.2. Post-optimization Assessment As mentioned, different technological, financial and environmental policies could be associated with the above-proposed parametric model (i.e. different discount rates in 118
levelization of costs, different energy-emission targets, etc.). This could result in several planning strategies (scenarios) to determine parameters of the model. By optimizing each scenario-based model, different objective values will be obtained. Therefore, there is a need for a post-optimization study to define the best compromise planning strategy. Optimality of each scenario is defined by evaluation of its achievements in terms of capacity planning objectives (i.e. minimum cost, minimum GHG emission and minimum internal energy consumption). In doing so, an approach based on fuzzy set theory, is developed: On the basis of section 2.3.1 definitions on fuzzy sets, assume that there are s = 1, 2, ..., S
different electricity generation planning scenarios. The optimality of scenario
‘x’ can be expressed via a fuzzy set consists of cost and environmental (energy and GHG) elements with their scenario-based optimality as membership function:
~ Opt (x) = {(Cost , µCost ( x )) , ( Environment , µ Environment ( x )) }
(3.5-22) where the scenario-based optimality of elements (with a minimization goal) can be defined as:
Max{TC ( s )}−TC ( x ) s µ Cost ( x ) = Max{TC ( s )}− Min{TC ( s )} s s
(3.5-23) and
119
µ Environment ( x ) = [ µ Environment ( x ) , µ Environment ( x )] = λ µ Environment ( x ) + (1 − λ ) µ Environment ( x )
(3.5-24) where µ Environment ( x ) = Min ( µ EM ( x ) , µ EN ( x ))
(3.5-25) µ Environment ( x ) = Max ( µ EM ( x ) , µ EN ( x ))
(3.5-26) and 1 µ EM ( x ) = Max{EM ( s )}− EM ( x ) s Max{EM ( s )}− TGHG s
EM ( x ) ≤ TGHG
EM ( x ) > TGHG
(3.5-27) 1 µ EN ( x ) = Max{EN ( s )}− EN ( x ) s Max{EN ( s )}−TEnergy s
EN ( x ) ≤ T Energy
EN ( x ) > TEnergy
(3.5-28)
To find out which scenario has attained the highest overall achievement (with respect to all capacity planning objectives), an optimality index should be assigned to scenarios. Considering decision-maker behavior, based on the above-determined fuzzy optimality set, for a completely pessimistic decision-maker the optimality index of scenario ‘x’ can be defined as the intersection of achievements (Slowinsky, 1998): 120
Opt (x) = µ Cost ( x ) ∧ µ Environment ( x )
(3.5-29) And for a completely optimistic decision-maker the optimality index of scenario ‘x’ is defined via the union of achievements (Slowinsky, 1998):
Opt (x) = µ Cost ( x ) ∨ µ Environment ( x )
(3.5-30) In this way, optimality index of scenario ‘x’ is then defined using the previously introduced concept of decision-maker degree of optimism:
Opt ( x ) = λ Opt ( x ) + (1 − λ ) Opt ( x )
(3.5-31)
The above proposed energy/environmental assessment methodology for capacity planning in electricity generation can be summarized as an algorithmic decision aid model (Figure 3.5.1). In Chapter 4, this methodology will be applied to a real-world case study for defining the optimal medium-term capacity planning scenarios for electricity generation in Canada.
121
Chapter 4 – CASE STUDY 4.1. Groundwater Remediation Policy Area: Compatibility Assessment For the case study, a residential area consisting of three different regions R1, R2 and R3 ( r = 1, 2, 3 ) in terms of water supply resources (Figure 4.1.1) was chosen. Groundwater in the area accounts for approximately 30, 60 and 85% of all water-uses in these regions, respectively. The area, especially the subsurface water, is environmentally threatened by pollution and hazardous chemicals from a petroleum processing plant (P) in the northwest, a manufacturing zone (I) in the northeast and a big land fill site (L) not very far from the south end of the area. To transfer some degree of responsibility to the producers of environmentally hazardous materials, the local government has forced the three pollution makers to cooperate in a remediation project to protect the groundwater resources from released contamination. In doing so, three grouting methods ( i = 1, 2, 3 ) have been proposed to effectively stop the dispersion of the contaminants (Table 4.1.1). The problem is to prioritize the remediation plans in terms of regional and overall compatibility measures in order to find the most compatible plan to implement. Grouting is a technique based on the injection of stabilizing liquid slurry under pressure into the soil that can create an impermeable wall (Lehr et al. 2002a). Generally for grouting methods, compatibility can be broken into the compatibility factors ( j = 1, 2 ) of ‘set time’ and ‘durability’ (Knox e al 1986). Set time is the time required for the wall to stabilize, and durability is a measure of the period of time that the grout wall will be effective in stopping the contaminants.
122
These two factors are affected by two types of parameters, the chemicals active in the contaminated site, and the characteristics of the soil in terms of permeability ( k = 1, 2 ). Highly permeable zones accept the grout more quickly than low permeable soil. Orders of magnitude changes (variation) in permeability through the soil layers can also cause gaps and poor sealing in grout walls. Taking into account the materials, equipments and processes used by the pollution producers, the possible chemicals of concern for the groundwater are listed in Table 4.1.2 ( n1 = 1, 2, ..., 10 ) along with their sources. Results of a survey done among the regional environmental remediation experts, in form of a linguistic evaluation, show that there are some interactions ( E~ni , j ) between these possible chemicals and the compatibility factors 1 (Table 4.1.2). Moreover, by categorizing all possible types of soil in terms of permeability features ( n 2 = 1, 2, ... , 9 ) in Table 4.1.3, and doing a similar survey on the impact of the soil types on technology set time and durability; Table 4.1.4 ( E~ni , j ) is 2 obtained. Sampling records in each region of the area are used to help experts assess the distributions of the chemicals (and their associated concentrations, W~n ) along with 1
determining soil types (Table 4.1.5). As seen, the soil type in each region is unique, and therefore, for the sake of simplicity, a unit weight cab be assigned to it ( W~n = {(1, 1.0)} ). 2
Descriptions of linguistic variables resulting from surveys are given in Table 4.1.6 along with the mean values of their associated fuzzy sets, which are illustrated in Figure 4.1.2 and Figure 4.1.3 based on the expert beliefs about the memberships. Although the selection of membership distributions is a key issue in problem modeling via fuzzy 123
frameworks, defining and measuring the membership functions is an empirical area of study. There have been some general directions for deriving these memberships from the experts’ knowledge and experience (Zimmermann 1990), and recently a number of heuristic methods have been developed for optimizing those initial expert-based estimations (Nakamura and Kehtarnavez 1995, Chen and Otto 1995, Mang et al. 1995, Figueiredo and Gomide 1999 and Simon 2002). The prioritization procedure is started by using (3.1-4) to find the joint effect sets (Table 4.1.7). Computation of these sets (by a simple subroutine in Visual Basic or MATLAB, etc.) will lead us to Figure 4.1.4, which presents the joint effect sets (factorial compatibility) of each plan in different regions for set time and durability factors ~
( JE i, j ( r ) ). For example, from (3.1-4), in region 1 ( r = 1 ), the second alternative technology ( i = 2 ) will have the following joint effect set for ‘set time’ factor ( j = 1 ):
10 ~ 9 wn* w*n ~ ~ JE 2,1 ( r = 1) = ∑ [ E n2,1 ] 1 + ∑ [ E n2,1 ] 2 1 2
n1 =1
n2 =1
(4.1-1) Thus, from Table 4.1.5 ( r = 1 → n1 = {1, 2, 3, 4, 5, 6, 7, 10} and n2 = {9} ), we have:
~ ~ 2,1 H * ~ 2,1 ~ 2,1 L* ~ 2,1 L* M* JE 2,1 ( r = 1) = [ E ] + [E ] + [E ] + [E ] n1 =1 n1 = 2 n1 = 3 n1 = 4 ~ 2,1 L* ~ 2,1 M * ~ 2,1 ~ 2,1 ~ 2,1 H* L* + [E ] + [E ] + [E ] + [E ] +E n1 = 5 n1 = 6 n1 = 7 n1 =10 n2 =9
(4.1-2) Now, from Table 4.1.2 and Table 4.1.4, we have:
124
~ L* M* H* L* JE 2,1 ( r = 1) = ( D + VD +D +D )+D
= (D
0.28
+ VD
0.50
+D
0.62
+D
0.28
)+D
(4.1-3) where D 0.28 , VD 0.50 , D 0.62 and D 0.28 sets are identified based on (2.3-13), and the summation will be calculated based on (2.3-16) (see (3.1-5) and (3.1-6) for more details) ~
By finding JE i , j ( r ) sets, JEi*, j ( r ) values will be calculated as shown in Table 4.1.8, which are used, in a regional basis, for the ranking of technologies in terms of compatibility factors (Table 4.1.8). By extending the aggregation procedure in relation to the importance of compatibility factors (obtained from the grouting technology experts in form of linguistic terms (Table 4.1.9), a regional compatibility score set is obtained for each technology (3.1-10). For example, for the second alternative technology ( i = 2 ) in the first region ( r = 1 ) we have:
v v ~ ~ ~ COM 2 ( r = 1) = g1 .[ JE 2,1 ( r = 1)] 1 U g 2 .[ JE 2, 2 ( r = 1)] 2
~ ~ 1/ H * 1/ M * = −[ JE 2,1 ( r = 1)] U [ JE 2,2 ( r = 1)]
(4.1-4) Note that ‘set time’ ( j =1 ) is a minimizing objective → g1 = −1 and ‘durability’ ( j = 2 ) is a maximizing objective → g 2 = +1 . Moreover, from Table 4.1.9 for the second alternative ~
~
technology ( i = 2 ) we have W1= H and W2 = M → v1=1 / H * and v2 =1 / M * . These COM~ i ( r ) fuzzy score sets are illustrated in Figure 4.1.5. The COM i* ( r ) values are then calculated to give us the regional rankings (Table 4.1.10) to highlight the most compatible alternative in each region. As mentioned before, groundwater resources have 125
different availability and patterns of consumption within the contaminated site regions. By adding the region population to the above criteria, Table 4.1.11 will give us the regional criticalities (necessity of groundwater remediation) described via linguistic attributes. Therefore, by using (3.1-13) an overall compatibility score (index) set is computed for alternative technologies (Figure 4.1.6). This will lead us to find the overall compatibility indicators ( COM i* ), and consequently the overall ranking of the remediation technologies (Table 4.1.12).
126
4.2. Transportation
Policy
Area:
Environmental
Optimality
Assessment A transportation company delivers a certain type of product, in two different forms of bulk and packed, from a production plant in Sweden to its major European customer with a demand of 400 tons/month for bulk products, and 150 tons/month for packed. Feasible transportation networks for packed and bulk products are presented in Figure 4.2.1 and Figure 4.2.2. The optimal transportation route is considered a route that provides a minimum overall transportation cost, while minimizing the environmental impacts resulted from transportation. To address the environmental impacts, three major European environmental strategies regarding the transportation activities have been considered as target policies; the EU target for transportation energy efficiency (COM 2003), the EU Kyoto target for GHG reduction (EEA 2003) and the EU acidification strategy (COM 1997). Transportation distances were obtained (Table 4.2.1) for land transport (ViaMichelin 2005) and sea transport (MTC 2005). Data on transportation cost (Table 4.2.2) were estimated based on prices provided by the transportation company without considering special discounts. Data on emissions and energy consumption (Table 4.2.3 to Table 4.2.6) for different types of vehicles originated from the Network for Transport and the Environment (NTM, 2005). These data includes the emissions resulted from production of fuel consumed for transportation. Moreover, emissions for different transportation means were recalculated according to the loading capacities. A number of assumptions were made after discussion with transportation experts in the company. A certain type of truck was selected for transportation - the truck of total 127
weight of 40 tons and the payload of 26 tons with a EURO2 engine, which uses MK1 diesel fuel. Truck loading factors depend on the transportation route and packaging type. Loading factors of 70% (short distance transportation) and 75% (long distance transportation) were used for packed products, 50% (short distance) and 80% (long distance) for bulk products. Other types of modality are boats and train. Boats with loading capacity of 2000 – 8000 tons are generally used to transport products between European ports. Electrical train (SJ) is responsible for transportation of products in Sweden. It should be noted that the environmental load of the SJ train is relatively low due to the use of the hydropower electricity. European trains with fuel mix of 60% of electricity and 40% of diesel are used for transportation outside Sweden. Considering the above assumptions and equations (3.2-5) to (3.2-9) constraints, upper and lower values of transportation planning objectives are obtained by solving the associated single objective optimization sub-problems (Table 4.2.7). Applying policy targets determined based on the minimum cost objective values (Table 4.2.8) will give us (based on (3.2-15) to (3.2-20)) the following constraints (in addition to equations (3.2-5) to (3.2-9), (3.2-17), (3.2-21) and those constraints defined based on objective functions with no policy target) in policy-based models with (3.2-16) as the single objective: In transportation planning model based on EU transportation energy efficiency policy (p=1) with a target of 14% reduction in transportation energy consumption ( f 4L =137.6 < l 4,1 = 428.4 < f 4U = 498.1 ): n −1 n
2
3
f 4 = ∑ ∑ ∑ ∑ ∆ ijrm ( 4 ) X ijrm ≤ 428.4 i =1 j = 2 r =1 m =1
i≠ j
(4.2-1)
128
In transportation planning model based on the EU Kyoto policy regarding GHG (p=2) with a target of 8% reduction in emission of CO2 and NO x as the major greenhouse gases resulted from transportation activities ( f1L = 7,069.1 < l1,2 = 22,293.1 < f1U = 24,231.6 and L U f 3 = 118 < l 3, 2 = 209.1 < f 3 = 227.3 ): n−1 n
2
3
f1 = ∑ ∑ ∑ ∑ ∆ ijrm( 1 ) X ijrm≤ 22,293.1 i =1 j =2 r =1 m=1
i≠ j
(4.2-2) n −1 n
2
3
f 3 = ∑ ∑ ∑ ∑ ∆ ijrm( 3 ) X ijrm≤ 209.1 i =1 j =2 r =1 m =1
i≠ j
(4.2-3) In transportation planning model based on EU acidification policy (p = 3) with a target of 35% reduction in emission of SO2 and NO
x
that are the main contributors of acid rain
( l 2,3 = 12.5 < f 2L = 19.3 and f 3L = 118 < l3,3 = 147.7 < f 3U = 227.3 ):
n−1 n
2
3
f 2 = ∑ ∑ ∑ ∑ ∆ ijrm( 2 ) X ijrm≤ 19.3 λ3 + 61.9(1− λ3 )
i≠ j
i =1 j =2 r =1 m =1
(4.2-4) n −1 n
2
3
f 3 = ∑ ∑ ∑ ∑ ∆ ijrm( 3 ) X ijrm≤ 147.7
i≠ j
i =1 j =2 r =1 m =1
(4.2-5) Optimal solutions of these policy-based models are provided in Table 4.2.9 (optimal routes and their associated amount of transported products) and Table 4.2.10 (optimal objective values). Optimality of objectives for each policy-based solution is defined 129
based on equations (3.2-24) to (3.2-26) (Table 4.2.11). Each set of optimality values shows that to what extent a policy-based solution is close to the optimal solutions of objectives. To assess the overall optimality of each policy-based solution, these objective-based optimality values should be aggregated in relation to the objective weights to obtain an overall optimality index. In doing so, as the environmental impacts from transportation activities are studied from global and local perspectives, it should be noted that some emissions mostly affect the global environment, and some others have the largest impact on the local environment where they take place. For example, CO2 has no significant impact on a local environment, but contributes to the global climate change wherever it is emitted. SO2, on the other hand, has its most significant effect in the local areas where the emission takes place. In this sense, an expert-based assessment of objective weights on a 1 to 5 scale is given for local and global scenarios (Table 4.2.12). Based on equations (3.2-27) to (3.2-29), the optimality indices of policy-based solutions are determined for different values of α. As shown in Table 4.2.13, for this particular transportation problem, from a local perspective, the second policy (EU Kyoto target) will have the highest mark while the third policy, which is the EU acidification strategy, has the highest rank from a global point of view.
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4.3. Waste Recycling Policy Area: Performance Assessment Canada has been a pioneer and a leading country in waste recycling programs. In 1981, the first ever blue-box recycling program was launched in Canada (PPO, 1997). From then on, most Canadian provinces have succeeded in expanding or strengthening regulations covering waste recycling programs including beverage containers. In recent years, Canada has moved towards a greater producer responsibility in beverage containers recycling. While producers responsibility in Canada typically manifest itself as a requirement for refundable deposits on beverage containers (with exception of Ontario), provinces employ a variety of approaches to meet their local needs. For example, there is a distinction that some provinces use a ‘return to retail’ approach, some employ ‘universal depots’ while others use both. Another variation is that Manitoba and Quebec apply a levy on certain producers to pay the large majority of curbside recycling costs. Table 4.3.1 shows these different approaches in beverage containers recycling (CRI, 1998). This variety of employed approaches along with the differences in demographic (Source: Statistics Canada) and production (Webb and Ladak, 2005) features (Table 4.3.2) has resulted in different trends in waste recycling, and generation in terms of quantity of waste, recovery rates, and costs for major categories of beverage containers as shown in Table 4.3.3, Table 4.3.4 and Table 4.3.5 (Morawski, 1998; 1999a; 1999b; 1999c; 1999d; 1999e; 1999f; 2000a; 2000b, Morawski and Felder, 2002a; 2002b; RCO, 2002; Felder and Morawski, 2003; SE, 2003; Morawski, 2003a; 2003b, Morawski and Felder 2004). In this table, to present all the generation and recovery information in
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tonnage, the conversion rates of Table 4.3.5 have been used for the average weight of beverage containers produced in or imported to Canada (Encorp Pacific, 2001). In this situation, assessment of the environmental performance of these various recycling programs could provide an opportunity to compare them in order to address their advantages and/or disadvantages, and to recognize the preferred approach in terms of performance criteria. It also enables us to investigate the opportunities for improvement of provincial beverage container recycling programs. As mentioned before, in such a study both local and global environmental objectives have been addressed, which are expressed by efficiency and effectiveness indicators. Life cycle analysis studies (Morawski and Felder, 2002a; Morris et al., 2005) estimate the major environmental benefits, which are obtainable by recycling every ton (or equivalent unit) of beverage containers (aluminum, glass and plastic) as shown in Table 4.3.7. From recovery information (Table 4.3.4), the annual environmental benefits of provincial recycling programs are estimated in five major categories of benefits (Table 4.3.8). For example, the annual energy saving for province of Saskatchewan is defined as:
Aluminum Glass Plastic 64 4744 8 644 744 8 64 4 744 8 E = ( 24.76)(1,610) + (0.18)(38,950) + ( 4.08)( 2,741) = 58,058
(Barrels of oil) (4.3-1)
Based on the calculated benefits and by considering waste generation statistics (Table 4.3.3) and system cost information (Table 4.3.5), the environmental performance indicators of Table 3.3.2 can be defined. For instance, for province of Saskatchewan, EF1 and ET 1 indicators are calculated as:
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EF1 = (58,058) / (13,156,294) = 0.0044
(Barrels of oil saving per $1 cost) (4.3-2)
ET1 = (58,058) / ( 46,583) = 1.2463
(Barrels of oil saving per each ton of generated waste) (4.3-3)
To express these indicators by (3.3-1) and (3.3-2) sets, the relative (dimensionless) values should be calculated (Table 4.3.9). For example, for province of Saskatchewan we have: µ
Maximum 678 ( ) = 0 . 0044 / 0.0187 = 0.2363 SK ~ EF1
(4.3-4) Maximum 678
µ
~ ( SK ) ET 1
= 1.2463 / 2.5276 = 0.4931
(4.3-5) where the maximum values for EF 1 and ET1 belong to province of Manitoba and province of Alberta, respectively.
According to these memberships, the provincial recycling systems can be assessed and ranked in terms of each individual efficiency and effectiveness indicator (Table 4.3.9). Furthermore, the overall efficiency and effectiveness (local and global) indices can be obtained via (3.3-8), (3.3-9) and (3.3-10). As previously mentioned, the fuzzy sets of the linguistic terms (weights) are represented by their α-cuts to calculate these overall indices. For example for α = 0.00 , from (3.3-11) and (3.3-12), the overall environmental
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efficiency of beverage container recycling programs for province of Saskatchewan is defined as:
µ
~ EF
( SK ) = [
0.2365.0 + 0.277 5.0 + 0.3435.0 + 0.406 2.5 0.2361.25 + 0.2601.67 + 0.2771.25 + 0.3431.25 + 0.406 , ] 5 5
=[
0.00 + 0.00 + 0.00 + 0.11 0.16 + 0.11+ 0.20 + 0.26 + 0.41 , ] = [0.02,0.23] 5 5
(4.3-6) Similarly, the overall environmental efficiency and effectiveness (local and global) indices can be calculated for α = 0.00 for all provinces (Table 4.3.10, Table 4.3.11 and Table 4.3.12). If this process is continued for other α-cuts (for example α = 0.20, 0.40, 0.60, 0.80 and 1.00 ), these indices can be determined and illustrated by fuzzy ~
~
~
sets of EF , ETL and ETG (see Figure 4.3.1). To rank the provincial systems based on the above fuzzy indices, their pair-wise preference relations have been calculated (Table 4.3.13, Table 4.3.14 and Table 4.3.15) by using (3.3-13). For example, when the province of Saskatchewan and the province of British Columbia are compared based on overall efficiency, from Figure 4.3.1 we have (numbers are rounded):
α} =0.00 α } =0.20 α } =0.40 α } =0.60 α = 0.80 α } =1.00 } L ~ (α )= 0.02 , 0.04 , 0.06 , 0.08 , 0.11 , 0.13 EF SK α} =0.00 α } =0.20 α } =0.40 α } =0.60 α } =0.80 α } =1.00 U ~ (α )= 0.23 , 0.20 , 0.18 , 0.16 , 0.15 , 0.13 EF SK
(4.3-7)
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α= 0.00 α } =0.20 α } =0.40 α } =0.60 α } =0.80 α } =1.00 } L ~ (α )= 0.01 , 0.02 , 0.03 , 0.05 , 0.07 , 0.09 EF BC =0.00 α } =0.20 α } =0.40 α } =0.60 α } =0.80 α } =1.00 α} U ~ (α ) = 0.19 , 0.16 , 0.14 , 0.12 , 0.10 , 0.09 EF BC
(4.3-8)
Then from (3.3-14) and (3.3-15): α} =0.00 α } =0.20 α } =0.40 α = 0.60 α } =0.80 α } =1.00 } UL ~ (α )= 0.22 , 0.19 , 0.15 , 0.11 , 0.08 , 0.04 ~ EF SK , EF BC α6 =7 08 .00 α6 =7 08 .20 α6 =7 08 .40 α6 =7 08 .60 α } =0.80 α } =1.00 LU ~ α ( ) = − 0 . 16 , − 0 . 12 , − 0 . 08 , − 0 . 04 , 0 . 01 , 0 .04 ~ EF SK , EF BC
(4.3-9) And: =7 08 .80 α6 =7 18 .00 α} =0.00 α } =0.20 α } =0.40 α } =0.60 α6 UL ~ (α )= 0.16 , 0.12 , 0.08 , 0.04 , −0.01 , −0.04 ~ EF , EF BC SK α6 =7 08 .00 α6 =7 08 .20 α6 =7 08 .40 α6 =7 08 .60 α6 =7 08 .80 α6 =7 18 .00 LU ~ (α ) = −0.22 , −0.19 , −0.15 , −0.11 , −0.08 , −0.04 ~ EF BC , EF SK
(4.3-10) Therefore from (3.3-13): P~
~ EF SK , EF BC
P~
~ EF BC , EF SK
= (0.22 + 0.19 + 0.15 + 0.11+ 0.08 + 0.04) + (0.04 + 0.01) = 0.84 ~ ~ ⇒ EF SK > EF BC = (0.16 + 0.12 + 0.08 + 0.04) = 0.40
(4.3-11) By doing such pair-wise comparisons for all provincial indices, the provincial ranks are obtained in environmental efficiency, and local and global environmental effectiveness (Table 4.3.16).
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4.4. Surface Water Policy Area: Water Quality Assessment The Chao Phraya Delta in Thailand contains three major basins, Chao Phraya, Tha Chin and Mekong. These rivers supply drinking water, support fisheries, transportation and recreation and receive wastewater discharge. Water pollution, through point and nonpoint sources, has become a major environmental concern in these basins. Major point sources of pollution to these river basins include domestic and industrial waste discharges, as well as some agricultural point sources such as pig, poultry, fish and other farms. Non-point sources include agricultural areas such as paddy fields, dry and vegetable farms, which are the main land uses in these basins (Simachaya et al. 1999 and Kaewkrajang 2000). The case study is focused on the investigation of water quality in Tha Chin river basin. It stretches from Chao Phraya River in the north, to the Thai Gulf in the south with a length of approximately 320 km and a basin area of around 12,000 km2, consisting of three reaches (1-upper, 2-middle and 3-lower) with multiple-water-uses. The main objective in the investigation is to assess the water quality by designing an index in terms of various quality factors, and correspondingly, to evaluate the potential treatment plans. The purpose of this evaluation is to prioritize treatment options, based on the amount of improvement they can provide for the quality of river water in order to determine the most effective plan. Based on the possible contaminations in the region (Simachaya et al. 1999 and Kaewkrajang 2000), the water quality is expressed by six different quality factors (Table 4.4.1). There have been certain measurements for these factors in different reaches of the river in 1999. The average values of these measurements are given in Table 4.4.1 136
(Simachaya et al. 1999). By considering different water-uses in river reaches (Table 4.4.2), a group of water quality experts at the University of Regina Environmental Informatics Laboratory (http://env.uregina.ca) are asked to assign their assessments for each of the measured factors (based on Table 2.4.1, Table 4.4.1and Table 4.4.2 information) via five linguistic quality attributes of “EC”, “VC”, “MC”, “FC” and “C” (Table 4.4.3), which are the abbreviated forms of “Extra Clean”, “Very Clean”, “Medium Clean”, “Fairly Clean” and “Contaminated”. To define the above linguistic attributes, according to (3.4-1), a type of fuzzy sets (see (4.4-1) and Figure 4.4.1) with membership distribution of µ S~ (s ) has been considered, i, r along with the quality scores ‘s’ ranging from 1 to 5 (as experts preferred). The main reason for selecting an exponential fuzzy distribution to illustrate the quality attributes, is to show the ability of max-min aggregation approach (section 2.3.3.3) in working with various types of fuzzy membership distribution, which is a major weakness in some aggregation methods that are limited to work with triangular or trapezoidal fuzzy distributions (Chen and Hwang 1992). Moreover, there was a green light from the experts to use the exponential function, as their consensus was that the use of this type of membership function in form of (4.4-1), like a triangular one, facilitates the classification of quality scores by giving a full membership to those that represent a group (1,2,3,4 and 5 for five different groups of linguistic attributes):
137
e − | s − 1| − |s − 2 | e µ~ ( s ) = e − |s − 3| S i, r e − |s − 4 | i = 1, 2, ..., 6 r = 1, 2 , 3 e − | s − 5 |
if if if if if
~ S i , r = EC ~ S i , r = VC ~ S i , r = MC ~ S i , r = FC ~ S i,r = C
(4.4-1)
With respect to different water-uses in river reaches, the experts have proposed their opinions about the importance and criticality of water quality factors in each reach. Table 4.4.4 shows these beliefs in terms of four categories of linguistic values; “VI”, “I”, “LI” and “UI”, which are abbreviated forms of “Very Important”, “Important”, “Less Important” and “Unimportant”. Fuzzy distributions and defuzzification rules (based on (3.4-3) for importance linguistic values are shown in Figure 4.4.2 and Table 4.4.5, respectively. Then, the experts’ judgments about the quality factors, and their importance degrees for different reaches of the river can be summarized in Table 4.4.6. Use of a triangular distribution is proposed to the experts to illustrate the importance of factors, because of its simplicity in terms of finding the representative value, and because there was no particular reason for using other types of fuzzy membership distributions. Although, the selection of membership distributions is a key issue in problem modeling via fuzzy frameworks, defining and measuring the membership functions is an empirical area of study. There have been some general directions for deriving these memberships from the experts’ knowledge and experience (Zimmermann 1990), and recently a number of heuristic methods have been developed for optimizing
138
those initial expert-based estimations (Nakamura and Kehtarnavez 1995, Chen and Otto 1995, Mang et al. 1995, Figueiredo and Gomide 1999 and Simon 2002). ~
From (3.4-4) and Table 4.4. information, reach quality sets QI r are obtained (see ~
Figure 4.4.3). As seen in this figure, QI r sets are defined as the following equations (by considering (3.4-5)): S~ w*3,1 ~ QI1 = 3,1 * ~ S 4,1 w 4 ,1
s ≤ QI1* s > QI * 1
(4.4-2) S~ w1*, 2 1, 2 ~ ~ w*6, 2 QI 2 = S 6, 2 S~ w*5, 2 5, 2
s ≤ QI 2 * * * ~ ~ QI 2 * < s ≤ S 6, 2 w 6, 2 I S 5, 2 w 5, 2 * * ~ ~ s > S 6 , 2 w 6 , 2 I S 5, 2 w 5, 2
(4.4-3) S~ w1*,3 ~ QI 3 = 1,3 * * ~ ~ S 3,3 w3,3 = S 6,3 w 6,3
s ≤ QI3 * s > QI3 *
(4.4-4) Therefore, reach quality indices QIr * (before proposed plans) are computed in Table 4.4.7. The QIr * trend per time illustrates the amount of effectiveness estimated for each plan in different reaches of the river. Table 4.4.7 shows these trends for the potential treatment plans of P1, P2 and P3 by comparing the improvements provided by these plans with those that are computed before plans. The after-plan indices are obtained based on the experts’ assessments and expectations about the water quality attributes after plan implementation, by performing the same procedure as it was done for before-plan 139
indices. Then, the effectiveness indicator + QIr * for each plan is defined with respect to (3.4-6) (Table 4.4.8). This is a basis for the prioritization of the plans in each reach, in terms of expected improvements in the water quality of that reach (Table 4.4.9). To get a water quality index for the river ( QI* ), it is required to aggregate the reach quality indices ( QIr * ) by considering the reach weights ( w r * ). This can be obtained from ~
(3.4-9). The reach weights in this equation are representative (centroid) values of Wr sets calculated based on (3.4-8) (see Table 4.4.10 and Figure 4.4.4). As illustrated in Figure 4.4.4, based on (3.4-8), a reach weight set covers and depicts all possible importance degrees and criticalities involved in that reach. In this way, the first reach (r = 1), the only reach with a ‘Very Important’ quality factor (see Table 4.4.4), has a weight set with larger memberships for higher weight values. Trend of the river index per time provides an effectiveness indicator for each water quality management plan in the river ( + QI * ). After estimating these indicators, a rank can be assigned to each plan (Table 4.4.11) in descending order of + QI * values.
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4.5. Electricity Generation Policy Area: Capacity Planning EnergyEnvironmental Assessment Evidence shows that Canada is one of the most energy-intensive industrialized countries (IEA 1996, NRCAN 1999, NRCAN 2005b). This is mainly because Canada has a vast land area with dispersed population and cold climate. In addition to energy intensity issue, studies (EC 2006) have shown that production of greenhouse gases has been dramatically increased by 26.6 % since 1990 despite the commitment Canada made to reduce these gases to 6% below 1990 level within the period of 2008-2012. In this sense, given the share of electricity generation in total energy consumption, 22 %, and in total produced greenhouse gases, 24% (NRCAN 2005c), design of medium-term planning strategies is a necessary step to advise a more energy-emission efficient power supply mix in Canada. The structure of electricity generation sector in Canada varies significantly among provinces. This is mainly due to regional differences in availability of electricity sources, and variations in provincial population, which is the main driver of electricity demand. Provinces like Quebec and British Columbia have abundant sources of hydro-electricity, while Prairie Provinces like Alberta and Saskatchewan are mostly relying on fossil fuels. A densely populated province like Ontario has been a pioneer in developing a wide-range of sources for electricity generation (hydro, nuclear, coal, natural gas, etc.), while remote and less populated areas like North Western Territories (NWT) and Yukon are generating electricity on a very small scale and from limited number of sources. It should be mentioned that in terms of governance, electricity supply is dominated by
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provincial/territorial monopoly utilities, which account for about 85% of Canada's generating capacity. On the basis of the proposed integrated capacity-planning model, and by setting the scope of planning to year 2020, information regarding the electricity generation parameters has been collected. Table 4.5.1 illustrates the current (year 2005) generation capacities in Canada’s provinces and territories (EIIT 1999, SC 2001, Emera 2004, GQ 2004, NLH 2004, OPG 2004, BCH 2005, CEA 2004, CNA 2005, MHEB 2005, NBP 2005, OPA 2005, ME 2006, NTPC 2006, YE 2006). Table 4.5.2, on the other hand, depicts the current and forecasted (for year 2020) region-based annual and peak electricity demands (NRCAN 1996). Considering previously mentioned uncertainties, optimistic and pessimistic estimations for cost (UEI 1997, OECD 1998, NRCAN 1998, CERI 2005) and technological (OECD 1998, OECD 2000, CBO 2003, NLH 2004, RERL 2004, CNA 2005, SP 2005, CANWEA 2006, ECES 2006) properties of electricity generation sources are shown in Table 4.5.3. The cost estimations are provided based on two different discount rates, considering the possibility of having a low-discount rate (5%) or a high-discount rate (10%) scenario in such a medium-term planning. On the basis of estimated service life for electricity generation facilities (Table 4.5.3) and from the catalogue of current working capacities (SC 2001), upper and lower approximations for regional retirement of capacities until 2020 are determined (Table 4.5.4). As Table 4.5.5 shows capacity development, especially for hydro and wind sources, is limited (Tampier 2002). For hydro sources this is because in Canada, most of the water bodies with appropriate geophysical characteristics have already joined the power generation network. For wind power, on the other hand, lack of wind (or wind below an
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acceptable level) and/or having a long period of extremely cold weather (below minimum operating temperature of wind turbines) limit the development of wind farms in many areas. Moreover, for remote and isolated areas like NWT and Yukon, development of electricity generation capacities such as natural gas, coal and nuclear are not a feasible option in medium term due to huge transportation/distribution costs (truck/ pipeline). Also, due to such isolation, these areas do not have electricity exchange (import/export) with other provinces (at least within the scope of a medium-term planning). Table 4.5.6 illustrates the possible electricity exchange routes, which are based on the idea that electricity is directly exchangeable only between neighboring provinces. In this way, some provinces could be an electricity-exchanger by buying the electricity from one province and selling it to another. Considering optimistic and pessimistic approximations (as a result of previously mentioned uncertainties) for energy and GHG intensities of electricity generation sources provided in Table 4.5.7 (Tampier 2002, NRCAN 2005c, Vestas 2005), the goal is to find the best compromise electricity generation scenario on the basis of cost, energy and GHG objectives. Current total energy consumption in electricity sector is about 3,807,135.47 TJ estimated based on 1990-2003 data (NRCAN 2005c). In an optimistic case, energy consumption will grow with “average annual growth rate” of energy consumption in Canadian industries, 1.35%, (NRCAN 2005a) which is an amount less than that of electricity industry, 1.67% (NRCAN 2005a) as a pessimistic case. In accordance to the Kyoto commitment, energy targets are determined as 6% reduction in the consumption levels estimated via the above growth rates. In terms of greenhouse gases, an extremely optimistic target to achieve would be the Kyoto protocol itself, 6% below 1990 level,
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which was 94,200 KT (NRCAN 2005c), while a pessimistic environmental target would be keeping the GHG in current level, 128,800 KT (NRCAN 2005c). In the light of the above facts, energy and GHG targets for year 2020 are summarized in Table 4.5.8. Given different options for constraining the least cost capacity planning model (no environmental constraints, GHG constrained, energy constrained and both GHG and energy constraints), two possible low and high discount rates, possibility of additional retirements (before service life) and permission for nuclear power development, there are 32 possible planning scenarios. Results of model optimization for these scenarios (linear programming using an excel solver engine developed by Frontline Systems Inc. (for more details see: www.solver.com) are presented in Table 4.5.9 based on a 50% degree of optimism. To determine the best compromise scenario, the objective values are then transformed to optimality scores using (3.5-23) to (3.5-31). For instance, for an energyconstrained planning scenario with discount rate of 5%, and availability of additional retirements and nuclear option, we have:
(3.5-23) ⇒ µ Cost =
Max − 21,044,939.4 32, 495,651− 21,044,939.4 = = 0.794 Max − Min 32,495,651−18,075,392.6
(least-cost optimality) (4.5-1)
(3.5-27) ⇒ EM = 217,863.9 > TGHG = 108,674 ⇒ µ EM =
Max − 217,863.9 485,318.5 − 217,863.9 = = 0.710 Max −TGHG 485,318.5 −108,674
(4.5-2)
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(3.5-28) ⇒ EN = TEnergy = 4,495,109.9 ⇒ µ EN = 1 (4.5-3)
(3.5-25) ⇒ µ Environment ( x ) = Min (0.710, 1.00) = 0.710 (4.5-4)
(3.5-26) ⇒ µ Environment ( x ) = Max (0.710, 1.00) = 1.00 (4.5-5) (3.5-24) ⇒ µ Environment ( x ) = [0.710,1.00] = 0.5 × 1.00 + (1 - 0.5) × 0.710 = 0.855 (environmental optimality) (4.5-6)
(3.5-29) ⇒ Opt = µ Cost ( x ) ∧ µ Environment ( x ) = Min (0.794, 0.855) = 0.794 (4.5-7)
(3.5-30) ⇒ Opt = µ Cost ( x ) ∨ µ Environment ( x ) = Max (0.794, 0.855) = 0.855 (4.5-8)
(3.5-31) ⇒ Opt = 0.5× 0.855 + (1− 0.5)× 0.794 = 0.825 (overall optimality) (4.5-9)
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These optimality scores are provided in Table 4.5.10 and Table 4.5.11 for different planning scenarios. The best compromise plans are then determined for each category of discount rates, 5% and 10%, by ranking the scenarios on the basis of their overall optimally scores (Table 4.5.12). For these optimal solutions, the associated values of model variables (electricity generations, capacity developments, additional retirements and electricity exports) are presented in Table 4.5.13, Table 4.5.14 and Table 4.5.15. Based on these values, the overview of electricity generation sector for year 2020 is given in Table 4.5.16.
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Chapter 5 – CONCLUSIONS 5.1. Discussion The groundwater remediation policy prioritizing decision aid presented in this dissertation (Nasiri et al. 2007) had some simplifications. The case study was simplified in order to draw a clear picture of the problem solving procedure, rather than to present a complex problem with several decision variables. It was believed that complexity was not a matter of validation for the proposed methodology. The main purpose was to demonstrate a semi-hypothetical case study to show the capability of a fuzzy logic multiple-attribute decision-making outline, for the prioritization of remediation policies based on linguistic assessments, which one might need to work with in real-world decision-making problems. Design of such a decision support system had two general objectives. First, it provided a fast tracking decision-making process to consider the criticality of time factor in decision-makings. Second, it integrated the decision elements by using a multipleattribute analysis outline to establish a unique and easy to understand decision index. This could be useful in real-world applications where one might encounter several decision parameters and sources of data. To aggregate fuzzy sets, summation was used in some steps of the procedure and unionization in the others. The type of aggregation depended on the nature of the sets. To combine the effective parameters, effects (impacts) were added to reach a joint effect. To aggregate compatibility parameters, the union of all possible values was used to form the possible compatibility set.
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In transportation planning, a major finding of this dissertation was that environmental policies might result in contradicted results in different domains. The EU Kyoto policy, which is considered as a recommendation for preserving the global environment, could serve as the best local policy for the given transportation network while EU acidification strategy, with local targets of SO2 and NOx reduction, had a better global performance. This dissertation also showed that implementing a direct energy efficiency target would have a lower performance in both local and global scenarios, compared to those policies with indirect targets on energy consumption. The EU Kyoto policy with just 8% target for GHG reduction could even provide higher energy efficiency than setting a direct target of 14%. As a result, there was a higher potential for SO2 and NOx reduction through planning of the transportation network, compared to what could be achieved through the energy efficiency policy (with the best cost performance). This caused the energy efficiency policy to reach a lower rank compared to the Kyoto policy with higher potential emission reductions for SO2 and NOx. Also, the acidification strategy with a high performance in terms of SO2 and NOx was dominated by the Kyoto policy because of having very poor cost efficiency. In the most pessimistic case, where DM decides based on the intersection of attainments (minimum value), the energy efficiency policy-based model got the lowest rank because of having a zero-optimality situation for NOx emission. By increasing the degree of optimism and moving towards the union of attainments (maximum value) effect of this poorly attained criterion decreased and therefore the solution obtained a better rank. Moreover, as the target for SO2 in acidification strategy was not a feasible
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value for the transportation network (12.5 < 19.3), even in the most optimal policy-based solution the planning could reach just 33% optimality for SO2 reduction. It should be mentioned that this environmental policy-based transportation planning (Nasiri et al. 2006c) would provide even higher emission reductions if combined with technological actions such as use of a low sulphur fuel for boats, as recommended by the recently finalized EU directive to reduce atmospheric emissions from seagoing ships (EUOJ 2005), or use of improved types of catalytic converters with less NOx emission for trucks. In the light of the above remarks, for the case study transportation network, the EU Kyoto policy, which required very few modifications in transportation planning (in terms of modality), could be adopted as a short-term strategy with the highest rank in local scenario and the second highest rank in global scenario. This plan could then gradually move towards a more train-based transportation in long-term, which would provide a more preferred global solution. In this dissertation, in terms of the results of the proposed waste recycle performance assessment decision aid model (Nasiri and Huang 2005), by comparing Table 4.3.1 and Table 4.3.16, the provincial systems fell into the following categories: •
Effectiveness-based systems, which included those provinces that had performed a deposit-return (DR) program. By comparison, DR programs were able to more effectively prevent the environmental consequences of the beverage container recycling programs.
•
Efficiency-based systems, which included provinces of Manitoba and Quebec, which had performed a curbside program. These provinces were able to provide a more
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efficient way to manage the associated environmental impacts, but were relatively poor on the effectiveness side. •
The provinces that were neither able to provide a relatively good environmental efficiency, nor a strong environmental effectiveness. This might be a result of applying a less developed approach (with a lower recovery ratio) such as Super BlueBox program in Ontario, or might be a consequence of a less productive system due to improprieties in system implementation and management.
In comparison with the traditional WQI and classical models developed to obtain such an index, this dissertation presents a fuzzy approach to deal with the uncertainties surrounding the water quality index making, including the problem of having multiplewater-uses. The approach was formed by collaboration of fuzzy set theory, and theory of multiple-attribute decision-making. The reason behind using a multiple-attribute analysis to calculate the water quality index was because a water quality index is derived from several water quality parameters, whereas these parameters were examined against a number of standards to provide the quality attainments. Fuzzy synthesis analysis and fuzzy clustering analysis techniques were able to adjust the water quality index with each water-use, but were not able to consider different uses together. Fuzzy multiple-attribute analysis utilized fuzzy set theory in a consistent way that targeted a broader range of uncertainties, including studies with multiple-water-uses. Prioritization of the water quality management plans was just one of the various applications for the water quality index. As this index was designed to trace, and to represent the status of water quality in a body of water, the proposed expert system for
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obtaining the WQI could be used in other types of decision-making problems where decisions need to be made based on the water quality, and/or its trends, such as impact analysis studies, forecasting works and so forth. The results of electricity generation capacity-planning model proposed in this dissertation (Nasiri and Huang 2006b) showed that in medium term to gain an optimal plan for electricity generation, there should be some significant changes in current capacities. In terms of capacity development, in both low and high discount rates natural gas had the biggest growth rate, while almost all hydropower capacities should be utilized. Such a development for natural gas capacities was mainly due to low capital costs of natural gas power generation facilities, which made them suitable to be used as peak load capacities. In addition, natural gas had a quite better environmental performance (energy and GHG) compared to coal-based facilities. Hydro source, on the other hand, had a low overall cost (especially for 5% discount rate) and a significantly low life cycle energy-GHG intensity, which mad it a suitable choice for base-load electricity generation in Canada. It could be seen from Table 4.5.16 that in a low discount rate scenario, hydro electricity would still have the biggest share of electricity generation, followed by natural gas. Nuclear option would also have significant developments in such a scenario due to a very impressive GHG performance. But this option as an energy intensive source of electricity with a big capital cost did not have a significant share in high discount rate scenario. In a high discount rate case, electricity generation was mainly shared by natural gas and hydro sources. It should be noted that the proposed model considered optimization of cost, GHG and energy objectives. Such an assessment recommended
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development of nuclear facilities in low discount rate case. But this decision could be a controversial issue considering other environmental objectives. In both scenarios, wind power as a clean energy with a very low energy intensity (compared to biomass) came at the forth place. Most capacity retirements belonged to coal and diesel power facilities, which were energy-intensive stations and the main producers of GHG in electricity sector. It should be noted that almost all of these retirements were within the context of scheduled retirements, as most of coal power generation facilities were approaching their service life before 2020. Biomass technology as a renewable source of electricity (by use of this type of technology for electricity generation we are reusing CO2 that has already been captured in biomaterials) would not be a major source of electricity in a medium-term plan, mainly due to current low energyefficient technologies. This source is currently usable in remote areas with availability of biomass materials. Upon the availability of more energy-efficient technologies this source of electricity could gain more share. It should be mentioned that almost all of cost and technological parameters in this study were approximated on the basis of a medium-term planning. Long-term planning for electricity generation would be a very complex issue due to the possibility of huge cost-technology shifts (ex. volatility of gas price, development of more energy-efficient biomass technologies, etc.). Also as mentioned before, annual demand and peak demand values that had been used in this planning were based on forecasts provided by Natural Resources Canada. It would be possible that in the long term, these forecasts could change with future implementation of broader, and more effective demand management efforts, and the development of more energy-efficient household appliances. Similarly,
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energy and GHG targets set for this planning could be affected by medium-term political changes, as these targets generally depict government agenda and public pressure for the environment (and in particular for Canada with government-based power generation system). However, the structure of model including objectives, variables and constraints would not be affected by all theses possible changes. A major finding of this dissertation was that lower discount rates could provide less expensive, more environmentally friendly capacity planning scenarios for electricity generation sector. This was mainly because this investment could be a key issue in electricity generation, especially when it comes to protecting the environment. A lower discount rate could ensure a lower levelized capital cost for electricity generation. This could create more opportunities for development of more wind, nuclear and biomass power generation facilities. It also should be mentioned that the solutions and optimal scenarios were obtained from a decision-maker with 50% degree of optimism (a completely compromise approach toward risk). Choosing a degree of optimism explicitly depended on the judgment of the decision-maker in attainability of parameter approximations. The more attainable the optimistic approximations were (recognized by decision-maker) the higher the degree of optimism was. Note that the methodologies proposed in this dissertation are flexible and open decision support architectures, which need extensive discussions with the users of the concerned organization and/or stakeholders involved in the area to understand their mental models, expectations and preferences. These methodologies highly rely on the experts’ patterns of thinking obtained by surveys or expert systems. Therefore,
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information gathered from the experts for the case studies is comprised of their beliefs and opinions. Information from other experts may have yielded different results. However, the important factor in implementing such systems is to design a suitable mechanism to gather, test and verify all data captured from the experts (see Linstone and Turoff 1975, Neufville 1990b and Porter et al. 1991). The proposed methodologies provide systematic structures to classify, and process the information required to reach a decision. If computer-based modules support these methods, by considering the criticality of time factor, this can provide helpful tools for non-experts involved at different levels of such decision-making processes.
5.2. Summary of Achievements In this dissertation research, innovative decision aid methodologies have been developed on the basis of multiple-attribute analysis, and theory of fuzzy sets to assist decisionmakers in complex multiple-criteria energy-environmental policy analysis. Using fuzzy set theory has provided the opportunity to treat the uncertainties embedded in such an analysis, especially those that arise from involvement of ill-defined or imprecise information. In the compatibility assessment for groundwater remediation policy analysis, a decision aid tool was introduced for the prioritization of remediation technologies based on the estimation of a compatibility index. As this model received data in form of linguistic judgments and experts’ opinions, fuzzy set theory was used to deal with such uncertainties. First, the concept of compatibility was broken down into measurable factors. Then by using a multiple-attribute decision-making outline, factorial, regional
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and overall compatibility indicators were computed for each remediation technology. Finally, by comparing these generated indicators, the remediation technologies were ranked. In the energy-environmental assessment for transportation planning policy analysis, a multiple-objective optimization approach was applied to define a least-cost solution for transportation network, while providing a preferred solution in terms of GHG emission and energy consumption targets. The optimization approach combined with a postoptimization assessment designed on the basis of multiple-attribute analysis. The optimality indicators in such an assessment were obtained using the logic of fuzzy sets and the concept of membership function. Aggregation of such indicators using decisionmaker degree of optimism approach was finally led to construction of a number of interesting planning scenarios. In the waste recycling policy area, a framework for environmental performance assessment of waste recycling programs was proposed. This assessment had three major steps, definition of performance indicators, measurement, and aggregation of them. Two categories of indicators were considered: The efficiency indicators introduced to compare the environmental achievements of a program with the required expenditures (benefits per unit cost) and the effectiveness indicators developed to compare the environmental benefits of a program, with the amount of generated wastes (benefits per unit waste). The simplicity and understandability were two major motivations for designing such an index. To obtain the performance indices, which were the weighted aggregation of performance indicators, a fuzzy multiple-attribute analysis was used based on the Bellman and Zadeh’s definition for decision and its expression by Yager and Kacprzyk. It provided the
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opportunity to express the criticalities of the indicators by fuzzy sets, and to formulate the aggregation process. It should be mentioned that the importance of each performance indicator should be defined based on the objective of performance assessment. For instance, from a global perspective, a reduction in greenhouse gases is generally a top priority, but is not a high priority in local scenarios. In case study, performance of Canada provinces was studied, as they have employed different categories of beverage container recycling systems. In the surface water policy analysis, a comprehensive tool was proposed for the assessment and evaluation of water quality based on development of water quality index (WQI). Due to complexities involved in obtaining such an index (i.e. multiple-wateruses, multiple river reaches, etc.) the proposed methodology was designed to be able to not only structure and identify information relevant to the problem, but also help nonexpert users reach a decision. This was achieved by constructing a multiple-attribute decision support expert system, which makes expert knowledge available to non-expert users. In doing so, qualitative or linguistic assessments were required in index making process. Thus, fuzzy set theory was employed to recognize such inherent fuzziness. The computed water quality index was used to provide an outline for the prioritization of alternative water quality management plans by defining the expected amount of improvements in WQI. At the end, applicability and usefulness of the proposed methodology was revealed by a real-world case study. In the energy-environmental policy analysis for electricity generation planning, an integrated capacity planning model for a network of electricity generation facilities was proposed. The model provided a multiple-objective planning approach, which was
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looking for a least-cost solution while considering minimization of environmental objectives of GHG and energy intensity. Uncertainties embedded in definition of model parameters were treated using lower and upper approximations. These approximations were utilized in optimization process by an approach based on concept of decision-maker degree of optimism. Given different possible planning scenarios, a post-optimization process was provided using fuzzy set theory concepts. In this way, the best compromise scenarios were determined for low and high discount rates. The result of model implementation in Canadian electricity generation sector showed that, in a medium term (year 2020), major capacity growth would be assigned to natural gas facilities, accompanied by retirement of most coal-burning facilities. The optimization also approves the fact that within this period of time hydro and natural gas sources will be the main producers of electricity followed by nuclear and wind technologies.
5.3. Recommendation for Future Research
a) In compatibility assessment, combining the compatibility assessment with a costbenefit analysis of alternatives and a risk assessment study could provide an opportunity for constructing a comprehensive site-technology assessment decision support system. b) In transportation planning, consideration of other environmental targets related to pollutions (ex. soil contamination, water quality impacts, etc.) caused by transportation activities, (directly or indirectly) will turn the model to an interesting broad environmental assessment tool.
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c) In performance assessment, equipping the methodology with a number of submodels, capable of analyzing the performance of individual steps of recycling programs, will provide an opportunity to target the whole program, as well as its components. In this way, the role of each component in performance of the whole system is definable, and thus the recommendations for performance improvement would be more detailed. d) In water quality assessment, designing a procedure to link the water sampling process with results of the water quality assessment could be a very interesting issue. In this way, for instance the number of required samples and distribution of sampling areas could be addressed by the evaluated water quality index. e) In energy-environmental assessment for capacity planning, to better manage the uncertainties embedded in such a planning process, a stochastic approach can be coupled with employed fuzzy concepts. This is especially useful when electricity demand and peak demand values are estimated based on possible climatic scenarios.
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183
Figures
184
Membership
1 0.8 0.6 0.4 0.2 0 a
m
b
Figure 2.3.1 – Triangular Fuzzy Set
185
x
Membership
1 0.8 0.6 0.4 0.2 0 a
b
c
d
Figure 2.3.2 – Trapezoidal Fuzzy Set
186
x
Identification
Alternatives
Of Attributes
Problem
Collection Identification Processing
Of Data
Defuzzification
Classical
Aggregation Models
MADM
Ranked Alternatives
Figure 2.3.3 – Fuzzy-Crisp Approach
187
Identification
Alternatives
Of Problem
Attributes
Identification
Collection
Of Processing
Data
Aggregation Models
Fuzzy MADM
Aggregation-Ordering Models Ordering Models
Ranked Alternatives
Figure 2.3.4 – Fuzzy-Fuzzy Approach
188
µU~
i
1 1 0.8 0.6 0.4
α0.20 ui
0
0.0.0.0.0.0.0.0.0.0.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2.2.2.2.2.2.3.3.3.3.3.3. 001020304050607080900010203040506070809000102030405060708090001020304050
uiL (α 0 )
uU i (α 0 )
~ Figure 2.3.5 – Estimation of membership distribution for U i
189
Membership
1 0.8 0.6 0.4 0.2 0
β }
α } a
b
x
Figure 2.3.6 – Bonissone trapezoidal fuzzy number
190
A1,1 A2,1 A3,1
Ah −1,1 Ah,1
A2,2 ……… A2, n 2 A3,2 …..…….…… A3, n3
Ah −1,2 ………………... Ah −1, n h −1
Ah,2 ……………………….……… Ah, nh
Figure 2.3.7 – An h-level decision-making problem
191
Effect Sets Effective Parameters
Joint Effect Sets
Joint Effect Indicators
Regional Compatibility Sets/Index
Region Criticality Weight
Overall Compatibility Sets/Index
192
Compatibility Factors
Figure 3.1.1 – Compatibility assessment decision support system
Importance
Environmental Targets
Upper and Lower Solutions
Optimality Indices
Optimal Solution
193
Transformed Single Objective Model
Figure 3.2.1 – Optimality assessment decision aid for transportation planning
MultipleObjective Transportation Model
Efficiency Indicators
Environmental Benefits
Efficiency Index
Local Criticalities Global Criticalities
Local Effectiveness Index
Effectiveness Indicators
Global Effectiveness Index
Performance Assessment
Figure 3.3.1 – Environmental performance assessment decision aid
194
M
L
H
Membership
1.00 0.80 0.60 0.40 0.20 0.00 0.00
0.20
0.40
0.60
0.80
1.00
Weight
Figure 3.3.2 – Membership distributions for fuzzy weights
195
Quality Factors
Quality Attributes
Reach Index River Index
Water Uses
Reach Weights
Factor Weights
Figure 3.4.1 - Fuzzy river-pollution decision support expert system
196
Technological Policies
Planning Strategies
Environmental Policies
Mathematical Model
System Characteristics
Optimization
Solutions
Post-Optimization Assessment
Optimal Strategy
Figure 3.5.1 - Integrated capacity planning decision aid model
197
Figure 4.1.1 - Remediation regions and contamination sources
198
1.00
VD
D
I
VI
0.80
µ
0.60 0.40 0.20 0.00 -80 -60 -40 -20 0 20 40 60 80
e nij
k
Figure 4.1.2 – Fuzzy distributions for effect linguistics
199
1
L
M H
0.8 0.6
µ 0.4 0.2 w
0
0.0 0.2 0.4 0.6 0.8 1.0
Figure 4.1.3 – Fuzzy distributions for weights
200
(a) i=3
i=2
i=1 1.00
1.00
1.00
0.80
0.80
0.80
µ 0.60
µ
0.60
µ
0.60
0.40
0.40
0.40
0.20
0.20
0.20
0.00 -100 -60 -20 20 60 100
e11
0.00 -100 -60
-20
20
60
e 21
0.00 -100 -20 60 140 220 300
e 31
(b) i=2
i=1
µ
i=3
1.00
1.00
1.00
0.80
0.80
0.80
0.60
0.60
µ
µ
0.60
0.40
0.40
0.40
0.20
0.20
0.20
0.00 e12 -100 -60 -20 20 60 100
0.00 -100 -60 -20 20 60 100
r=1
r=2
e 22
e 32 0.00 -100-70 -40 -10 20 50 80
r=3
~
~
Figure 4.1.4 - Joint effect sets for (a) set time JE i ,1 ( r ) and (b) durability JE i , 2 ( r )
201
i=2
i=1
µ
i=3
1.00
1.00
1.00
0.80
0.80
0.80
0.60
0.60
µ
0.40
µ
0.60
0.40
0.40
0.20
0.20
e2 0.00 -100-60 -20 20 60 100140
0.20
e1 0.00 -100-60 -20 20 60 100140
r=1
r=2
0.00 -320-240-160 -80 0
r=3
~
Figure 4.1.5 – Regional compatibility (score) sets COM i (r )
202
e3 80
P1 1.00
1.00
0.80
0.80
0.80
0.60
µ
P3
P2
1.00
0.60
0.60
µ
0.40
µ
0.40
0.20
0.20
0.20
e1 0.00 -150 -90 -30 30 90 150
0.40
0.00 -120 -70 -20 30 80 130
e2
0.00 -320 -220 -120 -20
~
Figure 4.1.6 – Overall compatibility (score) sets COM i ( r )
203
e3 80
City 1
City 2
Production plant
Termina
City 3
Customer
City 4
Figure 4.2.1 – Feasible transportation network for packed products
204
City 1
City 2
City 3
City 4 Production plant
Termina
Customer
City 5
City 6
Figure 4.2.2 – Feasible transportation network for bulk products
205
BC NB NS QC
1.00
1.00
0.90
0.90
0.90
0.80
0.80
0.80
0.70
0.70
0.70
0.60
0.50
0.40
Global Effectiveness Index
1.00
Local Effectiveness Index
Efficiency Index
AL MB NF ON SK
0.60 0.50
0.40
0.60
0.50
0.40
0.30
0.30
0.30
0.20
0.20
0.20
0.10
0.10
0.10
0.00
0.00
0.00
M embership
1.00
0.80
0.60
206
0.40
Figure 4.3.1 – Efficiency and Effectiveness indices
0.20
M embership
0.00
1.00
0.80
0.60
0.40
0.20
0.00
1.00
0.80
0.60
0.40
0.20
0.00
M embership
1.00
EC
VC
MC
FC
C
Membership
0.80 0.60 0.40 0.20 0.00 1
1.5
2
2.5
3
3.5
4
4.5
5
Score
Figure 4.4.1 - Membership distributions for quality attributes
207
Membership
1
UI
LI
I
VI
0.8 0.6 0.4 0.2 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
Weights
Figure 4.4.2 - Membership distributions for importance degrees
208
r=1 1 Membership
0.8 0.6 0.4 0.2 0 1
1.5
2
2.5
3 3.5 Score
4
4.5
5
4
4.5
5
4
4.5
5
r=2 1 Membership
0.8 0.6 0.4 0.2 0 1
1.5
2
2.5
3 3.5 Score
r=3 1 Membership
0.8 0.6 0.4 0.2 0 1
1.5
2
2.5
3
3.5
Score
i =1
i =2
i =3
i =4
~
i =5
i =6
Figure 4.4.3 - Reach quality index set QI r and the optimal index QIr *
209
r=1 1
Membership
0.8 0.6 0.4 0.2 0 0
0.2 0.4
0.6 0.8
1
Weight
r=2 1
Membership
0.8 0.6 0.4 0.2 0 0
0.2 0.4 0.6 0.8
1
Weight
r=3 1
Membership
0.8 0.6 0.4 0.2 0 0
0.2 0.4 0.6 0.8 Weight
1
~
Figure 4.4.4 - Reach weight sets Wr
210
Tables
211
Table 2.3.1 – A multiple-attribute decision-making table with fuzzy values
~ W1
~ W2
…
~ Wn
Attributes
i 1
~ r1,1
~ r1,2
…
r~1, n
2
~ r2,1
~ r2, 2
…
~ r2, n
Alternatives
…
m
~ rm,1
~ rm, 2
212
…
…
n
…
…
…
2
…
1
j
~ rm, n
Table 2.4.1 - Water quality standards for different water-uses* Water-use
Quality Factor (mg/L)
DO a BOD5 b Total Coliform Bacteria c Nitrate-nitrogen Ammonia-nitrogen Total phosphorus
Drinking
Aquatic Recreation Balanced life
use
-
>6
>6
>4
< 1.5
-
-
