An interval full-infinite programming method to supporting environmental decision-making

Abstract

An interval full-infinite programming (IFIP) method is developed by introducing a concept of functional intervals into an optimization framework. Since the solutions of the problem should be 'globally' optimal under all possible levels of the associated impact factors, the number of objectives and constraints is infinite. To solve the IFIP problem, it is converted to two interactive semi-infinite programming (SIP) submodels that can be solved by conventional SIP solution algorithms. The IFIP method is applied to a solid waste management system to illustrate its performance in supporting decision-making. Compared to conventional interval linear programming (ILP) methods, the IFIP is capable of addressing uncertainties arising from not only the imprecise information but also complex relations to external impact factors. Compared to SIP that can only handle problems containing infinite constraints, the IFIP approaches are useful for addressing inexact problems with infinite objectives and constraints. NomenclatureA ± ∈ {R ± } m×n , where R ± denotes a set of intervals A ± (s) ∈ {R ± } m×n , where R ± denotes a set of functional intervals a ± ij = the ith row and j th line element of A ± a ± ij = the ith row and j th line element of A ± (s i )where R ± denotes a set of intervals C ± (s) ∈ {R ± } 1×n , where R ± denotes a set of functional intervals c ± j = the j th element of C ± c ± j (s i ) = the j th element of C ± (s i ) c ± ij k = waste transportation costs from city i to facility j in period k ($/tonne) 710 L. He et al.d ± jk = residue transportation costs from waste-to-energy (WTE) facility j to landfill in period k ($/tonne) EF = energy price ($/m 3 ), an independent variable f ± = system cost f ± opt = the most desirable system cost i = index for cities (i = 1, 2, 3) j = index for facilities (j = 1, 2, 3) k = index for periods (k = 1, 2, 3) m = index for facility capacity expansion type (m = 1, 2, 3) RE ± k = revenue of unit municipal solid waste disposed by WTE facility in period k ($/tonne) RF = residue flow rate from WTE facility to landfill GP = gas price ($/liter), an independent variable TE ± jk = operational costs of facility j in period k ($/tonne) TL ± = existing capacity of the landfill (tonnes) WG ± ik = daily generation amount of solid waste in city i in period k (tonnes/day) WTE ± j = existing capacity of WTE facility j (j = 1, 2) X ± ∈ {R ± } n×1 , where R ± denotes a set of intervals x ± ij k = solid waste flow from city i to facility j in period k (tonnes/day) x − ijk,opt = the lower bound of solid waste flow from city i to facility j in period k ($/tonne)

Dual-Interval Linear Programming Model and Its Application to Solid Waste Management Planning

“…Different from a single interval, e.g., [a, b], with deterministic bounds a and b, a dual interval [[a, c], [d, b]] carries interval bounds [a, c] and [d, b], representing the uncertainties in lower and upper bounds of a single interval. Recently, a number of similar studies were reported on dealing with uncertainties in the boundaries of a single interval (Guo et al, 2008;He et al, 2008He et al, , 2009Lu et al, 2008). Nie et al (2007) introduced a concept of fuzzy-boundary interval (an interval with fuzzy-valued lower and upper bounds) and then developed an interval-parameter fuzzy robust programming (IFRP) model.…”

An interval full-infinite programming approach for energy systems planning under multiple uncertainties

“…An effective way of describing this uncertainty would be the functional intervals; this can be defined as a lower and an upper bound, which are both functions of its associated impact factor. When the coefficients of the objective functions and constraints are both allowed to be functional intervals and/or intervals, the ordinary linear programming problem becomes a more complicated problem [11,42,14,15,44,45]. Thus, interval full-infinite programming (IFIP) can be generated by introduced full-infinite programming (FIP) into an interval linear programming (ILP) framework.…”

DIPIP: Dual Interval Probabilistic Integer Programming for Solid Waste Management

“…Environmental protection and resources conservation are of major concerns along with increasing waste generation and decreasing waste-disposal capacity. In response to these, various optimization techniques were used for supporting effective management of the systems (Chang and Wang, 1997;Huang et al, 2007;Ahluwalia and Nema, 2007;Li, 2007;He et al, 2008). At the same time, uncertainties exist in many system components (e.g.…”

“…Therefore, incorporation of the joint probabilistic constraint programming (JPC) method and the dual-interval concept within an interval linear programming framework would be helpful for reflecting such dual uncertainties. Although several approaches were reported on dealing with uncertainties in the boundaries of interval inputs (Cai et al, 2007;Nie et al, 2007;Guo et al, 2008;Lu et al, 2008;He et al, 2008), limitations existed when the quality of information was not satisfactory enough to be presented as probability and/or possibility distributions for the boundaries. Few previous reports could be found on the development of a hybrid inexact probabilistic model that can simultaneously handle joint probabilistic constraints and dual uncertainties.…”