DEVELOPMENT OF INEXACT T2 FUZZY OPTIMIZATION APPROACHES FOR

SUPPORTING ENERGY AND ENVIRONMENTAL SYSTEMS PLANNING UNDER

UNCERTAINTY

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 Environmental Systems Engineering

University of Regina

by

Lei Jin

Regina, Saskatchewan

April, 2014

Copyright 2014: L. Jin

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Lei Jin, candidate for the degree of Doctor of Philosophy in Environmental Systems Engineering, has presented a thesis titled, Development of Inexact T2 Fuzzy Optimization Approaches for Supporting Energy and Environmental Systems Planning Under Uncertainty, in an oral examination held on April 22, 2014. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: *Dr. Lianfa Song, Texas Tech University

Supervisor: Dr. Guo H. Huang, Environmental Systems Engineering

Committee Member: Dr. Stephanie Young, Environmental Systems Engineering

Committee Member: **Dr. Liming Dai, Industrial Systems Engineering

Committee Member: Dr. Boting Yang, Department of Computer Science

Chair of Defense: Dr. Dongyan Blachford, Faculty of Graduate Studies & Research *via Teleconference **Not present at defense

ii

ABSTRACT

With the increase and expansion of environmental requirements and dwindling of

fossil fuel resources, current environmental and energy systems have aroused wide public

concern. In this dissertation research, several optimization modeling methodologies have

been developed for energy and environmental systems planning. They include: (a) a hybrid

dynamic dual interval model (DDIP) for irrigation water allocation; (b) a robust interactive

interval fully fuzzy model (RIIFFLP) for environmental systems planning; (c) a robust

interval type-2 fuzzy set model (R-IT2FSLP) to manage irrigation water resources, (d) a

robust inexact joint-optimal α cut interval type-2 fuzzy boundary model (RIJ-IT2FBLP)

for planning of energy systems, and (e) a pseudo-optimal stochastic dual interval T2 fuzzy

sets model (PD-IT2FSLP) for environmental pollutant control and energy systems

planning.

The DDIP has been developed by integrating dynamic programming (DP) with the

dual interval technique into a general optimal framework. It was applied to a hypothetical

case of irrigation water allocation in western Canada.

The RIIFFLP method has been developed to deal with fully fuzzy uncertainties by

using the fuzzy ranking method to find a balance between the necessity of constraints and

the objective function of a linear interval fuzzy sets programming as a technique for

optimal decision-making.

iii

The R-IT2FSLP method has been developed through integrating the concept of type-

2 fuzzy sets with an interval fuzzy boundary model to achieve maximum system profits

with limited environmental resources under uncertainties. The solutions obtained clearly

show that the type-2 fuzzy sets methodology can provide significantly improved results

that are more accurate by comparison to formal optimization methods.

The RIJ-IT2FBLP model has been developed by combining the join-optimal α cut

method, the interval RTSM solution method and the interval type-2 fuzzy sets boundary

method. The developed model was applied to issues concerning long-term energy sources.

The PD-IT2FSLP energy model has been developed to support energy system

planning and environmental pollutant control under multiple uncertainties for Xiamen City

in China. The solutions of the PD-IT2FSLP model will help energy authorities improve

current energy consumption patterns and ascertain an optimal pattern for energy utilization

in Xiamen City.

iv

ACKNOWLEDGEMENTS

I want to express my sincere appreciation to my supervisor, Professor Dr. Gordon

Huang, for his extremely precise guidance, selfless support and positive assistance from

the beginning of my graduate studies to the successful completion of this dissertation. My

thanks also extend to all the members of his family, particularly to Mr. Huang Guoping for

his encouragement, when I was in Xiamen.

I would also like to express my thanks to the committee members for their valuable

contributions and suggestions, which were very helpful in improving this dissertation.

I gratefully acknowledge the Faculty of Graduate Studies and Research and the

Faculty of Engineering at the University of Regina, and Xiamen University of Technology

for providing research scholarships, the Shen Kuo research exchange program and

travelling expenses while I was studying at the University of Regina.

My further appreciation goes to Dr. Dongyan Blachford for her generous help. Many

thanks also to my friends and classmates in the IEESC research team for their kind

assistance in many aspects of my research and for providing their warm friendship.

Finally, I would especially like to acknowledge my parents, wife, and many other

family members for their profound affection and spiritual support. Their unconditional love

has meant the whole world to me.

v

TABLE OF CONTENTS

ABSTRACT ........................................................................................................................ ii

ACKNOWLEDGEMENTS ............................................................................................... iv

TABLE OF CONTENTS .................................................................................................... v

LIST OF TABLES ........................................................................................................... viii

LIST OF FIGURES ............................................................................................................ x

CHAPTER 1 INTRODUCTION ........................................................................................ 1

1.1 Background ............................................................................................................... 1

1.2 Challenges in Environmental System Planning ........................................................ 2

1.3 Challenges in Energy System Planning .................................................................... 3

1.4 Challenges in Optimization Modeling for Energy and Environmental Systems ...... 5

1.5 Research Objectives .................................................................................................. 9

1.6 Organization ............................................................................................................ 12

CHAPTER 2 LITERATURE REVIEW ........................................................................... 14

2.1 Optimization Modeling of Energy and Environmental Systems ............................ 14

2.1.1 Planning for energy management systems ....................................................... 14

2.1.2 Planning for water resources management systems ......................................... 19

2.1.3 Planning for solid waste management systems ................................................ 23

2.2 Optimization Modeling under Uncertainty ............................................................. 27

2.2.1 Stochastic mathematical programming ............................................................ 27

2.2.2 Fuzzy mathematical programming ................................................................... 31

2.2.3 Interval mathematical programming ................................................................ 34

2.3 Optimization modeling under multiple uncertainties .............................................. 38

2.3.1 Integration of interval fuzzy and stochastic optimization methods .................. 38

2.3.2 Imprecise fuzzy boundary methods .................................................................. 41

2.4 Summary ................................................................................................................. 43

vi

CHAPTER 3 OPTIMIZATION PROGRAMMING FOR ENVIRONMENTAL SYSTEM

MANAGEMENT UNDER UNCERTAINTY ................................................................. 46

3.1 A Hybrid Dynamic Dual Interval Programming for Irrigation Water Allocation

under Uncertainty .......................................................................................................... 46

3.1.1 Background ....................................................................................................... 46

3.1.2 Methodology ..................................................................................................... 50

3.1.3 Application ....................................................................................................... 60

3.1.4 Result analysis and discussion .......................................................................... 67

3.1.5 Summary ........................................................................................................... 79

3.2 Robust Interval Fully-Fuzzy Programming with a Ranking Fuzzy Relation Method

for Solid Waste Management under Uncertainty .......................................................... 80

3.2.1 Background ....................................................................................................... 80

3.2.2 Methodology ..................................................................................................... 83

3.2.3 Application ....................................................................................................... 98

3.2.4 Result analysis and discussion ........................................................................ 110

3.2.5 Summary ......................................................................................................... 129

CHAPTER 4 INEXACT T2 FUZZY LINEAR PROGRAMMING FOR ENERGY AND

ENVIRONMENTAL SYSTEM MANAGEMENT UNDER UNCERTAINTY ........... 131

4.1 A Robust Inexact T2 Fuzzy Sets Linear Optimization Programming for Irrigation

Water Resources Management .................................................................................... 131

4.1.1 Background ..................................................................................................... 131

4.1.2 Methodology ................................................................................................... 134

4.1.3 Application ..................................................................................................... 149

4.1.4 Result analysis and discussion ........................................................................ 166

4.1.5 Summary ......................................................................................................... 181

4.2 A Robust Inexact Joint-optimal α Cut Interval Type-2 Fuzzy Boundary Linear

Programming (RIJ-IT2FBLP) for Energy Systems Planning ..................................... 183

4.2.1 Background ..................................................................................................... 183

4.2.2 Methodology ................................................................................................... 188

4.2.3 Application ..................................................................................................... 211

4.2.4 Results analysis and discussion ...................................................................... 229

vii

4.2.5 Summary ......................................................................................................... 248

CHAPTER 5 APPLICATION OF INTERVAL T2 LINEAR APPROACH FOR

ENERGY AND ENVIRONMENTAL SYSTEMS PLANNING IN THE CITY OF

XIAMEN......................................................................................................................... 249

5.1 A Pseudo-optimal Stochastic Dual Interval T2 Fuzzy Sets Approach for Energy and

Environmental Systems Planning in the City of Xiamen ............................................ 249

5.1.1 Overview of study system .............................................................................. 249

5.1.2 Study system ................................................................................................... 259

5.1.3 Methodology and modeling formulation ........................................................ 274

5.1.4 Results analysis and discussion ...................................................................... 311

5.1.5 Summary ......................................................................................................... 338

CHAPTER 6 CONCLUSIONS ...................................................................................... 340

6.1 Summary ............................................................................................................... 340

6.2 Research Achievements ........................................................................................ 344

6.3 Recommendations for Future Research ................................................................ 347

REFERENCES ............................................................................................................... 349

viii

LIST OF TABLES

Table 3.2.4 Waste flow from city i to facility j in period k under different α degrees . 112

Table 3.2.5 Integer solutions for landfill and composting facilities ............................. 113

Table 3.2.6 α-acceptable optimal solutions .................................................................. 115

Table 3.2.7 Membership of the fuzzy parameters under α-acceptable degrees ............ 122

Table 3.2.8 Solution of IIFLP and RIIFLP model under each α-acceptable degree ..... 125

Table 4.1.1 Water resources and economic data........................................................... 150

Table 4.1.2 Total available water with associated probabilities ................................... 151

Table 4.1.3 Maximum and minimum water demands for users ................................... 152

Table 4.1.4 Optimal solutions of R-IT2FSLP linear programming model ................... 168

Table 4.1.5 The optimal solutions of IT2FSLP linear programming model ................ 169

Table 4.1.6 The optimal solutions of R-IT2FSLP linear programming model ............ 177

Table 4.1.7 The optimal solutions of IT2FSLP linear programming model ................ 178

Table 4.1.8 Solutions of the type-2 fuzzy sets approach .............................................. 180

Table 4.2.1 Capacity expansion options for power generation facilities (GW)............ 222

Table 4.2.2 Supply cost for energy carriers in different periods (million $/PJ) ........... 224

Table 4.2.3 Power generation cost in different periods (million $/PJ) ......................... 225

Table 4.2.4 Cost of capacity expansions in different periods (million $/GW) ............. 227

Table 4.2.5 Scale capacity for power technologies in different periods (GW)............. 228

Table 4.2.6 Solutions of energy supply (PJ) ................................................................. 231

Table 4.2.7 Solutions of energy demand (PJ) ............................................................... 235

Table 4.2.8 Binary solutions of capacity expansion for conversion technologies ........ 238

Table 5.1.1 Energy consumption of Xiamen ................................................................ 265

Table 5.1.2 Relationship between GDP and energy consumption in Xiamen City ...... 270

Table 5.1.3 Energy cost in different periods (106 CNY/PJ) ......................................... 307

Table 5.1.4 Cost of power generation in different periods (106 CNY/PJ) .................... 308

ix

Table 5.1.5 Pollutant discharge coefficients (104tonnes/PJ) ......................................... 309

Table 5.1.6 Maximum pollution discharge amount in Xiamen (104 tonnes) ................ 310

Table 5.1.7 Results of PSD-IT2FSLP for Xiamen City................................................ 313

Table 5.1.8 Binary solutions of capacity expansion for conversion technologies ........ 327

x

LIST OF FIGURES

Figure 3.1.1 Flow chart of DDIP model with the solution algorithm ............................ 59

Figure 3.2.1 Flow chart of MSW RIIFFLP model with the solution algorithm ............ 97

Figure 3.2.3 Upper bound system costs for waste management .................................. 117

Figure 3.2.4 Lower bound system costs for waste management ................................. 118

Figure 3.2.5 Interval objective costs for waste management ....................................... 119

Figure 4.1.1 Deterministic membership of converntional interval boundary .............. 139

Figure 4.1.2 The membership of interval fuzzy sets boundaries ................................. 140

Figure 4.1.3 Random membership of T2 fuzzy boundaries......................................... 141

Figure 4.1.4 Flow chart of R-IT2FSLP model with the solution algorithm ................ 148

Figure 4.1.5 The lower membership function of the fuzzy set ................................ 158

Figure 4.1.6 The lower membership function of the fuzzy set 1S .............................. 159

Figure 4.1.7 The lower membership function of the fuzzy set 2S .............................. 160

Figure 4.1.8 The lower membership function of the fuzzy set 3S .............................. 161

Figure 4.1.9 The upper membership function of the fuzzy set ( ) .......................... 162

Figure 4.1.10 The upper membership function of the fuzzy set +

1S .......................... 163

Figure 4.1.11 The upper membership function of the fuzzy set +

2S ........................ 164

Figure 4.1.12 The upper membership function of the fuzzy set 3S

........................ 165

Figure 4.1.13 The lower bound water demand for each user ......................................... 170

Figure 4.1.14 The upper bound water demand for each user ......................................... 171

Figure 4.1.15 The optimal solution of lower bound for water shortage ........................ 172

Figure 4.1.16 The optimal solution of upper bound for water shortage ........................ 173

Figure 4.2.1 Interval type 2 fuzzy parameter membership function ............................ 191

xi

Figure 4.2.2 Interval type-2 fuzzy sets parameter with uncertain ∇ ............................ 194

Figure 4.2.3 Interval type-2 fuzzy sets parameter with join uncertain ∇=∆ ................ 195

Figure 4.2.4 Interval type-2 fuzzy sets parameter with join uncertain ∇&∆ ............... 196

Figure 4.2.5 Flow chart of RJ-IT2FLP approach model .............................................. 210

Figure 4.2.6 Solutions obtained through RJ-IT2FLP model......................................... 232

Figure 4.2.7 Lower bound α-cuts solutions obtained through RJ-IT2FLP model ....... 240

Figure 4.2.8 Upper bound α-cuts solutions obtained through RJ-IT2FLP model ....... 241

Figure 4.2.9 Optimal α-cut solutions of upper and lower boundaries ......................... 242

Figure 4.2.10 Lower bound solutions of from 1,tUP

to 6,tUP

of RJ-IT2FLP model ..... 244

Figure 4.2.11 Lower bound solutions of from 7,tUP

to 9,tUP

of RJ-IT2FLP model ..... 245

Figure 4.2.12 Upper bound solutions of from 1,tUP

to 6,tUP

of RJ-IT2FLP model ..... 246

Figure 4.2.13 Upper bound solutions of from 7,tUP

to 9,tUP

of RJ-IT2FLP model ..... 247

Figure 5.1.1 Geographical position of Xiamen ............................................................ 261

Figure 5.1.2 The interaction between energy and environmental systems .................. 268

Figure 5.1.3 Interval T2 fuzzy interval membership function with distance ∇ ........... 281

Figure 5.1.4 Features of stochastic dual interval T2 fuzzy sets random parameters ... 286

Figure 5.1.5 Interval solutions of energy supply for Xiamen City .............................. 314

Figure 5.1.6 Lower bounds of Xiamen City’s energy structure in the first period ...... 317

Figure 5.1.7 Lower bounds of Xiamen City’s energy structure in the second period . 318

Figure 5.1.8 Lower bounds of Xiamen City’s energy structure in the third period ..... 319

Figure 5.1.9 Upper bounds of Xiamen City’s energy structure in the first period ...... 320

Figure 5.1.10 Upper bounds of Xiamen City’s energy structure in the second period .. 321

Figure 5.1.11 Upper bounds of Xiamen City’s energy structure in the third period ..... 322

Figure 5.1.12 Energy system cost under various probability levels .............................. 330

Figure 5.1.13 Increase rate of different bounds ............................................................. 331

1

CHAPTER 1

INTRODUCTION

1.1 Background

As concern for environmental issues grows society is gradually realizing the

significant relationship between social progress, energy source management and

environmental protection (Tietenberg and Lewis 2000). In order to coordinate the

relationship among environmental requirements and energy consumption, many

optimization techniques have been developed (Hickman and Pitelka 1975, Perrin and Sibly

1993, Chan and Huang 2003, Lin and Huang 2008, Lv 2010, Cai, Huang et al. 2012).

Consumption of fossil energy aggregates the environmental damage, and the costs of

environmental protection are incalculable. In China, the consumption of petroleum

products increased nearly 8 times and the total energy production increased 3.5 times from

1971 to 2007 (Narayan and Prasad 2008, Dhakal 2009). Correspondingly, the emission of

air pollutants, for example, PM2.5, sulfur dioxide and carbon dioxide have been increased

many times per year (Lin, Huang et al. 2010). Meanwhile, complex energy structures and

supply factors lead to uncertain data and records. For instance, the generation of solid waste

is 100 to 150 tonnes per day, which is one of the major decision challenges for authorities

in their allocation of environmental and energy systems. These circumstances and others

have led to the development of modeling methodologies to respond to such issues in

environment and energy systems. Consequently, improving previous uncertain techniques

2

of system management is the keystone of current research in environmental protection and

energy source systems (Cai, Huang et al. 2009, Lin and Huang 2011). Thus, in this study,

the main efforts are focused on the development of optimization methodologies to tackle

uncertainties and interactive relationships among the economic, environmental and energy

systems.

1.2 Challenges in Environmental System Planning

The environmental system is a set of complex interactive relationships that includes

water, solid waste, air pollution and many other factors. Within each relationship are

various uncertain features, caused by uneven temporal and spatial distribution of sources.

For example, during the holidays, the generation of municipal solid waste is much higher

than during work days. In the farming season, irrigation water demand is greater than the

water demand during other seasons. Thus, the deterministic input cannot fully illustrate the

complexity of environmental systems. In other words, a crisp number cannot be presented

precisely with a finite number of digits (Moore 1966). However, interval-analysis

techniques can deal with such volatility. Interval mathematical method is known as an

effective tool for handling such uncertain problems. In the past decades, many interval

models were developed for dealing with such data fluctuation in environmental systems

(Huang 1992, Huang and DAN MOORE 1993, Wu, Huang et al. 1997, Huang 1998, Guo,

Liu et al. 2001, Huang, Sae-Lim et al. 2001, Cheng, Chan et al. 2003, Maqsood, Huang et

al. 2005, Li, Liu et al. 2006, Li, Huang et al. 2007, Guo and Huang 2008, Li, Huang et al.

2008, Liu and Huang 2008, Cheng, Huang et al. 2009, Huang, Sun et al. 2010). These

studies realized groundbreaking achievements. However, the complexity of the

3

uncertainties of environmental systems is far beyond the range of previous research. These

studies only partially represent a small part of the uncertainties and cannot fully reflect

uncertain relationships and interactions when higher levels of uncertainties, for example

“fully fuzzy parameters,” appear in the environmental systems. There are limited reports

in the literature on the study of such uncertainties. More importantly, there is a lack of

investigation of dual uncertain boundaries or multiple uncertainties when boundary

distribution functions are unknown within a general planning framework. Consequently, a

higher level study of uncertain boundaries is required to support management issues in

environmental systems.

1.3 Challenges in Energy System Planning

Along with rapid economic growth and the improvement of human living standards,

energy consumption has increased significantly, all of which has led to an imbalance

between supply and demand of energy. Fossil fuel energy consumption has contributed to

massive environmental pollution issues. The main task of energy system management is to

adjust the current energy structure, to optimize energy configuration, and to secure the

energy supply and demand balance. This task also requires energy authorities to use limited

energy sources to meet the various demands and to maximize economic benefits while, at

the same time, reducing environmental pollution. The energy optimization models have

been considered the most efficient tools for energy management. Therefore, numerous

energy system models have been developed based on optimization techniques, which can

provide alternative decision support between economic goals, finite energy sources, and

4

environmental pollution control targets (Lin and Huang 2008, Cai, Huang et al. 2009, Lin,

Huang et al. 2009, Lewis 2011, Lin, Huang et al. 2011).

Previously, most studies of energy system models analyzed the interaction between

environmental pollution policies and energy consumption strategies (Fishbone and Abilock

1981, Manne and Wene 1992, Naill 1992, Hashim, Douglas et al. 2005, Sellers 2011).

Within these studies, a few studies of energy models focused on energy activities

associated with energy use scale levels, such as community levels and regional levels.

Other energy studies focused on uncertain optimization techniques such as traditional

uncertainty ( the first level of uncertainty) in energy source demand, power consumption

and facilities expansion (Cai, Huang et al. 2009, Lin, Huang et al. 2009, Xie, Li et al. 2010,

Huang, Niu et al. 2011, Liu, Huang et al. 2011). In recent years, the primary research of

energy system models evaluates the relationship between climate change and energy

activities and corresponding energy decision patterns (Collier and Löfstedt 1997, Kasemir,

Dahinden et al. 2000, Mitigation 2011). Some studies have suggested that the energy

activities should minimize the influence of environmental impact (Ishimaru, Nakashiba et

al. 1995, Shrivastava 1995, Ocak, Ocak et al. 2004, Wang and Mauzerall 2006, Bektaş and

Laporte 2011). In recent years, most important studies on energy models focused on

multiple uncertain input in order to address the complexity of energy systems and generate

optimal decision-support processes. (Heinrich, Howells et al. 2007, Kim, Kim et al. 2008,

Cai, Huang et al. 2009). Although many studies of energy models extended previous

modeling methodologies, they have not been able to represent a higher level of boundary

uncertainties within their general modeling framework. These studies also could not reveal

5

the complicated multiple uncertain relations. However, no related studies have been

reported in energy planning areas. Thus, this study focuses on an improvement of uncertain

optimization techniques by allowing higher level uncertain information presented as “fuzzy

fuzzy intervals” to be directly communicated into the management processes of energy

systems. Consequently, the feasible decision alternatives of energy systems can be obtained

through integrating the analysis of a higher level of uncertain boundaries.

1.4 Challenges in Optimization Modeling for Energy and Environmental Systems

Previously, a number of studies indicated that uncertainties existed in energy and

environment systems. In order to deal with these uncertainty problems, optimization

models were largely used to handle such uncertainties. However, most of these optimal

models were applied to manage only a single process in the energy systems (Linnhoff and

Flower 1978, Ulleberg 1998, Abido 2002, Abido and Abdel-Magid 2002, Braun 2002,

Schiehlen 2005, Cai, Huang et al. 2009, Cai, Huang et al. 2009, Dhakal 2009, Ooka and

Komamura 2009, Yang, Wei et al. 2009, Liu, Pistikopoulos et al. 2010). In other words,

these optimal models just focused on an individual technology of the energy industry. Few

literature reports indicated that these studies could handle multiple uncertainties within

interrelated factors.

However, energy systems have complex and interconnected characteristics fraught

with all kinds of uncertainties. For example, daily power consumption is a fluctuating

interval. It is lower in the morning but higher in the evening. A determined amount of

power will only lead to unnecessary losses. This condition indicates that the energy systems

have a strong interaction with the social economy. More broadly, the energy system is

6

associated with the environmental system and the ecosystem. These correlations contain

uncertainties similar to the irrational numbers included in real numbers. The traditional

optimization models with a deterministic input method cannot fully represent the

complexity of the energy and environmental systems. For example, on one hand, local

impact of environmental pollutants is hardly evaluated through the interval value of various

energy activities, which may cause major air pollution from combustion of fossil fuels. On

the other hand, energy and environment systems are easily influenced by the social

economy through direct or indirect regulation policies, which has huge effects in energy

markets. The energy systems also impact ecosystems by production of energy such as oil

recovery. However, ecological restoration contributes uncertain factors to the energy and

environmental systems. These three correlations can lead to complex and multiple

uncertainties.

Previously, the uncertainties of optimization modeling were expressed as uncertain

parameters in the energy and environment systems. The fuzzy mathematical method, the

interval mathematical method and the stochastic mathematical method were commonly

used when dealing with the uncertainties in the process of optimization modeling. The

stochastic method has the ability to reflect a system’s stochastic disturbances. With a

known probability distribution, this method can effectively deal with fluctuating intervals

of components. The stochastic method was divided into two groups of studies. One group

was studies using a probabilistic method, and the other one was those using multi-stage

stochastic method. The common feature behind these two methods is the use of probability

information within optimization frameworks. However, this feature has also been

7

considered as one of the defects of the stochastic method. The probability distributions of

the parameters need to be known. However, in most uncertain cases, it is hard to find data

records. For example, the manager of solid waste management operation can record a

generation rate of waste with intervals in a city. However, these fluctuating intervals have

no sense of probability distribution (Huang, Anderson et al. 1994). Thus, the application

of the stochastic method is also limited.

The fuzzy mathematical method, different from probability theory, is another

effective tool (Zadeh 1965), used to tackle uncertainty. In particular problems, the

information may be incomplete, imprecise, fragmentary, unreliable, contradictory, or

deficient in some other way (Klir and Yuan 1995). However, the fuzzy methods can use

this ambiguous information to provide quantization information for decision makers. It has

been widely used in optimization systems. The fuzzy flexible method and the fuzzy

possibility method are two main applications of the fuzzy theory in energy and environment

management. The fuzzy flexible method can deal with uncertainties that contain fuzzy

objectives and flexible constraints in the function (Zimmermann 1985). The fuzzy

possibility method can solve the problems of parameters, which are regarded as possibility

distributions (Zadeh 1978). However, the fuzzy methods cannot represent independent

uncertainties in the left-hand sides of the constraints (Huang and DAN MOORE 1993).

Although there were a large number of studies of the fuzzy methods, they show that they

are still unable to deal with practical cases because they lead to complicated computational

processes and complex intermediate models (Inuiguchi, Ichihashi et al. 1990).

8

While probabilistic and fuzzy sets methods are widely accepted to represent

uncertainties, the interval methods (Moore 1966) have kindled wide attention. After the

probabilistic method was proved to produce incorrect results under uncertainty (Ferson and

Ginzburg 1996), the interval method emerged as the best solution in a situation without a

probability distribution. Intervals can be represented within the lower and upper bounds of

uncertain quantities. Consequently, the interval methods can handle those imprecise

estimations that exist in energy and environmental systems. With this advantage, the

interval methods have become widely accepted in the optimization process because the

imprecise number only can be represented as a fluctuation boundary without distribution

information. The uncertain parameters have been described as intervals, which are

unknowable but have lower and upper bounds in both sides of the functions for allocation

problems (Huang, Baetz et al. 1992). The interval method was applied to the interactive

problems of solid waste management. The uncertain data of the generation of solid waste

has been effectively communicated into the optimization processes. This creative

development triggered a wealth of interval mathematic methods wildly applied in energy

and environment management (Bass, Huang et al. 1997, Wu, Huang et al. 1997, Huang

1998, Liu, Huang et al. 2000, Guo, Liu et al. 2001, Cheng, Chan et al. 2003, Hu, Huang et

al. 2003, Huang, Huang et al. 2005, Li, Huang et al. 2007). The interval method has

improved the previous optimization methods by allowing interval inputs, which satisfied

requirements in both computation and data ambiguousness. However, when the range of

intervals became increasingly large, for example [1000, 1000000], the performance of

interval methods is meaningless given such wide uncertain bounds.

9

In order to achieve more accurate quantitative results under uncertainty, the models

developed later have integrated the advantages of interval, fuzzy, and stochastic methods

to present each of the levels of complex uncertainties (Huang, Baetz et al. 1994, Huang

and Loucks 2000, Yeomans, Huang et al. 2003, Maqsood, Huang et al. 2005, Li, Huang et

al. 2006). For example, an interval-parameter fuzzy stochastic missed integer linear

programming method has been developed to analyze regional waste management (Huang,

Sae-Lim et al. 2001). In addition, interval parameter fuzzy integer programming has been

developed to manage other municipal solid waste issues (Nie, Huang et al. 2007), and a

two-stage interval stochastic method has been applied to uncertain waste management

systems (Maqsood, Huang et al. 2005). These applications combined three methods to deal

with uncertainties. The uncertain information was characterized by either fuzzy sets or

intervals in the optimization modeling processes.

However, most of these studies focused on one or a few constraint parameters. They

could not completely describe the performance of multiple uncertainties as they exist in

real world problems. These previous studies could not reflect the uncertain relationship

between function goal and constraints. The ordinary intervals and fuzzy methods could not

handle situations such as indefinite boundaries. Thus, in order to improve the applicability

of the optimization models, a more realistic optimization modeling method is desired to

support the planning of energy and environment systems under multiple uncertainties.

1.5 Research Objectives

In this dissertation research, optimization modeling methodologies including a

dynamic dual interval programming of irrigation water allocation, a robust interactive

10

interval fully fuzzy linear programming of municipal solid waste management, a robust

interval Type-2 (T2) fuzzy set linear programming of Saskatchewan irrigation water

resource management, a robust joint-optimal α cut interval T2 fuzzy linear programming

of energy systems, and a pseudo-optimal dual interval T2 fuzzy sets linear programming

of Xiamen energy planning were developed to address robust planning issues in current

energy and environment systems under multiple uncertainties. These modeling methods

will combine interval fuzzy mathematic programming, dynamic stochastic programming,

interval T2 fuzzy sets programming and robust two step methodology to improve the

existing optimization approaches and to adjust current allocation patterns in energy and

environmental systems. The improved methods were be applied to both hypothetical and

real world cases to demonstrate their advantages. In detail, the specific objectives of this

research were as follows:

(a) To develop a hybrid dynamic dual interval programming (DDIP) to support

allocation of the irrigation water systems under dual interval uncertainties. The

DDIP approach improves the existing programming of dynamic intervals by

explicitly addressing the system uncertainties by using dual intervals. The results

of the proposed model indicate that it has an effective computational process and

its subjective variables are incorporated into the solutions for the final decision.

(b) To develop a robust interactive interval fully fuzzy linear programming (RIIFFLP)

model by using a fuzzy ranking method to find a balance between the requirements

of the constraints and the objective function of a fuzzy set function as a technique

for optimal decision-making under uncertainty. It was applied to municipal solid

11

waste management though a general optimization framework. This method

considerably improves previous interval fuzzy linear programming methods by

using a new solution method called the robust two-step method (RTSM). A case

study demonstrated that the solution obtained from the RIIFFLP model had more

feasible results by comparison to the existing fuzzy linear programming methods.

Through introducing the concepts of fully fuzzy linear programming, problems in

solid waste management can be clearly addressed and easily solved.

(c) To develop a robust interval T2 fuzzy set linear programming (R-IT2FSLP) for

management of irrigation water resources. It improves upon previous interval fuzzy

bound models by allowing uncertainties, presented as multiple fuzzy boundaries,

to create additional degrees of freedom and enable direct modeling of uncertainties

within an optimization framework. This method more explicitly reflects the

system’s uncertainties under the fuzzy membership function, because it can be

difficult to determine the system’s membership function, which changes the

system’s boundaries from a known number into multiple uncertain ones. The

solution of the R-IT2FSLP method was compared with formal optimal methods to

see how applicable it is to irrigation water systems under uncertainty. It is clear that

this R-IT2FSLP model, as an alternative reference tool, can provide more accurate

results to support decision makers.

(d) To develop a robust joint-optimal α cut interval T2 fuzzy linear programming (RJ-

IT2FLP) method for energy generation, conversion, and transition under multiple

uncertainties. Firstly, an interval T2 fuzzy method was be proposed for handling

vague linguistic interval data. Secondly, a robust joint-optimal α cut solution

12

process was be applied, and finally, the developed model was applied to a case

study of long-term energy resource planning. The solutions of the RJ-IT2FLP

method can help decision makers handle multiple ambiguity issues existing in

energy demand, supply and capacity expansions. The RJ-IT2FLP model not only

delivers an optimized energy scheme, but also provides a suitable way to balance

uncertain cost and profit parameters of an energy supply system.

(e) To develop a pseudo-optimal dual interval T2 fuzzy sets linear programming (PD-

IT2FSLP) method to support energy system planning and environmental

requirements under uncertainties in Xiamen City. This method was based on an

integration of interval T2 Fuzzy Sets (FS) boundary programming, dual interval

fuzzy linear programming, and stochastic linear programming techniques. It

enables the PD-IT2FSLP method and provides robust abilities to tackle

uncertainties, which are expressed as T2 FS intervals, dual intervals, and

probabilistic distributions within a general optimization framework. This method

can efficiently facilitate system analysis of energy supply and energy conversion

processes, and of environmental requirements as well as provide capacity

expansion options with multiple periods. Thus, the lower and upper solutions of

PD-IT2FSLP would help local energy authorities adjust current energy patterns,

and discover an optimal energy strategy for the development of Xiamen City.

1.6 Organization

The structure of this dissertation is as follows: Chapter 2 reviews the previous studies

on energy and environment systems planning models and optimization methods under

13

various uncertainties. Chapter 3 presents the highlights of hybrid dynamic dual interval

programming and its application in irrigation water resources management, and describes

development of robust interactive interval fully fuzzy linear programming through a fuzzy

ranking method to find a balance between the constraints and the objective function of

linear programming under fully fuzzy uncertainties. The first section of Chapter 4 contains

a robust interval T2 fuzzy set linear programming model. The interval T2 fuzzy set

programming method is a higher level of the formal interval fuzzy sets method to handle

ambiguous interval information existing in water resources management. The second

section of this chapter focus on the development of a robust joint-optimal α cut interval T2

fuzzy linear programming for energy system management. This model implies multiple

uncertainties of interval T2 fuzzy boundaries exist in energy systems. The third section of

Chapter 4 describes a real-world case study of a pseudo-optimal dual interval T2 fuzzy sets

linear programming method. Dual interval and stochastic interval T2 fuzzy optimization

methods were integrated into the development of this methodology. Chapter 5 summarizes

this dissertation research.

14

CHAPTER 2

LITERATURE REVIEW

2.1 Optimization Modeling of Energy and Environmental Systems

Energy and environmental systems are characterized by complex interrelationships

and various uncertainties. Optimization techniques have been employed to analyze and

support such complex issues in energy and environmental systems in the past decades

(Anderson and Nigam 1968, Bahn, Haurie et al. 1998). The efforts of these techniques were

mainly focused on developing models including fuzzy mathematical models, stochastic

mathematical models, interval mathematical models, interval stochastic models, fuzzy

stochastic models and interval type-2 fuzzy models. These models can help evaluate and

solve operation problems in the area of resource management. The results of these studies

have improved the efficiency of energy utilization and environmental conditions. This

chapter will review these previous studies on optimization modeling methods for both

environment and energy management.

2.1.1 Planning for energy management systems

In the area of energy management, many of the optimization models have been

developed for the allocation of energy sources in different areas (Linnhoff and Ahmad

1989, Frangopoulos and Von Spakovsky 1993, DeChaine and Feltus 1995, Ko, Balsara et

al. 1995, Totrov and Abagyan 1997, Mandel, Brezina et al. 1999, Brahma, Guezennec et

al. 2000, Paganelli, Ercole et al. 2001, Duleba and Sasiadek 2003, Fernández‐Recio,

15

Totrov et al. 2003, Smith, Sewell et al. 2004, Nazhandali, Zhai et al. 2005, Tang and

McKinley 2006, Liu, Yang et al. 2007, Ceraolo, di Donato et al. 2008, Bartman, Zhu et al.

2010, Banos, Manzano-Agugliaro et al. 2011, Beloglazov, Abawajy et al. 2012). These

methods were applied to minimize energy consumption, thereby reducing the economic

cost in the processes. The objectives of these studies were either to maximize systems

reliability or to minimize the risk of energy shortage, both needed for energy sources

management. For example, Smith and Verdinelli (1993) developed a linear programming

method for energy distribution and supply in New Zealand. This programming method was

an improvement on the traditional process analysis methods. It could consider more details

of energy activities. This model was also applied to identifying optimal policies for energy

demand and consumption. In 1984, Schulz and Stehfest proposed an optimization model

for multiple objectives for the free market in a certain region. This method considered three

targets, including minimization of system cost, maximization of reliability of the energy

supply systems, and environmental conditions. Later, in 1988, Satsangi et al. proposed a

linear model to analyze the optimal patterns of limited energy sources for India. It provided

an optimal solution for satisfying local demands on energy products. Kahane (1991)

summarized the advantages of optimization methods for management of energy systems,

and concluded that for the purpose of improvement of efficiency, energy systems should

involve every individual end-user rather than only the traditional supply and consumption

segments. Macchiato et al. (1994) introduced an optimization model to minimize cost of

emission reduction. They used this model to measure the minimum cost of energy systems

at each level of pollution control. Arivalagan et al. (1995) proposed an integer linear

optimization model to reveal an optimum pattern for processing companies. The solutions

16

achieved demonstrated that this method was successful in providing an optimization

strategy for these corporations. Schoenau et al. (1995) proposed an optimization model for

in-bin drying of canola grain. This method was applied to minimize fan energy and heat

energy consumption for energy systems. The results of this method indicated that the total

cost of the energy system could be fully estimated using different energy sources. In order

to minimize the capital cost of Swedish electricity and other relative industries, Henning

(1997) introduced a time-dependent model based on linear optimization programming.

Bojić and Stojanović (1998) developed mixed integer linear programming to optimize heat

and power generation patterns within a factory. They used this optimization technique with

a congregated heat and power system to produce the regional diagram. This application

indicated that the optimization model could actually save energy. Sayed (1999) introduced

an optimization model to analyze energy supply plans to meet energy demands. This model

was unique because it had features of both reliable and cost effective methods. An

optimization model was developed and applied to a real case of industrial planning in India.

The results indicated that their optimization strategy needed to identify the energy sources

for energy buildings. In the same year, Santos and Dourado (1999) proposed a multiple

objective model to coordinate the relationship between energy generation and consumption

on a global scale.

Iniyan and Jones (2000) applied a linear programming model to measure the energy

demand and pollutant emissions for energy and environmental management. Through their

optimal methods, limited energy sources, particularly renewable energy sources, were

effectively allocated, and a minimized system cost was achieved under different energy

17

structures. Carlson (2002) proposed an optimization model for long-term operations. In his

research, the system of energy structure and the environmental penalties were considered

for the first time. To achieve maximum productivity of heat sources, Drozdz (2003)

developed a linear model to manage geothermal power and its conversion processes. Due

to the complex criteria and cost coefficients in energy relationships, Sinha and Dudhani

(2003) employed an optimal model to allocate energy sources. Using the same research

procedure, Koroneos et al. (2004) introduced a multi-objective method to maximize the

benefits among environment conditions, energy processes, and energy shortage. This

method also combined renewable energy sources with traditional energy structures, and it

was used for energy management issues in Lesvos Island, Greece. Avetisyan et al. (2006)

developed a linear model to choose optimal extension schemes for local power systems.

Considering that nonlinear relationships exist in energy systems, Ostadi et al. (2007)

introduced a nonlinear programming method to analyze optimal energy patterns within

factories. This method integrated power loading modes with consumption constraints of

energy systems. Ordorica-Garcia et al. (2008) developed an optimization model to

minimize the total annual cost of supply energy with air emission constraints. Rentizelas

(2009) developed multi-biomass energy conversion applications with an optimal method.

This method combined holistic modelling of the system to maximize the financial benefit

of the investment. By applying this method, the energy demand of the customers was fully

met. The results of this optimal model generated a planning scheme with lower air emission.

Mirzaesmaeeli (2010) employed a multi-period linear programming model to allocate

electric systems. This method was developed with the objective of minimizing cost and air

pollutants to meet a specified power demand. The optimal model was successfully applied

18

to two cases to examine whether the economic and environmental requirements could be

realized.

In the meantime, large-scale optimization modeling methods were introduced into the

area of energy management. For example, Fishbone and Fishbone (1981) developed one

of the first large-scale market allocation models (MARKAL). To devise this model, a group

of people from 17 countries was employed. This model can evaluate the effects of energy

and environmental policies in national, regional and provincial levels. It provides an

optimal allocation pattern to achieve the lowest cost between energy sources and

technology constraints. Similarly, the Brookhaven Energy System Optimization Model

(BESOM) was developed to investigate optimal patterns of mixed energy sources and

conversation technologies (Brock and Nesbitt 1977, Kydes 1980, Fishbone and Abilock

1981, Manne and Wene 1992, Messner 1997). In 2000, an advanced version of the

MARKAL model was developed to analyze the influences of carbon-emission trading

among countries. This version of the optimal model examines energy trade problems and

maps them into geographic information systems. Based on this model, energy generation,

conversion, and relative activities are incorporated within an optimization framework

(Cosmi, Cuomo et al. 2000, McDonald and Schrattenholzer 2001, Seebregts, Goldstein et

al. 2002, Goldstein, Kanudi et al. 2003, Loulou, Goldstein et al. 2004, Chen 2005, Endo

and Ichinohe 2006, Endo 2007, Strachan, Kannan et al. 2008, Gül, Kypreos et al. 2009, Hu

and Hobbs 2010).

19

2.1.2 Planning for water resources management systems

Water resources management has been the subject of extensive exploration to support

allocation of water distribution, reservoir operation and flood control for many years (Yeh

1985, Başaḡaoḡlu, Mariño et al. 1999, Bastiaanssen, Noordman et al. 2005, Lopez, Fung

et al. 2009). Skaggs (1978) developed a water management model to evaluate infiltration,

subsurface drainage, surface drainage, potential evapotranspiration and soil water

distribution. The application of this model demonstrated that water management systems

can achieve optimization results by considering the performance of this alternative method.

Manoutchehr (1982) formulated a linear programming method to maximize the amount of

water that could be pumped from resources to the physical capability of institutional

constraints for ground water management. This method was applied to the surface water

management of the Pawnee Valley of south-central Kansas. It gives optimal suggestions

for the water management of the Pawnee Valley area over the next decade. By using the

constraint linear programming, Peter et al. (1984) introduced, for water authorities, a multi-

objective optimization model to establish a more unified basin-wide allocation plan by

considering water supply, water quality control and prevention of undesirable overdraft of

the basin. They also linked the linear programming model with water simulation models,

which achieved a good representation. Walski et al. (1987) applied optimization models to

optimally size water distribution pipes. The results achieved from their models were helpful

in sizing water pipes and provided a good assessment or gauge for decisions in water

management. Because leakage form water networks can cause a significant loss from a

water supply, Jowitt and Xu (1990) formulated a linear programming model to minimize

the leakage in water distribution systems. William (1992) used optimization linear

20

programming methods for the planning and management of ground water systems. Ellis

(1993) presented a stochastic dynamic programming method for water quality management

from multiple point sources, which included parameter uncertainties. These models were

linked with WASP4 and similar models to generate management options. By connecting

both model and parameter uncertainty in the modeling processes, trade-offs were found

between the two factors on control decisions. Booker and Young (1994) developed an

optimization model to investigate the performance of alternative market agencies for

Colorado River water resources management. They used this method to analyze the amount

of water shortage to consumers and concluded that water allocation would require large

transfers from other water resources, and that its deliveries to Mexico would exceed

requirements. In order to solve the conflict between environmental protection and social

economic requirements, Chang et al. (1995) used multi objective linear programming

methods to manage reservoir resources by considering local economic income,

employment level and water quality related to the discharge targets for the Tweng-Wen

watershed system in Taiwan. By using simplex methods, a feasible option indicated that

the residential area of this area can be allowed to increase.

Recently, Boccelli et al. (1998) formulated an optimization model for disinfectant

injections with dynamic scheduling. This model minimized the total dose needed to satisfy

residual constraints. Optimal scheduling, obtained from the application of the model, could

reduce the average disinfectant concentration within the water supply system, and also,

booster disinfection could reduce the amount of disinfectant. Huang and Loucks (2000)

developed an interval two-stage stochastic programming for water resource management

21

under uncertainty. This method combined the inexact optimization and the two-stage

stochastic programming techniques to solve probability distribution issues raised in water

resource management. The results of this method indicated that reasonable solutions could

be obtained. Since there are nonlinear relations in existing water resource management,

Cai et al. (2001) described a nonlinear model with combined genetic algorithms to solve

water resources management problems. Using this approach allows sufficient details in the

modeling process to be retained so that it could explore the long-term sustainability

problems. By considering the hedging rules along with the rule curves, Tu et al. (2003)

formulated a mixed integer linear programming model to analyze both the traditional

reservoir rule curves and the hedging rules to manage a multi-purpose and multi-reservoir

system in the southern region of Taiwan. The results minimized the impact of drought and

reduced the water supply to meet the distribution target, thereby proving the applicability

of this model. Chu et al. (2004) presented a robust counterpart optimization method of

wastewater reuse. Based on the analysis of this method, water resources price changes can

be simulated and evaluated. This method provided useful suggestions regarding China’s

water and wastewater management. Reis et al. (2006) integrated the genetic algorithm and

linear optimization programming to determine reservoir systems management over given

periods. This method identified allocation parameters, initially used for linear

programming, to determine operational decisions. This proposed method generated

comparable results to operate water resources without a priori imposition in the larger

reservoir of Roadford Hydrosystem. Bartolini et al. (2007) evaluated the impacts of water

policies by using multi-attribute linear programming models in Italy. This model combined

five main scenarios to reflect the reactions of provincial agricultural policy, world markets,

22

and the local community. The results indicated that the diversity of irrigation water systems

depended on the balance of water conservation, agricultural policy, and development

targets. Lu et al. (2008) developed an inexact two-stage fuzzy-stochastic optimization

method to represent numerous punishment policies for water resource management. This

method extended the traditional two-stage stochastic method by introducing the fuzzy sets

theory into the general optimal framework. The desired allocation pattern’s results

provided a maximized system benefit, which was higher than the previous planning

methods given feasible levels for water resources. With the same purpose, Guo and Huang

(2008) developed a two-stage fuzzy chance-constrained programming method under dual

uncertainties. An improvement upon the previous methods, this model could deal with dual

uncertainties with stochastic and fuzzy features. Solutions using this method provided

optimal water planning strategies, and could give a more stabilized system feasibility. Li

et al. (2008) formulated an inexact multistage stochastic integer programming method for

water resource management under uncertainties. This optimization technique incorporated

multistage stochastic programming within an inexact programming framework. It could

analyze uncertain issues expressed as probabilities and discrete intervals for water

allocation over a multistage context. The results of this model indicated the applicability

of binary and continuous variables. It is able to help water authorities to identify an optimal

strategy against water shortage under complex uncertainties. By considering urban water

consumers, national benefits, and social hazards, Fattachi and Fayyaz (2010) used an

optimization multi-objective programming model for studying the Hamedan, Iran, potable

water network to evaluate the abilities and efficiency of the proposed model. The results

23

of this model indicated that this optimization method could be used as an efficient

technique for water resource allocation.

2.1.3 Planning for solid waste management systems

Optimization methods have been broadly utilized in the area of solid waste

management and are outlined chronologically as follows: In the early 1980s, after

analyzing the influence of waste on the environment and associated assimilation capacities,

a waste management system was designed by Panagiotakopoulos (1972) with the

development of an optimization linear programming method and network analysis. This

optimization programming method was also employed to examine the optimal

management schemes concerning multiple environmental aspects by Greenberg et al.

(1976). Since resource exploration generates water, air and solid waste pollution, Bishop

and Narayanan (1979) illustrated optimum strategies determined by a planning model to

control these environmental issues. Further development in this area occurred in the 1990s

when Perce and Davidson (1982) investigated how the costs were related to different

hazardous waste management schemes, and economical solutions were provided

accordingly. Jennings et al. (1984) introduced a transportation routing optimization method

for regional hazardous waste management. The waste generation sources, waste treatment

processes, and waste disposal sinks were connected to detect weaknesses in the waste

network. Optimal solutions were achieved and demonstrated that this method could test the

value of improvement of the current waste disposal planning patterns. Applicable planning

models developed by Kirca and Erkip (1988) were employed to determine of the location

of the solid waste management transfer stations used in Istanbul, Turkey. Comparison

24

between recycling programs was studied by introducing optimization methods by Lund

(1990) in whose paper the most cost-effective lifetime of the landfill was proposed. Huang

et al. (1992) developed a linear optimization analysis method to manage municipal solid

waste in the civil engineering area. This method allowed that uncertainties existed in model

coefficients and stipulations. The results of this method indicated that reasonable solutions

could be obtained by the two-sub objective functions. For evaluating the complex and

conflict alternatives in solid waste management systems, Caruso et al. (1993) used an

optimization approach with heuristic techniques for managing of the waste issues in

regions of Italy. The results obtained from this model, proved an optimal strategy for

comparison between the waste system and possible alternatives. Chang and Wang (1995)

included a mixed integer programming model with dynamic optimization features to solve

the problems of growing metropolitan regions with increasing environmental concerns.

This dynamic method showed that limiting traffic congestion could influence the optimal

flow pattern of solid waste management in the Kaohsiung territory of Taiwan. Given the

focus on minimizing the transportation costs of waste, Kulcar (1996) used linear

programming to optimize the collection processes. Later that year, Chang and Wang (1996)

suggested decision making systems to facilitate the planning and alleviate the conflict

between different waste disposal methods.

The research of recent years has shown that optimization technologies applied to

municipal solid waste management continue to develop and evolve. Chang and Chang

(1998) explored a new operational program that involved the energy and material recovery

requirements in solid waste systems. This proposed method, applied to the Taipei

25

Metropolitan region, illustrated that waste inflows with various physical and chemical

compositions and heating values with different generation rates could be analyzed to meet

energy recovery and other requirements of the waste incinerators in the Taipei metropolitan

region. Weitz et al. (1999) formulated a life-cycle assessment method to evaluate integrated

municipal solid waste management in the United States. The developed methods included

environmental and cost analysis sections. Each section calculated different types of data

within the whole waste management process. All respective data were applied into the

decision support tools to give an optimal pattern of municipal solid waste management.

Wilson et al. (2001) introduced a more sustainable approach to allocate solid waste, which

involved 11 managers from leading-edge European municipal solid waste programs in 9

different countries working together. The key “system drivers” were identified, which

provided more sustainable strategies of waste management. Huang et al. (2002) introduced

a violation analysis approach for regional solid waste management under uncertainty. This

method was based on the method of interval-parameter fuzzy integer programming. In their

approach, the tolerable violations were permitted so that the model’s decision space could

be enlarged. Consequently, this model’s results helped to produce multiple decision

alternatives under different system conditions and increased the system’s reliability.

Maqsood and Huang (2002) introduced a two-stage interval-stochastic programming

model for management of solid waste systems in which the linkage of predefined policies

was undertaken and observed under both probabilities and interval uncertainties. The

results provided desired patterns with the lowest system cost for generating decision

alternatives. In order to solve the problem of inefficiency of solid waste management, Najm

and Fadel (2004) formulated optimization models to address these concerns. Sumathi et al.

26

(2008) addressed a multi-criteria decision analysis method combined with geographic

information systems. The outcome indicated that high efficiency could be achieved by

using this method for issues in the selection of waste systems. Zhu et al. (2009) described

a waste generation analysis to provide optimal suggestions for the Pudong area in Shanghai,

China. Xu et al. (2010) introduced a stochastic robust interval linear programming model

to support management problems of municipal solid waste management. His method

attempted to improve the conventional stochastic robust optimization programming

methods for long-term management issues. The results of this approach disclosed that

waste alternatives could be generated by adjusting the parameters within constraint

intervals.

Overall, optimization modeling methods have been formulated for the decision

support of complex energy and environmental issues. However, innumerable uncertainties

exist in this process. Within the reality of real-world problems, these uncertainties be have

parastically on system components and seriously affect the rationality of decisions. These

uncertainties may further exacerbate the complexity among systems’ parameters, which

could influence the objective function and become associated with higher systems’ risks.

However, the conventional optimization methods were not able to solve these

interconnected uncertainty issues. Thus, as one kind of compensation method, the uncertain

optimization methodologies could better tackle such problems to enhance the applicability

of optimization methods.

27

2.2 Optimization Modeling under Uncertainty

In section 2.1, most of applications gave priority to the deterministic optimization

modeling methods. However, in real world planning problems of energy and environmental

systems, the uncertainty either exists in the parameters or else in the interactive

relationships of the systems. The information about a particular optimization problem may

be imprecise, fragmentary, and contradictory (Klir and Yuan, 1995). Thus, in order to more

accurately describe these ambiguous processes or relations or parameters of system issues,

uncertain optimization methods are usually involved to solve such problems. The previous

study of uncertain optimization methods are mainly dominated by stochastic mathematical

programming, fuzzy mathematical programming and interval mathematical programming.

Greater details about these methods will be provided in the following sub-sections.

2.2.1 Stochastic mathematical programming

To solve problems with random input parameters, Beale (1955) used stochastic

mathematical programming methods in an optimization framework. Actual engineering

problems almost invariably include a few unknown parameters, and thus, this method can

be a supplement to deterministic optimization methods. The characteristics of this method

are input parameters that are known only within probability distributions, and these

parameters can manifest themselves throughout the modeling process as stochastic

elements among coefficients of the objective function, constraints and the right-hand-side

stipulations (Li and Huang 2007). The main advantages to the stochastic mathematical

programming method are that it does not simply reduce the complexity of the problems,

rather it can provide a complete view of the effects of uncertainties and relationships

28

between uncertain inputs and resulting solutions for decision makers (Huang 1994).

Normally, stochastic methods can be replaced by deterministic equivalents, and solutions

of the deterministic model can be extended to represent the stochastic results (Garstka and

Wets 1974, Olsen 1976). This method is mainly classified into two categories, chance-

constrained programming and two-stage stochastic programming.

Chance-constrained programming is used for analyzing the reliability of a model’s

constraints under uncertainty (Cai, Huang et al. 2007). According to previous research

(Ellis, McBean et al. 1985), the constraints of chance-constrained methods are needed to

be partly for satisfaction to solve problems. In other words, these given constraints require

being proportionally satisfied under certain probabilities. When probability distributions

(the risk of violation) are known, the chance-constraints method can effectively handle

uncertain parameters in the right hand side of the programming model. This method was

first introduced into management science by Charnes and Cooper (1958). Since then, this

method has been widely used in many aspects (Miller and Wagner 1965, Kirby 1967,

Charnes, Cooper et al. 1971, Hogan, Morris et al. 1981, Charnes and Cooper 1983,

Jagannathan 1985, Olson and Swenseth 1987, Morgan, Eheart et al. 1993, Iwamura and

Liu 1996, Liu and Iwamura 1998, Liu 2001, Lei, Wang et al. 2002, Cooper, Deng et al.

2004, MA, WEN et al. 2005, Talluri, Narasimhan et al. 2006, Yang, Yu et al. 2007, Li,

Arellano-Garcia et al. 2008, Pagnoncelli, Ahmed et al. 2009, Meng and Wang 2010,

Küçükyavuz 2012). Chandi et al (1978) formulated two chance-constrained linear

programming models to reflect the stochastic nature of monthly inflows for the Mayurakshi

irrigation project of India. Keown and Taylor (1980) approached chance-constrained

29

capabilities to reflect uncertainty in product demand. Yakowitz (1982) formulated chance-

constrained linear programming for water resource problems. Terry et al. (1984) suggested

a chance-constrained approach to production planning, which allowed decision makers to

specify both probabilistic produce demands and production line operating characteristics

in keeping with actual situations. Fujiwara et al. (1987) proposed a chance-constrained

model to deal with random variables included the main stream, tributaries and storm water

to determine the most economical level of wastewater treatment at each discharge city.

Morgan et al. (1993) developed a mixed-integer-chance-constrained programming method

to find the global optimal trade-off curve for maximum reliability versus a minimum

pumping objective for groundwater remediation problems. Roush et al. (1994) used

chance-constrained programming to formulate commercial feeds for animals. By applying

this method, more than $ 250,000 could be saved every year. For an insurer’s aspiration

level of return on risk levels, Li (1995) formulated the ‘P-Models’ of chance constrained

programming to test the industry’s aggregated data. Iwamura and Liu (1998) employed

chance-constrained integer programming method to simulate capital budgeting problems

in a fuzzy environment. The genetic algorithm was designed to solve this chance-

constrained integer model with fuzzy parameters.

In the last few years, Mohammed (2000) presented a chance-constrained linear

programming method to consider stochastic fuzzy goal programming problems when the

right-hand-side coefficients were random variables. Liu (2000) constructed a general

framework of fuzzy random chance-constrained programming. Chen (2002) employed a

chance-constrained programming method to measure the technical efficiency of 39 banks

30

in Taiwan. Through a chance-constrained method, Sakawa and Kato (2003) could analyze

random or ambiguity variable coefficients, which existed in objective functions and

constraints. For multi-period operation of a multi-reservoir system, Azaiez et al. (2005)

developed a chance-constrained multi-period model to release the planned amount of

surface water. Calafiore and Ghaoui (2006) converted probability distributions into convex

second-order cone constraints to solve the chance-constrained linear programming. Li et

al. (2007) developed an inexact two-stage chance-constrained linear programming method

for planning waste management systems. To address highly uncertain problems of

chemical composition in secondary steel production, Rong and Lahdelma (2008)

represented a fuzzy chance constrained linear programming method. The final solutions of

their method based on realistic data showed that the failure risk could be managed. Liu et

al. (2008) formatted an inexact chance-constrained linear programming model to improve

water quality with minimum economical cost for Qionghai watershed in China. Cai et al.

(2008) studied an inexact community-scale energy model to plan renewable energy

systems under uncertainty. Sakawa et al. (2009) used a probability constrained model to

maximize the probability for multi objective linear programming approaches. Guo et al.

(2010) proposed an inexact fuzzy-chance-constrained two-stage mixed-integer linear

programming approach for flood diversion planning. With fuzzy boundaries, the multiple

uncertainties of this method are expressed as a combinations of intervals, fuzzy sets and

probability distributions. Solutions from this method give an optimal pattern with a

minimized system cost and maximized safety level.

31

2.2.2 Fuzzy mathematical programming

The first study of fuzzy methods was Max Black (1937), and after that Lotfi Zadeh

(1965) profoundly impacted the field by introducing the theory of fuzzy sets to represent

uncertainty to solve problems associated with intuitive information. From then, fuzzy

mathematic programming was widely used in procedures of optimal decision support

(Pawlak 1997, Kuo, Chen et al. 2001, Kuo, Chi et al. 2002, Lin and Hsieh 2004, Kulak

2005). Fuzzy programming was actually derived from incorporation of fuzzy sets theory

and cooperated with optimal mathematical programming frameworks. This fuzzy method

was divided into two major types, which are fuzzy flexible programming and fuzzy

possibility programming. On the one side, fuzzy flexible programming can provide

flexibility in both objectives and constraints (Inuiguchi, Ichihashi et al. 1990, Dubois,

Fargier et al. 2003, Mula, Poler et al. 2006). On the other side, fuzzy possibility

programming provides fuzzy regions for parameters with possibility distributions (Zadeh,

1975). In short, fuzzy mathematical programming has proven able to handle both

vagueness and ambiguity uncertain problems (Zimmermann 1983, Inuiguchi, Ichihashi et

al. 1993, Inuiguchi and Ramı́k 2000, León, Liern et al. 2003, Mula, Poler et al. 2006). Base

on this fuzzy method, other hybrid types of fuzzy programming were generated such as

fuzzy linear programming, fuzzy dynamic programming and fuzzy multi-objective

programming.

As the typical extension method, fuzzy linear programming can deal with uncertain

parameters in fuzzy membership functions from both optimization models’ right hand sides.

It allows vague information to be represented as fuzzy numbers in the parameters (Tanaka

32

and Asai 1984). Delgado et al. (1985) described the important problems in fuzzy linear

programming, and proposed a method for resolution. Carlsson and Korhonen (1986)

developed a fuzzy model in which the parameters are not known. This method shows the

optimal solution could convert to a function of the degree of precision with a numerical

example. To determine a compromise solution, Rommelfanger et al. (1989) presented a

method for solving fuzzy parameters in the objective function. Inuiguchi et al. (1990)

introduced continuous piecewise linear membership functions incorporated with fuzzy

linear programming. Tomsovic and Cheung (1992) formulated a fuzzy linear programming

approach to voltage control. It also incorporated some heuristic concepts of the system

approach. The solution in this case represented the compromise bewteen the objectives and

constraints. Lee and Wen (1996) purposed fuzzy linear programming methods and applied

these methods to the assimilative capacities of a river basin in Taiwan. The solutions of

these fuzzy methods were compared with a crisp linear programming. The results indicated

that the capacity of fuzzy linear programming is better than the deterministic approach.

Pendharkar (1997) used fuzzy linear programming to solve scheduling problems for the

coal industry in Virginia, United States. The results of this model indicated the fuzzy

method has the potential to optimize the work schedule of the coal industry. Shih (1999)

employed three types of fuzzy linear programming methods to resolve cement

transportation planning problems. This research was focused on transportation constraints

including demand fulfillment, operation capacity, and traffic congestion.

After 2000, fuzzy linear programming became more widely applied as the result of

the practical abilities of this method. Zou (2000) proposed an independent variable

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controlled by grey fuzzy linear programming method. It improved ordinary grey fuzzy

linear programming by embedding an independent control variable into general optimal

model formulations. The solutions indicated the improved method could provide more

realistic results than other methods. Pal et al. (2003) formulated an approach with the

highest membership value of fuzzy goals and two numerical examples showed the

advantage of fuzzy linear programming. For the optimal production of seasonal crops in a

given year, Biswas and Pal (2005) modeled land-use planning problems in agricultural

systems. This approach involved many factors in agriculture, which are the utilization of

total land, supply of productive resources, and aspiration levels of production of various

crops in the district of Nadia, in the Indian state of West Bengal. This method minimized

the under-deviational variables of the membership goals with the highest membership

value as achievement levels. It also compared favourably with the existing cropping plans

of this district. To deal with uncertainties caused by fuzziness, Sadeghi and Hosseini (2006)

demonstrated the application of fuzzy linear programming for optimization of energy

systems in Iran. This study revealed fuzzy linear programming is a flexible approach that

could compete favourably with other uncertainty approaches. Chen et al. (2007) used fuzzy

linear programming to minimize the supply chain of warehouses and distribution centers.

A numerical example demonstrated the effectiveness of fuzzy linear programming in

providing interactive solutions in an uncertain supply network. In order to determine the

optimal cell configuration in each period, Safaei et al. (2008) developed an extended fuzzy

linear programming model of dynamic cell formation problems. Chen and Kou (2009)

proposed fuzzy linear models to determine the fulfillment levels of parts characteristics for

customer satisfaction. To illustrate the advantages of proposed methods, a numerical

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example was provided. Lu et al. (2010) developed an interval-valued fuzzy linear

programming method for water resource management. This method was based on an

infinite α-cuts solution algorithm and led to the ability of dealing with individual

uncertainty and dual uncertainties in real world cases. It proved the system benefit could

be enlarged with the growth of violation risk. Sun et al. (2012) introduced an inexact

piecewise-linearization-based fuzzy programming method for solid waste management.

The factors of operation costs, aspiration levels and capacities tolerance of waste treatment

were reflected. To prove the ability of this method, two relative models were developed.

By comparing the results between the two models, the optimized waste amounts were

found to be similar in both models. Li and Wan (2013) developed a fuzzy approach with

multiple types of attribute values and incomplete weight information. In their study, the

proposed method demonstrated its superior through a strategy partner selection.

2.2.3 Interval mathematical programming

Regarding bounding and truncation errors, interval analysis was first introduced by

Ramon Moore (1966). In engineering applications, this method can provide rigorous

enclosures of solutions to equations, so decision makers can determine whether the results

adequately represent reality (Moore 1966). As a branch of interval analysis methods, the

interval approach was first introduced to optimization framework and named interval

mathematic programming (Huang, Baetz et al. 1992). According to research, the major

improvements over the previous optimal