Inexact Programming Research Articles

An integrated decision support method for strategic planning and tactical management of regional biomass power plants under uncertainties

In the context of global climate change and energy security pressures, biomass power generation has been regarded as an effective way to supplement the energy supply, reduce fossil fuel consumption, and alleviate environmental pollutants. This paper presents an integrated modeling framework to address the regional biomass power generation planning to facilitate biomass utilization in developing countries. By combining the geographic information system, multi-criteria decision-making approach, as well as the type-2 fuzzy inexact programming method, the proposed integrated framework could assist the optimal decision-making for the specific site and size selection of biomass power plants in the whole region as well as the optimal operation strategies under deep uncertainties. The proposed modeling framework is applied to the particular context of Henan, an agricultural province in China with great biomass potential. The results suggested that biomass power plants with a total capacity of 2.4 GW could be established, which could satisfy 6% of local electricity consumption and avoid 18.51 Mt of carbon emissions. The sensitivity analysis of the main economic parameters indicated that lower investment and operation costs were essential to stimulate the utilization of local biomass energy. A higher feed-in tariff is not encouraged due to the limited incentive effect but is a heavy burden for the government. Furthermore, the optimal strategies for biomass power generation planning under uncertainties could be suggested for different decision makers with various risk preferences to achieve a tradeoff between system cost and risk.

Planning of best management practices for watershed-scale pollution control: A simulation-based inexact fractional programming approach

The implementation of best management practices (BMPs) is crucial for protecting and restoring watersheds from nonpoint source (NPS) pollution. BMP placement plans with a high pollution reduction to cost ratio are highly desired by decision makers. Nevertheless, few studies tackled such a ratio optimization problem in their efforts to identify BMP placement plans. BMP planning also faces challenges in handling inherent uncertainties and estimating NPS pollution loads. To address these gaps, a simulation-based inexact credibility-constrained mixed-integer fractional programming (SICMFP) model was proposed for supporting the placement of BMPs. This model was developed by integrating the export coefficient model (ECM), compound topographic index (CTI), inexact programming, and mixed-integer programming within a fractional programming framework. SICMFP extends the capability of previous BMP placement models by optimizing the ratio of conflicting objectives and handling uncertain parameters based on interval theory and credibility measure. It can also conveniently simulate the spatiotemporal distribution of pollution loads through the ECM, and efficiently identify potential wetland sites using CTI. SICMFP was applied to plan BMP placement in the Luan River Basin, located in northern China. The results showed that SICMFP could provide BMP placement plans with the optimal system efficiency between pollution reductions and cost savings under different credibility levels. These plans could help decision makers gain in-depth insights into the tradeoffs between expected system efficiency and the risk of constraint violation. As the credibility level rises from 0.6 to 0.9, the optimal ratio between pollution reduction and cost declined from [0.88, 1.66] kg/103 yuan to [0.70, 1.15] kg/103yuan, whereas the costs and pollution reductions increased. Compared with the least-cost single-objective model, SICMFP could lead to more pollution reductions and higher system efficiency. The developed SICMFP approach can also be extended to address the ratio optimization problems of water pollution management in other watersheds.

Water management policy analysis: Insight from a calibration-based inexact programming method

Predicting the impacts of water management policies under uncertainty can support policy-makers in making sound decisions. Most previous methods for policy-impact simulation lacked a calibration process that would allow model results to match observed values, and did not address the inherent uncertainties. To fill this methodological knowledge gap, an interval credibility-constrained positive mathematical programming (ICPMP) approach was first developed in this study for simulating the impacts of water management policies. This method improved over traditional positive mathematical programming in terms of uncertainty treatment. It also improved upon inexact programming methods through activating a calibration process. ICPMP was then coupled with a multi-attribute decision-making approach, leading to a simulation-evaluation method that could support the analysis and screening of policy alternatives. This method has been applied to an agricultural water management problem in northwestern China. Various policy alternatives corresponding to different water prices and varied fuzzy credibility levels of water availability were investigated. The impacts of these alternatives over farmers’ income, irrigation water consumption, benefit per cubic meter of water, planting area, and fertilizer use were simulated through ICPMP. Forty policy alternatives with better indicator performances were selected to be comprehensively evaluated using the multi-attribute decision-making approach. The results indicated that when the decision-makers have a positive preference for water availability, the water price at the turning point of water use efficiency (i.e. benefit per cubic of water) was recommended. When decision-makers have conservative attitude towards water availability, it was recommended to keep the water price unchanged. Comparisons of ICPMP against traditional methods indicated that, ICPMP could generate more reliable results through incorporating uncertain information as inputs and reproducing actual situation. As well, solutions of ICPMP could provide in-depth analysis of the trade-offs between violation risk and system benefits. ICPMP and the associated simulation-evaluation method are also applicable to analyze other types of policies.

A Two-stage Inexact Programming with Value-at-risk under Uncertainty

Abstract Measuring risks for two-stage programming model has certain theoretical and actual significance. Therefore, a model of two-stage inexact programming with value-at-risk (TIPV) has been developed. The highlights of TIPV are (1) it can reflect the effects of uncertain parameters including random variables and intervals of fuzzy boundary; and (2) the potential risks of recourse can be revealed. A solution algorithm is developed through considering uncertainties, two-stage and VaR comprehensively.

Sustainable agricultural water management incorporating inexact programming and salinization-related grey water footprint

In arid and semi-arid regions, improper irrigation activities not only exacerbate water shortages, but also lead to environmental pollution such as soil salinization that hinders crop growth and agricultural sustainability. There has been a lack of agricultural water management tools that could support agricultural water management with salinization-related grey water footprints being considered and associated uncertainties being addressed. In this study, salinization-related grey water footprints were measured through accounting for relationships among irrigation, soil salinity, evapotranspiration and crop yield, and then incorporated into an agricultural water management model for supporting environmentally sound irrigation decisions. Such an agricultural water management model was also characterized by a newly proposed generalized fuzzy interval fractional programming (GFIFP) method that could address ratio problems of two objectives and dual uncertainties. The developed methodology coupling the GFIFP method and grey water footprints was applied to an irrigation region in northwestern China where water scarcity and soil salinization hindered local development. Five credibility levels corresponding to decision makers' varied satisfactory degrees over water availability, and nine weight coefficients of possibility and necessity measures were considered. Results showed that, in order to reduce negative environmental impacts while increasing economic benefits, more irrigation water should be allocated to wheat and sunflower rather than corn. When the credibility level is 0.5 and weight coefficient is 0.9, the system efficiency would be the highest. Compared to the benchmark year of 2018, this scenario would generate [0.330, 6.647] billion yuan more benefits and [16.0, 133.9] million m3 less grey water footprints. Compared to three conventional approaches, GFIFP could provide decisions with more flexibility and less environmental impacts. The developed approach could also be applied to agricultural water management problems in other areas aiming at reducing grey water footprints and enhancing environmental benefits under uncertainty.

A two-stage inexact programming with value-at-risk for water resources management

In this study, the application of a two-stage inexact programming with value-at-risk (TIPV) model in water resources system planning has been developed. The TIPV method is intended to tackle the inexact parameters and the risks of economic loss. The application of case study shows that more alternatives under multiple levels of risks could be generated. The amount of water shortages and the width of system benefit intervals would decrease as the risk increases. TIPV could provide more effective information for stakeholders to recognize social policies with maximized system benefits under various risk levels.

Data-driven interval credibility constrained quadratic programming model for water quality management under uncertainty

Although integrated simulation-optimization modeling can provide a comprehensive and reliable analysis for water quality management (WQM), it is usually not easy to implement in practice. This study proposed a new efficient simulation-optimization modeling approach by leveraging the power of data-driven modeling, to support WQM under various uncertainties. A water quality simulation model is integrated with the optimization model, and then substituted by a series of numerical surrogate models based on inexact linear regression. The transformation can significantly reduce the computational burden and make it possible to implement uncertainty quantification through hybrid inexact programming. The proposed model incorporates interval quadratic programming and credibility constrained programming to deal with nonlinearity and various uncertainties associated with the management system. The proposed approach is applied to a real case study of the Grand River watershed in Canada for controlling phosphorus concentration in river water. The Grand River Simulation Model (GRSM) is employed as the physical simulation model to estimate the total phosphorus concentration in the river. Interval solutions under different confidence levels of violating the effluent standards were obtained, which can be used to generate optimal phosphorus control strategies. The results indicate the proposed data-driven interval credibility constrained quadratic programming (DICCQP) model is able to provide reliable and robust solutions for WQM by considering nonlinearity and various uncertainties while maintaining a high computational efficiency. The proposed new framework can be extended and applied to the other watersheds. The high efficiency of the proposed model makes it possible to solve large-scale complex water quality management and planning problems.

A sustainable road pricing oriented bilevel optimization approach under multiple environmental uncertainties

The rapidly-increasing urban transportation contributes to a number of externalities such as congestion and pollution; moreover, environmental uncertainties widely exist and bring challenges to creating traffic emission control strategies. To address these problems, this paper proposes an inexact bilevel programming approach under stochastic and fuzzy uncertainties (BLP-SF) for a toll scheme design with the considerations of the externalities of vehicular emission and road pricing policy. The BLP-SF approach can deal with multiple environmental uncertainties by (1) specifying the uncertainties as probability distributions and/or fuzzy sets, and (2) integrating chance-constrained programming and fuzzy possibility programming into the bilevel programming framework. A road pricing problem is exemplified under various policy scenarios to demonstrate the applicability of the BLP-SF approach. It is shown that the proposed BLP-SF approach can achieve optimal charging schemes with improvements in total emission reduction and congestion alleviation compared with traditional models. This study contributes to urban transportation management associated with congestion and emission.

Managing the uncertainty problems of municipal solid waste disposal

In waste management systems, several parameters such as the rate of waste production, disposal facility, treatment cost, and their relations may be uncertain and can influence the associated optimization processes. These uncertainty problems in waste management were addressed by using various inexact programming methods. For example, fuzzy, stochastic programming, and interval programming techniques were generally used for solving the uncertainty-related waste management problems. The analysis revealed that the efficiency of waste management system can be maximized by the proper use of these optimization techniques. In this approach, an uncertainty problem is reduced into several subproblems with sureness by using the minimax regret optimization technique. And these subproblems are focused on a calculation where the lament of not getting the goal is minimized. The analysis also revealed that the fuzzy-stochastic method was increasingly used for dealing with the waste management system uncertainty in recent times. This paper gives an overview of dealing with the uncertainty problems of waste disposal in urban areas. An indication of the solid waste disposal problems and its management in conjunction with the repercussion of the investigation is described. The rationale and setting of the uncertainty issues in proper waste management are detailed. The applications of fuzzy analysis approach and integrated waste management in dealing with the uncertainty problems are presented. The applications of these techniques in diverse case studies worldwide are discussed and finally, the conclusions of the literature analysis are summarized.

A Stochastic Inexact Robust Model for Regional Energy System Management and Emission Reduction Potential Analysis—A Case Study of Zibo City, China

In this study, in order to improve regional energy system adjustment, a multistage stochastic inexact robust programming (MSIRP) is proposed for electric-power generation planning and structure adjustment management under uncertainty. Scenario-based inexact multistage stochastic programming and stochastic robust optimization were integrated into general programming to reflect uncertainties that were expressed as interval values and probability distributions in the objective function and constraints. An MSIRP-based energy system optimization model is proposed for electric-power structure management of Zibo City in Shandong Province, China. Three power demand scenarios associated with electric-power structure adjustment, imported electricity, and emission reduction were designed to obtain multiple decision schemes for supporting regional sustainable energy system development. The power generation schemes, imported electricity, and emissions of CO2 and air pollutants were analyzed. The results indicated that the model can effectively not only provide a more stable energy supply strategies and electric-power structure adjustment schemes, but also improve the balanced development between conventional and new clear power generation technologies under uncertainty.

A multi-objective optimization model for hub network design under uncertainty: An inexact rough-interval fuzzy approach

This article proposes a multi-objective mixed-integer model to optimize the location of hubs within a hub network design problem under uncertainty. The considered objectives include minimizing the maximum accumulated travel time, minimizing the total costs including transportation, fuel consumption and greenhouse emissions costs, and finally maximizing the minimum service reliability. In the proposed model, it is assumed that for connecting two nodes, there are several types of arc in which their capacity, transportation mode, travel time, and transportation and construction costs are different. Moreover, in this model, determining the capacity of the hubs is part of the decision-making procedure and balancing requirements are imposed on the network. To solve the model, a hybrid solution approach is utilized based on inexact programming, interval-valued fuzzy programming and rough interval programming. Furthermore, a hybrid multi-objective metaheuristic algorithm, namely multi-objective invasive weed optimization (MOIWO), is developed for the given problem. Finally, various computational experiments are carried out to assess the proposed model and solution approaches.

An inexact programming approach for urban electric power systems management under random-interval-parameter uncertainty

In electric power management problems, a number of system parameters are subjected to many powerful and unpredictable influences (e.g. economic situation, technologies in energy conservation). This leads to the difficulty in accurately qualifying them through conventional approaches. In this paper, a concept of random-boundary interval (RBI) whose upper and lower bounds are random variables is introduced to account for the high-uncertain components, like electricity demand, in electric power systems. A fuzzy integer programming model with RBI parameters is developed to support effective planning of urban electric power systems (UEPS). A case study is then provided for demonstrating the applicability of the proposed model. The obtained solutions can be used for generating efficient electricity-supply and capacity-expansion schemes under various situations. Compared with the existing methods, the proposed model could tackle the complex parameter uncertainty in electric power systems, and the potential system-failure risks arising from such uncertainty could be quantified by means of violation levels and satisfaction levels.

Applications of inexact programming methods to waste management under uncertainty: current status and future directions

Waste management problems are subject to uncertainties presented as intervals, random variables and/or fuzzy sets. During the past 20 years, inexact programming methods have been developed and applied increasingly to waste management problems under uncertainty. To obtain a snapshot of these studies, this paper gives a review on recent developments, applications, challenges, and barriers associated with inexact programming techniques in supporting waste management. The results indicate that the majority of inexact programming methods can be categorized as two-stage stochastic programming, chance-constrained programming, fuzzy flexible programming, fuzzy robust programming, interval-parameter programming, mixed-integer programming, multiple-objective programming, and nonlinear programming. The demanding areas for future research efforts would include: expansion of conventional concepts to quantify uncertainties, integration of single inexact programming method with other programming methods to deal with multiple uncertainties and even complexities (e.g. nonlinearities and interactions), integration of inexact programming with other modeling techniques (e.g. life cycle assessment, multiple-criteria decision analyses, and waste flow simulation) to support sustainable waste management, development of more efficient algorithms to solve the proposed methods, linkage of waste management with its environmental impacts (e.g. air pollutants and GHG emissions as well as leachate pollution) within an inexact optimization framework, and applications of the developed methods to novel (e.g. specific types of wastes) or real-world waste management cases in different countries.

An Inexact Fuzzy-Queue Programming Model for Coupled Coal and Power Management

In this study, a coupled coal and power management model is developed based on an inexact fuzzy-queue programming method. The model is applied to a case study of coupled coal and power management system. The results indicate that the developed model can afford reasonable measures for solving coal-blending and coal-resources-allocation problems in coupled coal and power management system. It is useful for (a) standardizing coal supply, (b) improving efficiencies of the boilers, and (c) reducing emissions of air pollutants for meeting local air-quality targets. Potential risks associated with coal shortages and instability due to property variations from multiple sources can also be highly lowered through diversifying coal sources. Moreover, it can handle queue problems and multiple uncertainties exist in coupled coal and power management activities. Thus, it can help decision makers obtain optimal coal-blending schemes and coal-allocation patterns.

A Genetic-Algorithms-Based Approach for Programming Linear and Quadratic Optimization Problems with Uncertainty

This paper proposes a genetic-algorithms-based approach as an all-purpose problem-solving method for operation programming problems under uncertainty. The proposed method was applied for management of a municipal solid waste treatment system. Compared to the traditional interactive binary analysis, this approach has fewer limitations and is able to reduce the complexity in solving the inexact linear programming problems and inexact quadratic programming problems. The implementation of this approach was performed using the Genetic Algorithm Solver of MATLAB (trademark of MathWorks). The paper explains the genetic-algorithms-based method and presents details on the computation procedures for each type of inexact operation programming problems. A comparison of the results generated by the proposed method based on genetic algorithms with those produced by the traditional interactive binary analysis method is also presented.

An Inexact Mix-Integer Two-Stage Linear Programming Model for Supporting the Management of a Low-Carbon Energy System in China

In view of the great contribution of coal-fired units to CO2 emissions, the coupled coal and power system with consideration of CO2 mitigation is a typical sub-system of the highly emitting Chinese energy system for low-carbon studies. In this study, an inexact mix-integer two-stage programming (IMITSP) model for the management of low-carbon energy systems was developed based on the integration of multiple inexact programming techniques. Uncertainties and complexities related to the carbon mitigation issues in the coupled coal and power system can be effectively reflected and dealt with in this model. An optimal CO2 mitigation strategy associated with stochastic power-generation demand under specific CO2 mitigation targets could be obtained. Dynamic analysis of capacity expansion, facility improvement, coal selection, as well as coal blending within a multi-period and multi-option context could be facilitated. The developed IMITSP model was applied to a semi-hypothetical case of long-term coupled management of coal and power within a low-carbon energy system in north China. The generated decision alternatives could help decision makers identify desired strategies related to coal production and allocation, CO2 emission mitigation, as well as facility capacity upgrade and expansion under various social-economic, ecological, environmental and system-reliability constraints. It could also provide interval solutions with a minimized system cost, a maximized system reliability and a maximized power-generation demand security. Moreover, the developed model could provide an in-depth insight into various CO2 mitigation technologies and the associated environmental and economic implications under a given reduction target. Tradeoffs among system costs, energy security and CO2 emission reduction could be analyzed.

An inexact programming approach for supporting ecologically sustainable water supply with the consideration of uncertain water demand by ecosystems

This paper presents the development and the first application of an inexact quadratic programming (IQP) approach for sustainable water supply under multiple uncertainties. The developed IQP improves conventional nonlinear programming by tackling multiple uncertainties within an individual parameter; IQP is also superior to existing inexact methods due to its reflection of economies of scale and reduction of computational requirements. An interactive solution algorithm with high computational efficiency was also proposed. The application of IQP to long-term planning of a multi-source multi-sector water supply system demonstrated its applicability. The close reflection of system complexities, such as multiple uncertainties, scale economies and dynamic parameters, could enhance the robustness of the optimization process as well as the acceptability of obtained results. Corresponding to varied system conditions and decision priorities, the interval solutions from IQP could help generate a series of long-term water supply strategies under a number of economic, environmental, ecological, and water-security targets.

Inexact Fuzzy Full-Infinite Mixed-Integer Programming Method for an Integrated Air and Waste Management System

Inexact fuzzy full-infinite mixed-integer programming (IFFMIP) was proposed as an extension of inexact full-infinite programming. It can deal with the uncertainties that exist in most of the parameters as intervals, economic data as functional intervals, fuzzy set in capacity constraint, and discrete variables in a facility-expansion plan. Although many models for dealing with waste planning and the air quality management problem exist, this is the first attempt to integrate waste flow allocation and an air quality control system. The proposed model was applied in an integrated air and waste management system. It can solve the waste allocation plan and air quality control, and reflect the interaction between these two systems. The solutions from the proposed IFFMIP model indicate that a landfill would be the first choice in period 1, whereas an incinerator would become the main approach for waste disposal at the end of the planning horizon. This is because of the increasingly strict environmental standard...

Integrated Modeling for Optimal Municipal Solid Waste Management Strategies under Uncertainty

In this study, an inexact fuzzy-probabilistic programming (IFPP) method is advanced for developing optimal municipal solid waste (MSW) management strategy with uncertain information. The IFPP can support the assessment of risk of violating constraints associated with fuzzy and random features. The developed method is applied to a case study of long-term MSW management planning in the city of Changchun, China. Violations for transfer-station capacity constraints are allowed under a range of probability and possibility levels, which are related to trade-offs between the system cost and the constraint-violation risk. The results indicate that useful solutions for planning the MSW management practices have been generated. They are valuable for supporting the identification of efficient waste-flow allocation patterns, the long-term capacity planning of the city’s waste management system, and the formulation of local policies regarding waste management under uncertainty.

Model-based decision support system for water quality management under hybrid uncertainty

Water quality management is inevitably complicated since it involves a number of environmental, socio-economic, technical, and political factors with dynamic and interactive features. In planning water quality management systems, uncertainties exist in many system components and may affect the system behaviours. It is thus desired that such complexities and uncertainties be effectively addressed for providing decision support for practical water quality management. The objective of this study is to develop a model-based decision support system for supporting water quality management under hybrid uncertainties, named FICMDSS, which is based on a hybrid uncertain programming (HFICP) model with fuzzy and interval coefficients. The system provides an effective tool for the decision makers in dealing with water quality management problems and formulating desired policies and strategies. The user can easily operate the system and obtain the decision support through user-friendly graphical interfaces. The HFICP model improves upon the existing inexact programming methods through incorporation of hybrid fuzzy and interval uncertainties into the optimization management processes and resulting solutions. Results of a water quality management case study indicated that the developed FICMDSS can facilitate the decision making in planning agricultural activities for water quality management in agricultural systems. Feasible decision alternatives for cropping area, amounts of manure and fertilizer application, and sizes of livestock husbandry can be generated for achieving the maximum agricultural system benefit subject to the given water-related constraints. The user can better make the decisions for water quality management under hybrid uncertainties with the help of FICMDSS.