ITSSIP: Interval-parameter two-stage stochastic semi-infinite programming for environmental management under uncertainty

Abstract

In this study, an interval-parameter two-stage stochastic semi-infinite programming (ITSSIP) method is developed for municipal solid waste (MSW) management under uncertainty. In order to better account for uncertainties, the uncertainties are expressed with discrete intervals, functional intervals and probability distributions. The ITSSIP method integrates the two-stage stochastic programming (TSP), interval programming (IP), chance-constrained programming (CCP) and semi-infinite programming (SIP) within a general optimization framework. ITSSIP has infinite constraint because it uses functional intervals with time (s) being an independent variable. At the same time, ITSSIP also presents probability distribution information. The ITSSIP method can incorporate pre-regulated MSW management policies directly into its optimization process to analyze various policy scenarios having different economic penalties when the promised amounts are not delivered. The model is applied to a MSW management system with three waste treatment facilities, three cities and three periods. As an extension of mathematical programming methods, the developed ITSSIP approach has advantages in uncertainty reflection and policy analysis. Firstly, ITSSIP can help generate optimal solutions for decision variables under different levels of waste-generation rate and different levels of constraint-violation probability, which are informative for decision makers; secondly, it has the capability in addressing the parameter's dynamic feature, i.e., variations of the parameters with time; this could hardly be reflected in the previous methods. The obtained solutions are useful for decision makers to obtain insight regarding the tradeoffs between environmental, economic and system-reliability criteria.

Introduction

Waste-disposal demands have been increasing throughout the world due to rapid population growth and socio-economic development. Waste managers often face challenges of environmental protection and resources' conservation. In response to these concerns, decisions with sound economic and environmental efficiencies are desired in managing the waste (Maqsood and Huang, 2003). For example, in municipal solid waste (MSW) management systems, there are many processes that should be considered by the decision makers, such as waste collection, transportation, treatment and disposal. Moreover, many system parameters, impact factors and their interactions are associated with uncertainties. The spatial and temporal variations of many system components may further multiply these uncertainties (Thompson and Tanapat, 2005). Therefore, to deal with these complexities, inexact system analysis techniques are desired. With raised solid waste generation, reinforced environmental regulations and improved waste-disposal technologies, the waste-management systems are becoming increasingly sophisticated under uncertainity(Rao and Kumar, 2007). Consequently, a number of stochastic, fuzzy, and interval programming methods were proposed to deal with these complexities (Zeng and Trauth, 2005, Wu and Chang, 2004, Maqsood and Huang, 2003, Huang et al., 1997, Chang and Wang, 1997).

Two-stage stochastic programming (TSP) is effective for problems where an analysis of policy scenarios is desired and when the right-hand side coefficients of the constraints are random with known probability distributions. The TSP can deal with recourse, where corrective actions can be taken after a random event has taken place. In TSP, a decision is firstly made before values of random variables are known; after the random events have happened and their values are known, a second-stage decision can be made to minimize “penalties” that may appear due to any infeasibility. However, TSP is not effective in handling independent uncertainties of the left-hand side coefficients in each constraint or the objective function. Thus, interval linear programming (ILP) methods were incorporated within the TSP framework (Maqsood et al., 2004, Li and Huang, 2006, Guo and Huang, 2008). For instance, Maqsood et al. (2004) developed an inexact two-stage mixed integer linear programming model for waste management under uncertainty. Li and Huang (2006) developed an inexact two-stage mixed integer linear programming method for solid waste management in the City of Regina. In these methods, interval numbers were directly incorporated into the TSP optimization processes. However, the interval TSP methods can hardly account for risks of violating uncertain system constraints.

The chance-constrained programming (CCP) method can reflect the reliability of satisfying system constraints under uncertainty. The CCP method requires that all of the constraints being satisfied in a proportion of cases under given probability levels (Loucks et al., 1981). Recently, a number of research works for incorporating CCP and interval TSP methods were undertaken (Maqsood and Huang, 2003, Li et al., 2006, Li et al., 2007). For example, Maqsood and Huang (2003) developed a two-stage interval-stochastic programming (TISP) model for the planning of solid waste-management systems under uncertainty. Li et al. (2006) developed an inexact two-stage chance-constrained programming method for planning solid waste-management systems. However, these methods require probabilistic specifications for uncertain parameters while, in many practical problems, the quality of information that can be obtained is mostly not satisfactory enough to be presented as probabilities.

Despite the successes of the previous studies in inexact programming based on interval parameters and probability distributions under a finite number of inequality constraints, many parameters in practical problems may be more complex, and can hardly be expressed as intervals or probability distributions. For example, a crisp interval parameters (R) can be defined as [a, b] where a and b are both constants, representing lower and upper bounds of R, respectively (Moore and Yang, 1959). If R varies with the change of its independent variables (say, time), it becomes no longer deterministic. Under this situation, a concept of functional interval [a(s), b(s)] can be proposed to describe this type of uncertainty (He et al., in press). When parameters in the programming problems are expressed as functional intervals, the dynamic feature, i.e., variations of the parameters with time, can be reflected. Each constraint with functional interval parameters implies infinite deterministic constraints. This can be named as an inexact semi-infinite programming problem. There have been a number of studies on semi-infinite programming (SIP) methods (Lo′pez and Still, 2007, Guo et al., 2007, Guo et al., 2008, Go′mez et al., 2005, Vaz et al., 2004, León et al., 2000, León and Vercher, 2004, Geletu and Hoffmann, 2004, He and Huang, 2004, Goberna and López, 2002, Stein and Still, 2002, Žakovi and Rustem, 2002, Wang and Kuo, 1999, Fang et al., 1999; Lin et al., 1998). For example, He et al. (in press) proposed an interval-parameter mixed-integer semi-infinite linear programming model where interval and functional informations were both addressed. Weber and Uğur (2007) presented a generalized semi-infinite programming approach with intervals of optimizing gene-environment networks. Weber and Tezel (2007) developed an approach on generalized semi-infinite optimization of genetic networks. However, none of the previous studies simultaneously handled both uncertainty and policy analyses, especially for analyses parameters expressed as functional intervals and probability distributions.

One potential approach to better account for uncertainties expressed as functional intervals and probability distributions is to integrate the TSP, CCP and interval semi-infinite programming (ISIP) within a general optimization framework. This research aims to develop an interval-parameter two-stage stochastic semi-infinite programming (ITSSIP) method for solid waste management under uncertainty. ITSSIP is an extension of the conventional interval linear programming for handling uncertainties with modeling parameters. The method can be used for reflecting uncertainties in left-hand sides presented as intervals and those in right-hand sides as intervals, functional intervals and probability distributions. The method will then be applied to a case study of regional waste management planning to demonstrate its applicability. Through comparing ITSSIP with parametric programming and interval-parameter two-stage stochastic programming (ITSP) methods, the advantages of the developed method in dealing with various uncertainties will then be demonstrated.