Inexact Programming Research Articles

A Superiority-Inferiority-Based Inexact Fuzzy Stochastic Programming Approach for Solid Waste Management Under Uncertainty

A superiority–inferiority-based inexact fuzzy stochastic programming (SI-IFSP) model was developed for planning municipal solid waste management systems under uncertainty. The SI-IFSP approach represents a new attempt to tackle multiple uncertainties in objective function coefficients which are beyond the capabilities of existing inexact programming methods. Through introducing the concept of fuzzy random boundary interval, SI-IFSP is capable of reflecting multiple uncertainties (i.e., interval values, fuzzy sets, probability distributions, and their combinations) in both the objective function and constraints, leading to enhanced system robustness. The developed SI-IFSP method was applied to a case study of long-term municipal solid waste management. Useful solutions were generated. A number of decision alternatives could be generated based on projected applicable conditions, reflecting the compromise between system optimality and reliability as well as the tradeoffs between economic and environmental objectives. Moreover, the consequences of system violations could be quantified through introducing a set of economic penalties, reflecting the relationships between system costs and constraint violation risks. The results suggest that the proposed SI-IFSP method can explicitly address complexities in municipal solid waste management systems and is applicable to practical waste management problems.

An inexact fuzzy-chance-constrained two-stage mixed-integer linear programming approach for flood diversion planning under multiple uncertainties

In this study, an inexact fuzzy-chance-constrained two-stage mixed-integer linear programming (IFCTIP) approach is developed for flood diversion planning under multiple uncertainties. A concept of the distribution with fuzzy boundary interval probability is defined to address multiple uncertainties expressed as integration of intervals, fuzzy sets and probability distributions. IFCTIP integrates the inexact programming, two-stage stochastic programming, integer programming and fuzzy-stochastic programming within a general optimization framework. IFCTIP incorporates the pre-regulated water-diversion policies directly into its optimization process to analyze various policy scenarios; each scenario has different economic penalty when the promised targets are violated. More importantly, it can facilitate dynamic programming for decisions of capacity-expansion planning under fuzzy-stochastic conditions. IFCTIP is applied to a flood management system. Solutions from IFCTIP provide desired flood diversion plans with a minimized system cost and a maximized safety level. The results indicate that reasonable solutions are generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of flood flows.

An inexact optimization model for evacuation planning

Purpose – The purpose of this paper is to develop an interval method for vehicle allocation and route planning in case of an evacuation.Design/methodology/approach – First, the evacuation route planning system is described and the notations are defined. An inexact programming model is proposed. The goal of the model is to achieve optimal planning of vehicles allocation with a minimized system time under the condition of inexact information. The constraints of the model include four types: number of vehicles constraint, passengers balance constraints, maximum capacity of links constraints and no negative constraints. The model is solved through the decomposition of the inexact model. A hypothetical case is developed to illustrate the proposed model.Findings – The paper finds that the interval solutions are feasible and stable for evacuation model in the given decision space, and this may reduce the negative effects of uncertainty, thereby improving evacuation managers' estimates under different conditions....

An Interval Mixed-Integer Semi-Infinite Programming Method for Municipal Solid Waste Management

This study proposed an interval mixed-integer semi-infinite programming (IMISIP) method for solid waste management under uncertainty. The uncertainty can be expressed as various constants, intervals, and functional intervals. The method is mainly based on the previous efforts on interval mixed-integer linear programming (IMILP) and semi-infinite programming. The method is applied to a solid-waste management system to illustrate its effectiveness in handling complex inexact programming problems. Two scenarios are considered: one is a case with only expansions of waste-to-energy (WTE) facilities being considered, and the other is associated with potential expansions for both the WTE and the existing landfilling facilities. The results obtained can assist in identifying optimal waste management policies under uncertainties associated with interval and functional-interval parameters. Compared with conventional IMILP methods, the solutions obtained from IMISIP could be “globally” optimal because the dynamic fluctuations of the system inputs could be reflected effectively.

Flexible interval mixed-integer bi-infinite programming for environmental systems management under uncertainty

A number of inexact programming methods have been developed for municipal solid waste management under uncertainty. However, most of them do not allow the parameters in the objective and constraints of a programming problem to be functional intervals (i.e., the lower and upper bounds of the intervals are functions of impact factors). In this study, a flexible interval mixed-integer bi-infinite programming (FIMIBIP) method is developed in response to the above concern. A case study is also conducted; the solutions are then compared with those obtained from interval mixed-integer bi-infinite programming (IMIBIP) and fuzzy interval mixed-integer programming (FIMIP) methods. It is indicated that the solutions through FIMIBIP can provide decision support for cost-effectively diverting municipal solid waste, and for sizing, timing and siting the facilities’ expansion during the entire planning horizon. These schemes are more flexible than those identified through IMIBIP since the tolerance intervals are introduced to measure the level of constraints satisfaction. The FIMIBIP schemes may also be robust since the solutions are “globally-optimal” under all scenarios caused by the fluctuation of gas/energy prices, while the conventional ones are merely “locally-optimal” under a certain scenario.

An inexact programming method for agricultural irrigation systems under parameter uncertainty

Rapid population growth and economy development have led to increasing reliance on water resources. It is even aggravated for agricultural irrigation systems where more water is necessary to support the increasing population. In this study, an inexact programming method based on two-stage stochastic programming and interval-parameter programming is developed to obtain optimal water-allocation strategies for agricultural irrigation systems. It is capable of handling such problems where two-stage decisions need to be suggested under random- and interval-parameter inputs. An interactive solving procedure derived from conventional interval-parameter programming makes it possible for the impact of lower and upper bounds of interval inputs to be well reflected in the resulting solutions. An agricultural irrigation management problem is then provided to demonstrate the applicability, and reasonable solutions are obtained. Compared to the solutions from a representative interval-parameter programming model where only one decision-stage exists, the interval of optimized objective-function value is narrow, indicating more alternatives could be provided when water-allocation targets are rather high. However, chances of obtaining more benefits exist in association with a risk of paying more penalties; such a relationship becomes apparent when the variation of water availability is much intensive.

Fuzzy Inexact Mixed-Integer Semiinfinite Programming for Municipal Solid Waste Management Planning

Based on the concept of functional intervals, fuzzy inexact mixed-integer semiinfinite programming (FIMISIP) method is developed for municipal solid waste management planning. The method allows the uncertainties in parameters expressed as fuzzy, interval, and functional interval numbers to be directly communicated into the programming problem. The FIMISIP problem is solved by dividing it into two interactive semiinfinite programming (SIP) subproblems. Solutions reflecting the inherent uncertainties can then be generated by combining the SIP solutions into a set of decision intervals. The method is applied to a municipal solid waste management planning system for demonstrating its effectiveness in dealing with uncertain and dynamic complexities. Compared to the previous inexact programming methods, FIMISIP has the advantages as follows: (1) the dynamic complexity can be addressed by introducing the functional-interval parameters associated with time into the programming problem; (2) the FIMISIP solutions p...

Identifying optimal regional solid waste management strategies through an inexact integer programming model containing infinite objectives and constraints

The previous inexact mixed-integer linear programming (IMILP) method can only tackle problems with coefficients of the objective function and constraints being crisp intervals, while the existing inexact mixed-integer semi-infinite programming (IMISIP) method can only deal with single-objective programming problems as it merely allows the number of constraints to be infinite. This study proposes, an inexact mixed-integer bi-infinite programming (IMIBIP) method by incorporating the concept of functional intervals into the programming framework. Different from the existing methods, the IMIBIP can tackle the inexact programming problems that contain both infinite objectives and constraints. The developed method is applied to capacity planning of waste management systems under a variety of uncertainties. Four scenarios are considered for comparing the solutions of IMIBIP with those of IMILP. The results indicate that reasonable solutions can be generated by the IMIBIP method. Compared with IMILP, the system cost from IMIBIP would be relatively high since the fluctuating market factors are considered; however, the IMILP solutions are associated with a raised system reliability level and a reduced constraint violation risk level.

ISMISIP: an inexact stochastic mixed integer linear semi-infinite programming approach for solid waste management and planning under uncertainty

An inexact stochastic mixed integer linear semi-infinite programming (ISMISIP) model is developed for municipal solid waste (MSW) management under uncertainty. By incorporating stochastic programming (SP), integer programming and interval semi-infinite programming (ISIP) within a general waste management problem, the model can simultaneously handle programming problems with coefficients expressed as probability distribution functions, intervals and functional intervals. Compared with those inexact programming models without introducing functional interval coefficients, the ISMISIP model has the following advantages that: (1) since parameters are represented as functional intervals, the parameter’s dynamic feature (i.e., the constraint should be satisfied under all possible levels within its range) can be reflected, and (2) it is applicable to practical problems as the solution method does not generate more complicated intermediate models (He and Huang, Technical Report, 2004; He et al. J Air Waste Manage Assoc, 2007). Moreover, the ISMISIP model is proposed upon the previous inexact mixed integer linear semi-infinite programming (IMISIP) model by assuming capacities of the landfill, WTE and composting facilities to be stochastic. Thus it has the improved capabilities in (1) identifying schemes regarding to the waste allocation and facility expansions with a minimized system cost and (2) addressing tradeoffs among environmental, economic and system reliability level.

Cutting plane algorithms for robust conic convex optimization problems

In this paper, we study some well-known cases of nonlinear programming problems, presenting them as instances of inexact or semi-infinite linear programming. The class of problems considered contains, in particular, semi-definite programming, second-order cone programming and special cases of inexact semi-definite programming. Strong duality results for the nonlinear problems studied are obtained via the Lagrangian duality. Using these results, we propose some dual algorithms for the studied classes of problems. The proposed algorithms can be interpreted as cutting plane or discretization algorithms. Finally, some comments on the convergence of the proposed algorithms and on preliminary numerical tests are given.

Duality for inexact semi-infinite linear programming

The aim of this work is to generalize strong duality theorems for inexact linear programming and to derive duality results for inexact semi-infinite programming problems. We give a detailed proof of the general result, using the Dubovitskii–Milyutin approach. The last section contains applications to inexact problems and a few comments for further developments.

An inexact multi-objective programming approach for strategic environmental assessment on regional development plan*

Abstract This paper presents the development of an inexact multi-objective programming (IMOP) model and its application to the strategic environmental assessment (SEA) for the regional development plan for the Hunnan New Zone (HNZ) in Shenyang City, China. Inexact programming and multi-objective programming methods are employed to effectively account for extensive uncertainties in the study system and to reflect various interests from different stakeholders, respectively. In the case study, balancing-economy-and-environment scenario and focusing-industry-development scenario are analyzed by the interactive solution process for addressing the preferences from local authorities and compromises among different objectives. Through interpreting the model solutions under both scenarios, analysis of industrial structure, waste water treatment plant (WWTP) expansion, water consumption and pollution generation and treatment are undertaken for providing a solid base to justify and evaluate the HNZ regional development plan. The study results show that the developed IMOP-SEA framework is feasible and applicable in carrying comprehensive environmental impact assessments for development plan in a more effective and efficient manner.

Duality for a Class of Minmax and Inexact Programming Problem

Tanimoto type duality results are presented for a class of minmax programming problems by employing minmax theorems of Game Theory which are more general than the celebrated theorem of von Neumann. This, in turn, enables one to costruct a Mond-Weir type dual and thereby weaken the convexity requirements on the objective and the constraint functions. Applications of this duality to a class of inexact programming problems are also presented.

Some generalization of inexact linear programming

In this paper the generalization of the inexact programming is introduced and next its properties are investigated. The special case of the inexact linear programming is the well known linear programming

Solution and investigation of one class of inexact integer programming problems

We give a formulation of an integer programming problem with inexact constraint matrix coefficients and an inexact objective function. An approach to solution and investigation of such inexact integer programming problems is proposed.

Linear programming with imprecise data

Abstract Conventional linear programming requires the deterministic specification of all the relevant data but generally this is only known imprecisely. Several ways in which imprecision may be incorporated into the programs are discussed. These include proximate programming, inexact programming and fuzzy programming. A simple illustrative example concerned with water quality is reworked using some of the described techniques. Fuzzy programming is a particularly useful model which can handle imprecision with respect to all the parameters and can also incorporate multiple goals.