Optimization Models

Optimization models are the core of the systems engineering modeling approach. Single objective programming (SOP) models aim to search for the optimal solution associated with a well-defined SWM problem in which there is a single objective and several technical and managerial constraints in the context of MCDM (Edwards-Jones et al., 2000). These models were often applied to help solve cost minimization issues and were normally formulated by deterministic methods, including linear programming, non-linear programming, dynamic programming and mixed-integer programming models. Along these lines, these optimization models are capable of optimizing economic issues like the minimization of total costs or to maximize the total benefits to help the vehicle routing (Liebman et al., 1975), to decide what type of SWM system should be designed and the location of landfill facilities, incinerators, transfer stations (Anderson and Nigam, 1967; Anderson, 1968; Esmaili, 1972; Helms and Clark, 1974; Marks and Liebman, 1970, 1971; Gottinger, 1986, 1988; Kirka and Erkip, 1988).

Environmental issues such as materials and energy management (Caruso et al., 1993; Chang and Chang, 1998a; Hokkanen and Salminen, 1997), greenhouse gas emissions, acidification compounds, and other pollutant emissions (Hokkanen and Salminen, 1997; Seo et al., 2003;

Nasiri and Huang, 2008) were of significance. Others were social issues like public acceptance, employment (Hokkanen and Salminen, 1997), labor issues (Chang et al., 1997b) and public consensus and participation (Hung et al., 2007). To harmonize both aspects, Berger et al. (1999) developed an optimization model to help decision-makers arrive at a long-term planning of SWM activities by which both environmental and social issues were brought into the optimization context.

Approaches applied to improve results might concern uncertainty associated with either data or the waste management decision making itself. The methods that were applied for addressing the uncertainty impacts mainly consist of fuzzy set theory, grey system theory, and probabilistic theory. Some of these techniques are used alone or in combination with others.

Stochastic programming requires large data sets for the identification of the probabilistic distributions, and its application is helpful to effectively reflect the probability distributions of a single right-hand side value in a constraint of optimization models (Huang et al., 2001; Li et al., 2006c). Combination of several right-hand-side values makes the algorithm numerically

43

intangible. Fuzzy sets theory that refers to the absence of sharp boundaries in the information was applied to support decision analysis of SWM systems in the context of various optimization models. A subjective continuous membership function is usually used for the description of this kind of vague information (Chang et al., 1997a). It enables one to deal with uncertainties connected with vague linguistic expressions in decision making when probabilistic data are not available. Grey systems theory applied to support optimization analysis in SWM systems is capable of dealing with several uncertain parameter values while at the same time addressing the vagueness of its intrinsic characteristics in the information during parameter estimation (Chang et al., 1997a). Such parameters are most likely expressed as interval numbers linked with the environmental or economic factors in objective functions and constraints. It was applied to handle a variety of uncertainty concerns associated along with costs minimization in different SWM systems with respect to construction and expansion planning of waste management facility and waste flow allocation planning (Huang et al., 1992, 1994, 1995a, b). Huang et al. (1993) first conducted cost minimization using a grey fuzzy integer programming model. Huang et al. (2001) pointed out that integrated methods with various combinations of the three uncertainty theories above can produce answers concerning types, times and sites for SWM practices with improvements in uncertainty, data availability and computational requirements. Such integration enables us to handle uncertainty of different sources at the same time (Zou et al., 2000). With such a philosophy, the interval-parameter fuzzy-robust programming model was developed and applied to a SWM system to minimize the total system cost through optimal waste flow allocation (Nie et al., 2007).

Facing the need to include multiple objectives, such as the need for minimization of total cost and maximization of recycling efforts at the same time, multi-objective programming (MOP) models were often formulated and applied. These deterministic MOP solution procedures may search for the compromised or satisfactory solution via a variety of methods.

They include, but are not limited to the AHP, TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution), ELECTRE (Elimination and Choice Translating Algorithm), PROMETHÉE (Preference Ranking Organization Method for Enrichment Evaluation), and NAIADE (Novel Approach to Imprecise Assessment and Decision Environments), to aid in SWM decision making (Caruso et al., 1993; Hokkanen and Salminen, 1997; Chang and Lu, 1997; Chang et al., 2009). When uncertainty becomes a major concern, probability theory, fuzzy set theory, and grey systems theory may also be applied to supplement the model formulation of MOP. For example, Chang and Wang (1997a) applied the fuzzy set theory to help in SWM system assessment with several objective functions that intended to minimize

44

costs, traffic impacts, noise impacts and air pollution impacts simultaneously. Findings indicate that the use of fuzzy sets allows a better comparison between these objectives reflecting economic, environmental and social nature. Further, the fuzzy sets and grey systems theories can be combined with each other in a MCDM formulation with respect to environmental performance indicators in the sense that uncertainties related to fuzzy goals and inexact or grey parameter values may be properly integrated as a valid part of the MOP models (Chang and Wang, 1997a; Chang et al., 1997a).

In decision sciences, however, an important issue that can affect the consequences of a SWM system is related to the violation of policies pre-defined by authorities, such as capacity expansion limits within a defined time period, or capacity limitations from installations which would have economic consequences. This unique type of uncertainty in decision making was brought into optimization context through the use of a two-stage stochastic programming (TSP) model – an extension of the stochastic programming model. In the TSP, a decision is first undertaken before values of random variables are known; then, after the random events have happened and their values are known, a second-stage decision can be made in order to minimize ‗penalties‘ that may appear due to any infeasibility (Loucks et al., 1981; Birge and Louvenaux, 1988, 1997; Ruszczynski, 1993). Many models for handling the SWM issues were formulated as two-stage stochastic programming models with the aim of including economic penalties based on the degree of violation of policy targets. The remaining features that characterize such models are related to combinations of methods for addressing uncertainty. To the greatest extent, the fuzzy interval two-stage stochastic mixed-integer linear programming model is capable of including uncertainties described in terms of probability density functions, fuzzy membership functions and discrete intervals (Li et al., 2006b). It allows some penalty violation under a range of significance levels, being capable of facilitating dynamic analysis of decisions for expansion capacity planning for multi-region, multi-facility, multi-period and multi-option (Li et al., 2006b), resulting in alternative solutions with respect to environmental, socio-economic and system reliability conditions. If nonlinearity exists at the planning stage, fuzzy two-stage stochastic quadratic programming model is capable of reflecting uncertainties expressed as probability distribution, interval values, and fuzzy-membership functions, respectively, providing better general satisfaction after tackling the nonlinearity in the context of optimization, and generating more robust solutions (Li et al., 2006b). Overall, Table 2.2 summarizes the spectrum of application addressing ramifications of uncertainty analyses.

45

Table 2.2 A summary of optimization models applied for solid waste management

Reference Scope Methodology

Li et al., 2007 Uncertainty analysis through optimization of waste flow and/or capacity expansion

Inexact two-stage chance-constrained LP

Li et al., 2006a Interval fuzzy two-stage stochastic mixed integer LP

Nie et al., 2007 Interval-parameter fuzzy-robust programming

He et al., 2009a,2009b Inexact mixed integer bi-infinite programming

Huang et al. 1995a, 1995b Grey integer programming (GIP)

Huang et al. 1995c Grey fuzzy integer programming

Huang et al., 2001 Interval-parameter fuzzy stochastic LP

Li et al., 2006b Interval-parameter two-stage stochastic mixed integer programming

Huang et al., 1994 Grey dynamic programming

Zou et al., 2000 Independent variable controlled grey fuzzy LP

Li et al., 2008a Two-stage fuzzy robust integer programming

Chang and Wang (1996a) Grey fuzzy multiobjective mixed integer programming (MOMIP)

Li et al., 2009b Inexact fuzzy-stochastic constraint-softened

programming

Li et al., 2009a Interval-fuzzy two-stage chance constrained integer programming

Guo and Huang, 2009 Inexact fuzzy chance-constrained two-stage mixed-integer LP

Guo et al., 2009 Interval-parameter fuzzy-stochastic semi-infinite

mixed-integer LP

Jing et al., 2009 Interval-parameter two-stage chance- constraint mixed

integer LP

Xu et al., 2010 Stochastic robust interval LP

Zhang et al., 2009 Hybrid interval-parameter possibilistic programming

Xu et al., 2009 Stochastic robust chance-constrained programming

Cai et al., 2009 Interval-valued fuzzy robust programming

Li and Huang, 2009b Interval-parameter robust optimization

Li and Huang, 2009ª Inexact minimax regret integer programming

Liu et al. 2009 Dual-interval parameter LP

Wu et al., 2006 Uncertainty analysis through optimization of waste flow allocation, considering effects of economies of scale

Interval non-linear programming

Chang et al., 1997ª Location-allocation model considering economic costs and environmental issues

Fuzzy interval multiobjective mixed integer programming approach

Chang and Lu, 1997 Fuzzy global criterion approach

Cheng et al., 2003 Waste landfill siting and waste flow allocation

Inexact MILP

Cheng et al., 2009 Uncertainty analysis to Random-boundary-interval LP

46

Reference Scope Methodology

determine optimized waste allocation

Grunow and Gobbi, 2009 WEEE management system

Mixed integer programming

Badran and El-Haggar, 2006 Optimized municipal SWM system concerning collection stations location

Mixed integer programming

Mitropoulos et al., 2009 To determine the number, sizes and locations of the SWM facilities

Mixed integer programming

Seo et al., 2003 Uncertainty analysis helps select the preferred solid waste management system

Fuzzy and AHP

Tseng, 2009 To evaluate different MSW management solutions

ANP (analytical network process) and DEMATEL (decision making trial and evaluation laboratory) Huang et al., 2007 Include public for

sustainable decision making

AHP with consensus analysis model

Nasiri and Huang, 2008 Environmental

performance assessment of waste recycling programs

Fuzzy multiple attribute decision analysis

Huang et al., 2002 Violation analysis constraint through

optimization of waste flow allocation and/or capacity expansion

Interval-parameter fuzzy integer programming

Li and Huang, 2007 Fuzzy two-stage quadratic programming

Li et al., 2008b Two-stage programming

Li and Huang, 2006 Inexact two-stage mixed integer programming

Maqsood et al., 2004 Inexact two-stage MILP

Otegbeye et al., 2009 To assess recycling system form SWM

LP Berger et al., 1999 To help regional

decision-makers in the long-term nature, present different features, scales and complexity. From an environmental point of view, they may significantly help address the forcing of human-induced impacts, identify the responses in the environmental systems, and assess consequences due to such disturbances in our society (Huang and Chang, 2003). From the perspective of MSW management, the use of IMS can be helpful to understand the driving forces that are responsible for the SWM system behavior and the consequences of that outside the systems. Models used in the context of IMS therefore may cover the integration or coupling of simulation, forecasting, and optimization analyses. This is, however with a higher uncertainty, most of the time, since data from SWM