Interval-parameter Linear Programming Research Articles

An Improved Solving Approach for Interval-Parameter Programming and Application to an Optimal Allocation of Irrigation Water Problem

In this study, an improved single-step method (SSM) is developed based on two-step method (TSM) to solve the interval-parameter linear programming (ILP) model of which the right-hand sides are highly uncertain. Two numerical examples are presented to ascertain appropriate value of λ in SSM. The risk preference degree of λ could be 0.8 for maximum objective function type. To demonstrate the applicability of the developed method, an agricultural water management problem has been provided in the case study section. The results show that SSM is more effective than TSM for complete solutions. There is only partial solution obtained from the first submodel of TSM, because the right-hand side of the wheat output constraint is highly uncertain. Finally, local farmers’ net benefit reaches to [8.949, 12.442] × 108 RMB (the unit of Chinese currency). The priority order of crops that are needed to be irrigated by surface water is maize > wheat > cotton.

An Interval-Parameter Fuzzy Linear Programming Approach for Accounting and Planning of Energy-Environmental Management Systems

In this paper, an energy systems planning (ESPM) model was developed for supporting the planning of an energy management system where the environmental protection was considered under uncertainty. This model was solved based on the interval-parameter fuzzy linear programming, which incorporated the fuzzy linear programming and interval-parameter linear programming into a general optimization framework. Through the conduction of this study, (a) complexities regarding energy activities and environmental conservation in energy management systems could be identified and analyzed, (b) uncertainties of parameters in energy man-agement systems could be incorporated into the planning process on the basis of interval numbers and fuzzy sets, (c) the interval form solutions could enhance the adaptation of ESPM to uncertain conditions, (d) the trade-off between energy activities and environmental protection measures as well as that between optimality and reliability of energy sys- tems couldbe reflected through the introduction of satisfaction degrees, and (e) desirable cost-minimized schemes for energy activitiessuch as energy resources allocation, technological development, facility capacity expansion and pollutant emission control could be provided for decision makers.

Development of an Interval-Based Evacuation Management Model in Response to Nuclear-Power Plant Accident

Nuclear-power-plant emergency management is very important, which can be used to avoid nuclear accident and radiation leak accident and the emergency action swiftly taken beyond the normal work procedure to mitigate the accident consequences. Evacuation management for large crowds after accidents involves a number of processes and factors with socio-economic and environmental implications. These processes and factors, as well as their interactions, are associated with a variety of uncertainties. In this study, an interval-based evacuation management (IBEM) model is developed in response to such challenges, based on interval-parameter linear programming (ILP) technique that can tackle uncertainties presented as interval values. The IBEM model is applied to a case study and then solved through an interactive algorithm that does not lead to more complicated intermediate submodels and has a relatively low computational requirement. Two scenarios are analyzed based on different policies of total capital considerations. A number of decision alternatives could be directly generated based on results from the IBEM model, which provide bases for in-depth analyses of tradeoffs among evacuation population, system cost, and constraint-violation risk.

A coupled factorial-analysis-based interval programming approach and its application to air quality management

In this study, a coupled factorial-analysis-based interval programming (CFA-IP) approach is developed through incorporating factorial analysis within an interval-parameter linear programming framework. CFA-IP can tackle uncertainties presented as intervals that exist in the objective function and the left- and right-hand sides of constraints, as well as robustly reflect interval information in the solutions for the objective-function value and decision variables. Moreover, CFA-IP has the advantage of investigating the potential interactions among input parameters and their influences on lower- and upper-bound solutions, which is meaningful for supporting an in-depth analysis of uncertainty. A regional air quality management problem is studied to demonstrate applicability of the proposed CFA-IP approach. The results indicate that useful solutions have been generated for planning the air quality management practices. They can help decision makers identify desired pollution mitigation strategies with minimized total cost and maximized environmental efficiency, as well as screen out dominant parameters and explore the valuable information that may be veiled beneath their interrelationships. Implications: The CFA-IP approach can not only tackle uncertainties presented as intervals that exist in the objective function and the left- and right-hand sides of constraints, but also investigate their interactive effects on model outputs, which is meaningful for supporting an in-depth analysis of uncertainty. Thus CFA-IP would be applicable to air quality management problems under uncertainty. The results obtained from CFA-IP can help decision makers identify desired pollution mitigation strategies, as well as investigate the potential interactions among factors and analyze their consequent effects on modeling results.

A generalized fuzzy linear programming approach for environmental management problem under uncertainty

In this study, a generalized fuzzy linear programming (GFLP) method was developed to deal with uncertainties expressed as fuzzy sets that exist in the constraints and objective function. A stepwise interactive algorithm (SIA) was advanced to solve GFLP model and generate solutions expressed as fuzzy sets. To demonstrate its application, the developed GFLP method was applied to a regional sulfur dioxide (SO2) control planning model to identify effective SO2 mitigation polices with a minimized system performance cost under uncertainty. The results were obtained to represent the amount of SO2 allocated to different control measures from different sources. Compared with the conventional interval-parameter linear programming (ILP) approach, the solutions obtained through GFLP were expressed as fuzzy sets, which can provide intervals for the decision variables and objective function, as well as related possibilities. Therefore, the decision makers can make a tradeoff between model stability and the plausibility based on solutions obtained through GFLP, and then identify desired policies for SO2-emission control under uncertainty. Implications: The GFLP method is effective in addressing air quality management planning associated with fuzzy parameter in the constraints and objective. The stepwise interactive algorithm proposed to solving the GFLP model would not lead to complicated intermediate submodels. Thus it would be much applicable for large-scale air quality management problems. The applications of the method can help decision makers in (i) generating allocation schemes for SO2 emissions; (ii) identifying the plausibility of each allocation alternative; and (iii) estimating the total costs of different SO2 emission control policies.

An Interval-Parameter Chance-Constraint Mixed-Integer Programming for Energy Systems Planning Under Uncertainty

Energy management systems are fraught with uncertainties. Such uncertainties may be expressed by interval numbers or probability distributions. In addition, issues of capacity expansion related to timing, sizing, and siting under such uncertainties need to be addressed. In this article, an interval-parameter chance-constraint mixed-integer programming (ICCP) is developed to tackle highly uncertain problems in energy management systems through integrating interval-parameter linear programming and chance-constraint programming. The developed model is then applied to a regional energy system. The results indicate that ICCP can effectively deal with uncertain information in energy management systems.

An approach to interval programming problems with left-hand-side stochastic coefficients: An application to environmental decisions analysis

An interval programming with stochastic coefficients (IPSC) model is developed for planning of regional air quality management. The IPSC model incorporates stochastic coefficients with multivariate normal distributions within an interval parameter linear programming (ILP) framework. In IPSC, system uncertainties expressed as stochastic coefficients and intervals are addressed. Since stochastic coefficients are the left-hand-side (LHS) parameters of the constraints in IPSC, a left-hand-side chance-constrained programming (LCCP) method is developed to solve the problem. The developed IPSC model is applied to a regional air quality management system. Uncertainties in both abatement efficiencies expressed as stochastic coefficients and environmental standards expressed as intervals are reflected. Interval solutions associated with different violation probability levels and/or different environmental standards have been obtained. Air quality managers can thus analyze the solutions with appropriate combinations of the uncertainties and gain insight into the tradeoffs between the abatement costs and the risks of violating different environmental standards.

Radial-Interval Linear Programming for Environmental Management under Varied Protection Levels

In this study, a radial-interval linear programming (RILP) approach was developed for supporting waste management under uncertainty. RILP improved interval-parameter linear programming and its extensions in terms of input reasonableness and output robustness. From the perspective of modeling inputs, RILP could tackle highly uncertain information at the bounds of interval parameters through introducing the concept of fluctuation radius. Regarding modeling outputs, RILP allows controlling the degree of conservatism associated with interval solutions and is capable of quantifying corresponding system risks and benefits. This could facilitate the reflection of interactive relationship between the feasibility of system and the uncertainty of parameters. A computationally tractable algorithm was provided to solve RILP. Then, a long-term waste management case was studied to demonstrate the applicability of the developed methodology. A series of interval solutions obtained under varied protection levels were compared, helping gain insights into the interactions among protection level, violation risk, and system cost. Potential waste allocation alternatives could be generated from these interval solutions, which would be screened in real-world practices according to various projected system conditions as well as decision-makers’ willingness to pay and risk tolerance levels. Sensitivity analysis further revealed the significant impact of fluctuation radii of interval parameters on the system. The results indicated that RILP is applicable to a wide spectrum of environmental management problems that are subject to compound uncertainties.

Waste management with recourse: An inexact dynamic programming model containing fuzzy boundary intervals in objectives and constraints

The existing inexact optimization methods based on interval-parameter linear programming can hardly address problems where coefficients in objective functions are subject to dual uncertainties. In this study, a superiority–inferiority-based inexact fuzzy two-stage mixed-integer linear programming (SI-IFTMILP) model was developed for supporting municipal solid waste management under uncertainty. The developed SI-IFTMILP approach is capable of tackling dual uncertainties presented as fuzzy boundary intervals (FuBIs) in not only constraints, but also objective functions. Uncertainties expressed as a combination of intervals and random variables could also be explicitly reflected. An algorithm with high computational efficiency was provided to solve SI-IFTMILP. SI-IFTMILP was then applied to a long-term waste management case to demonstrate its applicability. Useful interval solutions were obtained. SI-IFTMILP could help generate dynamic facility-expansion and waste-allocation plans, as well as provide corrective actions when anticipated waste management plans are violated. It could also greatly reduce system-violation risk and enhance system robustness through examining two sets of penalties resulting from variations in fuzziness and randomness. Moreover, four possible alternative models were formulated to solve the same problem; solutions from them were then compared with those from SI-IFTMILP. The results indicate that SI-IFTMILP could provide more reliable solutions than the alternatives.

A fuzzy linear programming approach for municipal solid-waste management under uncertainty

In this study, a fuzzy linear programming (FLP) method is developed for dealing with uncertainties expressed as fuzzy sets that exist in the constraints’ left-hand and right-hand sides and the objective function. A direct transforming algorithm is advanced for solving the FLP model that improves upon the existing method through provision of a quantitative expression for uncertain relationships among a large number of fuzzy sets. The proposed solution method can greatly reduce computational requirements, which is particularly meaningful for the application of FLP to large-scale practical problems with many fuzzy sets. The developed FLP method is applied to a case of long-term waste-management planning. The results indicate that reasonable solutions have been obtained. They can be used for generating decision alternatives and to help managers identify desired policies for waste management under uncertainty. Compared with the conventional interval-parameter linear programming approach, FLP can provide more information for solutions, containing not only the lower and upper bounds but also the most possible value for decision variables and objective function.

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

A dual-interval parameter linear programming (DILP) model was developed and applied to the planning of municipal solid waste (MSW) management systems under uncertainty. The DILP can address system uncertainties with complex presentations. Parameters in the DILP can be expressed as interval numbers; however, due to the complexity of the real world, highly uncertain information may exist in the boundaries of interval parameters. When sufficient information for these boundaries is available to access intervals, the novel concept of dual interval (being an interval-boundary interval) can be developed for handling such uncertainties and be introduced into the existing interval-parameter linear programming (ILP) framework; this leads to the DILP method where the uncertain parameters are represented as single or dual intervals. The DILP approach improves upon the ILP method by allowing dual uncertainties (presented as dual intervals) to be incorporated into the optimization processes. Decision alternatives can be generated through analysis of the single- and dual-interval solutions according to projected applicable conditions. Applicability of the developed model was demonstrated through a case of long-term MSW management planning. Reasonable solutions can be obtained, which are useful for generating desired decision alternatives and providing more information to decision makers.

Planning of municipal solid waste management systems under dual uncertainties: a hybrid interval stochastic programming approach

In this study, a random-boundary-interval linear programming (RBILP) method is developed and applied to the planning of municipal solid waste (MSW) management under dual uncertainties. In the RBILP model, uncertain inputs presented as interval numbers can be directly communicated into the optimization process; besides, intervals with uncertain lower and upper bounds can be handled through introducing the concept of random boundary interval. Consequently, robustness of the optimization process can be enhanced. To handle uncertainties with such complex presentations, an integrated chance-constrained programming and interval-parameter linear programming approach (ICCP) is proposed. ICCP can help analyze the reliability of satisfying (or risk of violating) system constraints under uncertainty. The applicability of the proposed RBILP and ICCP approach is validated through a case study of MSW management. Violations for capacity constraints are allowed under a range of significant levels. Interval solutions associated with different risk levels of constraint violation are obtained. They can be used for generating decision alternatives and thus helping waste managers to identify desired policies under various environmental, economic, and system-reliability constraints.

IPEM: An Interval-parameter Energy Systems Planning Model

Energy systems planning models are specifically developed for effective planning of energy activities in a regional, national, or global context. However, the planning process is fraught with uncertainties that may affect the effectiveness of the planning. In this study, an interval-parameter linear programming approach is introduced to develop an interval-parameter energy systems model (IPEM) for supporting effective regional energy systems planning under uncertainty. The developed methodology is then applied to a hypothetical regional energy system. The results strongly suggest that this innovative approach can effectively handle the uncertain information expressed as intervals in the energy planning process and provide more satisfactory solutions for the optimization problem of energy allocation and capacity expansion within a regional jurisdiction. Compared with other energy systems models, this model generates two schemes corresponding to the upper and the lower bounds of system objective, which represent two extreme decisions regarding environmental-economic trade-off. The interval solutions allow for detailed interpretation of the trade-off between environmental pollution risks and economic objectives.

Municipal Solid Waste Management Under Uncertainty: A Mixed Interval Parameter Fuzzy-Stochastic Robust Programming Approach

A mixed interval parameter fuzzy-stochastic robust programming (MIFSRP) model is developed and applied to the planning of solid waste management systems under uncertainty. The MIFSRP can explicitly address system uncertainties with multiple presentations. It can be used as an extension of the existing interval-parameter fuzzy robust programming, interval-parameter linear programming, and chance constraint programming methods. In this MIFSRP model, the hybrid uncertainties can be directly communicated into the optimization process and resulting solution through representing the uncertain parameters as interval numbers and fuzzy membership functions with random characteristics. Highly uncertain information arising from simultaneous appearance of fuzziness and randomness for the lower and upper bounds of interval parameters can be effectively addressed through integrating chance constraint programming, interval linear programming, and fuzzy robust programming methods into a general optimization framework. Th...

IFRP: A hybrid interval-parameter fuzzy robust programming approach for waste management planning under uncertainty

In this study, an interval-parameter fuzzy-robust programming (IFRP) model is developed and applied to the planning of solid waste management systems under uncertainty. As an extension of the existing fuzzy-robust programming and interval-parameter linear programming methods, the IFRP can explicitly address system uncertainties with complex presentations. Parameters in the IFRP model can be represented as interval numbers and/or fuzzy membership functions, such that the uncertainties can be directly communicated into the optimization process and resulting solution. Furthermore, highly uncertain information for the lower and upper bounds of interval parameters that exist due to the complexity of the real world can be effectively handled through introducing the concept of fuzzy boundary interval. Consequently, robustness of the optimization process and solution can be enhanced. Results of the case study indicate that useful solutions for planning municipal solid waste management practices can be generated. They reflect a compromise between optimality and stability of the study system. Willingness to pay higher costs will guarantee the system stability; however, a desire to reduce the costs will run the risk of potential instability of the system. The results also suggest that the proposed hybrid methodology is applicable to practical problems that are associated with highly complex and uncertain information.

An Interval-Parameter Fuzzy Approach for Multiobjective Linear Programming Under Uncertainty

An interval-parameter fuzzy linear programming method (IFMOLP) is proposed in this study for multiple objective decision-making under uncertainty. As a hybrid of interval-parameter and fuzzy methodologies, the IFMOLP incorporates interval-parameter linear programming and fuzzy multiobjective programming approaches to form an integrated optimization system. The method inherits advantages of interval-parameter programming, and allows uncertainties and decision-makers’ aspirations to be effectively communicated into its programming processes and resulting solutions. Membership functions for both objectives and constraints are formulated to reflect uncertainties in different system components and their interrelationships. An interactive solution procedure has been developed based on solution approaches of the interval-parameter and fuzzy programming techniques, plus necessary measures for handling the multiobjective feature. A didactic example is provided in the paper to illustrate the detailed solution process. Possibilities of further improvements by seeking Pareto optimum and incorporating flexible preference within constraints are also discussed.