Interval Linear Programming Model Research Articles

Integrated multi-objective chance-constrained fuzzy interval linear programming model with principal component analysis for optimizing agricultural water resource management under uncertainties

ABSTRACT This study addresses the pivotal challenge of water resource allocation in urban environments by introducing a novel approach – a multi-objective chance-constrained fuzzy interval linear programming model integrated with principal component analysis (PCA). This innovative model aims to alleviate subjectivity in urban water management processes, particularly in adjusting water demands across various sectors. The proposed model incorporates correlation analysis to identify dimensionality-reducing factors of multitarget components, determining the proportion of each target component relative to the total components. Fuzzy sets are applied to irrigation water resource allocation quantity, segmented into six levels of fuzzy membership to analyze the stochasticity of water supply. Results demonstrate the model's efficacy, revealing that variations in risk probabilities impact water supply, necessitating positive water management strategies to enhance agricultural efficiency and negative strategies to mitigate the risk of inadequate water supply. Key findings emphasize the significance of agricultural water availability and the structure of irrigation water use in optimal resource allocation. Importantly, the study showcases the enhanced precision achieved through the proposed multi-objective chance-constrained fuzzy interval linear programming with PCA, thereby refining the optimization outcomes for water management under multifaceted objectives.

Predictive simulation and optimal allocation of surface water resources in reservoir basins under climate change

Predicting and allocating surface water resources are becoming increasingly important tasks for addressing the risk of water shortages and challenges of climate change, especially in reservoir basins. However, surface water resource management includes many systematic uncertainties and complexities that are difficult to address. Thus, advanced models must be developed to support predictive simulations and optimal allocations of surface water resources, which are urgently required to ensure regional water supply security and sustainable socioeconomic development. In this study, a soil and water assessment tool-based interval linear multi-objective programming (SWAT-ILMP) model was developed and integrated with climate change scenarios, SWAT, interval parameter programming, and multi-objective programming. The developed model was applied to the Xinfengjiang Reservoir basin in South China and was able to identify optimal allocation schemes for water resources under different climate change scenarios. In the forecast year 2025, the optimal water quantity for power generation would be the highest and account for ∼60% of all water resources, the optimal water quantity for water supply would account for ∼35%, while the optimal surplus water released from the reservoir would be the lowest at ≤5%. In addition, climate change and reservoir initial storage would greatly affect the surplus water quantity but not the power generation or water supply quantity. In general, the SWAT-ILMP model is applicable and effective for water resource prediction and management systems. The results from different scenarios can provide multiple alternatives to support rational water resource allocation in the study area.

Optimum strategies of regional kitchen waste treatment against a background of carbon mitigation

Reducing carbon emissions of kitchen waste treatment (KWT) is regarded as an effective way for achieving the goal of carbon mitigation. In this study, a hybrid method, combining life cycle analysis (LCA) and an interval linear programming model, was proposed. The carbon emissions of four technologies (i.e., landfilling, incineration, anaerobic digestion, and composting) for KWT were evaluated based on the LCA framework. Strategies for the multiple technologies were obtained using the interval linear programming model. This method can effectively identify the potentials for carbon mitigation in the KWT technologies with consideration of their eco-economic performance. To confirm the effectiveness of the method, a case study was conducted in the urban agglomeration of the Pearl River Delta (PRD) region in China. Three scenarios (i.e., S1 to S3) were proposed for multiple KWT modes from single-city and urban-agglomeration perspectives. The results of LCA showed that the carbon emissions were smallest (i.e., 95.77 kg CO 2 eq/tonne) for anaerobic digestion among the technologies. In terms of desired strategies in the KWT management, the application proportions of anaerobic digestion and composting would significantly increase under S2 and S3, while the proportion of incineration would be smaller than that under S1. Considering variations of eco-economic performance under the multiple scenarios, the KWT strategies under S3 would be recommended in terms of their contributions in carbon mitigation. It was recommended that the capacity of KWT should be improved 12.75–15.37 Mt and 6.86–7.95 Mt in order to adopt the strategies under S2 and S3.

Managing irrigation water resources with economic benefit and energy consumption: an interval linear multi-objective fractional optimization model under multiple uncertainties

At present, water shortage has increasingly become one of the key factors restricting the sustainable development of agriculture. And during the process of agricultural production, the energy consumption has also been paid more and more attention. Therefore, in order to deal with the above problems, an improved interval linear multi-objective fractional programming model (ILMOFPM) was developed, which has the characteristic is that it not only focuses on the optimal allocation of the amount of irrigation water resources, but also on improving irrigation water resources use efficiency. To demonstrate its effectiveness and applicability, ILMOFPM was applied to a case study in Jinghuiqu irrigation district, Shaanxi Province, China. Maximum economic benefit per unit of water resource and minimum energy consumption per unit of water resource were selected as the planning objectives. A series of irrigation water resources optimal allocation schemes were obtained, which could give the decision makers the desired optimal scheme according to the actual situation. Based on the results of the established model, economic benefit and energy consumption generally decreased as the representative hydrological years changed from wet to dry, and the total crop planting area also showed a decreased trend. Moreover, a lot of water resources was saved by introducing crop evapotranspiration and effective precipitation in the developed model, 57.45%, 44.17%, 39.33% of total water resources consumption were saved under different representative hydrological years, respectively.

Optimization of residents' dietary structure with consideration of greenhouse gas mitigation and nutritional requirements

Food-related GHG emissions are ultimately driven by residents' dietary structures. Influenced by socio-economic development and consumer performances, modification for residents' dietary structures should reflect the variations in food demand and nutritional requirements. An integrated approach was developed by incorporating autoregressive integrated moving average (ARIMA), multiple linear regression, and an interval linear programming model into a general life cycle analysis (LCA) framework. In detail, (a) the GHG emissions from agricultural systems were assessed in the LCA framework; (b) the variations in food demand influenced by socio-economic development were identified; and (c) a specific model for dietary structure optimization that focused on GHG emission reduction and dietary nutritional balance was established. The variations of food supply in different seasons and food demands in different age and gender groups were considered in the model. A case study was proposed to illustrate the application of the approach in Guangdong Province, China. Compared with the predicted food demands based on multiple linear regression models, per capita cereal, fruit, and meat intake under the optimized dietary structure would be slightly smaller than the predicted demands, as well as vegetable, bean and nut intake would be more than the predicted demands. Among different age and gender groups, adult and adolescent males would contribute the biggest GHG emissions of diets, due to their high daily energy requirement. Conversely, adult and older women would contribute the smallest GHG emissions in food consumption.

A Copula-based interval linear programming model for water resources allocation under uncertainty

Water scarcity tends to be aggravated by increase in water demand with the trend of socio-economic development. Thus, non-stationary characteristics of water demand should be identified in water resources allocation (WRA) to alleviate the potential influences from water shortages. In this study, a Copula-based interval linear programming model was established for regional WRA. Through combining correlation analysis and an interval linear programming model, this model can: 1) identify interactions between water demand and socio-economic development levels based on Copula functions, 2) explore variations in water shortage with consideration of multiple risk tolerance levels of decision-makers based on Copula sampling, and 3) obtain desired strategies for WRA through an interval linear programming model. Also, Dalian City in China was selected as a case study area to verify the effectiveness of the model for WRA to five water users (i.e., agricultural sector, industrial sector, public service sector, domestic residents, and ecological environment). Considering multiple tolerance levels of decision-makers to water shortage risk, three scenarios (i.e., S1 to S3), indicating 20%, 40%, and 60% of their low, medium, and high tolerance levels, were proposed. The results showed that the correlation between the amount of water demand and indicators of socio-economic development can be described by Clayton and Gaussian Copula functions. The total water supply of Dalian in 2030 would increase by 2.06%–2.65%, compared with the one in 2025. The allocation of water resources across districts was influenced by varied water demand, energy consumption, and risk tolerance levels. Compared with the amount of water allocation in 2025, the contribution of transferred water sources would increase by 6.71% and 7.04% under S1 and S2 in 2030, respectively, and decrease by 14.31% under S3. With the increase of risk tolerance levels of decision-makers, the amount of water supply in Dalian City would gradually decrease.

The solution for type-2 fuzzy linear programming model based on the nearest interval approximation

Linear programming is an important branch of operations research. The model is mature in theory and widely used in real life. However, various complex realistic scenarios involve fuzzy information. In this paper, we consider a fuzzy linear programming (FLP) model in which all parameters are trapezoidal interval type-2 fuzzy numbers (IT2FNs) and propose a solution method based on the nearest interval approximation and the best-worst cases (BWC) method. We prove the nearest interval approximation interval of trapezoidal IT2FNs, then the trapezoidal IT2FNs in the model are transformed into interval numbers which both upper and lower limits are interval numbers. With the help of best-worst cases (BWC) method, the sub-models of the transformed interval linear programming model are proposed, and four sub-solutions with different specific meanings can be obtained by solving them respectively. Finally, an application example is presented to show the rationality and practical significance of the method.

Synergic management of crop planting structure and biomass utilization pathways under a food-energy-water nexus perspective

With the advantages of avoiding land-use competition, biomass from agricultural residues is a promising solution to alleviate the energy crisis and environmental problems. In the traditional biomass utilization management problem, few studies have considered that land and water are key resources for the sustainable development of biomass from a life cycle view. Concerning the complex food-water-energy nexus, this paper focuses on the synergetic management of crop planting structure and biomass utilization as a whole system. An integrated framework is proposed for the co-optimization of cropland distribution and biomass utilization pathway under multiple uncertainties. Base on interval programming and fuzzy set theory, a type-2 fuzzy interval linear programming model is developed to handle multiple uncertainties with various characteristics and provide the optimum strategies for decision makers with different risk preferences. The proposed framework is verified by a case study of Hebei Province, China. Moreover, the impacts of water shortage and the implementation of the carbon price on the optimized strategies, economic and environmental benefits are investigated to provide deeper management insights when facing complex external factors. The obtained results suggest that corn and wheat will still be the primary crops, and bioethanol production will gain priority in the biomass utilization pathways for economic purposes. Water resources availability will greatly affect the crop planting allocation as well as total benefits but barely influence the biomass utilization pathways. Furthermore, with the implementation of the carbon trade market, a higher carbon price will stimulate biomass pellets production to replace fossil fuel consumption and improve the economic benefits directly.

A novel interval linear programming based on probabilistic dominance

This study investigates a novel interval linear programming based on probabilistic dominance. Firstly, the definition of interval linear programming is briefly reviewed. Then a new interval linear programming model is presented based on probabilistic dominance. The probabilistic dominance index treats the intervals as uniformly distributed variables and the interval inequality relation is further defined by probability. To deal with non-linearity in probabilistic dominance index, sequential quadratic programming is used to solve the problem and the performance measure approach is proposed to overcome the convergence difficulties. The determination and sensitivity analysis of the target performance measure are discussed to assess the sequential quadratic programming algorithm. Meanwhile, the extension of the proposed method to fuzzy interval linear programming is discussed. Furthermore, the proposed method is applied to the design of the plane truss structure with interval parameters. Finally, the effectiveness and rationality of the developed method are demonstrated by two mathematical examples and one interval parametric plane truss structure optimization example.

Time Variant Interval Linear Programming for Environmental Management Systems

Optimization technology is widely applied to maximize economic profit under ecology constrains in environmental management systems. To tackle the inherent uncertainties, inexact optimization methods have been proposed. Interval linear program- ming (ILP) model has drawn increasing scholarly attention. ILP model describe uncertainty by one coarse scaled stochastic process. However, uncertainty often involves multiple stochastic processes when zooming into high resolution. ILP model may not satisfied fine scale constraints. A time variant interval linear programming (TVILP) model is developed to implement temporal downscaling, and likewise, a heuristic algorithm integrating dynamic programming is proposed for Markov chained TVILP. Dynamic programming can converts time complexity exponential to polynomial. In the current paper, the performance of TVILP model is analyzed based on the following three metrics: maximal profit (M_profit), constraint violation risk (CVR), and maximal profit path risk (MPR). The perfor- mance of TVILP is further compared with the performance of Best and Worst method, the classic ILP model, Interval linear programming contractor, and Interval-parameter multi-stage stochastic linear programming. Experimental results reveal that TVILP provides refined solutions on a smaller granularity whose decision space contracts based on the most possible transition paths. Unable to obtain the maxi- mum profit, though, TVILP does pose decreased constraint violation risk and maximal profit path risk, facilitating more feasible and reliable decision-making on environmental management.

Finding efficient solutions in the interval multi-objective linear programming models

The aim of our paper is to obtain efficient solutions to the interval multi-objective linear programming (IMOLP) models. In this paper, we propose a new method to determine the efficient solutions in the IMOLP models by using the expected value and variance operators (EVV operators). First, we define concepts of the expected value, variance, and uncertainty distributions, and present some properties of the EVV operators. Then, we introduce the IMOLP model under these operators. An IMOLP model consist of separate ILPs, but using the EVV operators and the uncertainty distributions, it can be converted into the interval linear programming (ILP) models under the EVV operators (EVV-ILP model). We show that optimal solutions of the EEV-ILP model are the efficient solutions of IMOLP models with uncertainty variables. The proposed method, which is called EVV, is not hard to solve. Finally, Monte Carlo simulation is used to show its performance assessment.

Explicit flow-risk allocation for cooperative maximum flow problems under interval uncertainty

Many decision-making problems in transportation networks can be defined as maximum flow problems. During the last five decades, several efficient solution approaches have been proposed for the deterministic maximum flow problems. On the other hand, arc capacities of networks cannot be precisely defined in many real life settings. These networks are called uncertain. In this case, it becomes challenging to maintain a stable flow on the network. This paper presents a new approach based on the framework of interval analysis for the solution of maximum flow problems. We address a multiple-owners network problem by presenting a risk explicit interval linear programming model for the desired value of the system aspiration level. Afterwards, we employ a well-known collaborative game theoretic approach (the Shapley value) in a multiple-owners network under interval uncertainty in order to solve the maximum flow problem. A detailed numerical example is provided to present the suitability of the proposed approach in devising a stable network flow. The obtained numerical results and the trade-offs between decision risk and network flow information would be very valuable for supporting decision makers in resolving maximum flow problems when facing uncertainty.

An Optimization-Evaluation Agricultural Water Planning Approach Based on Interval Linear Fractional Bi-Level Programming and IAHP-TOPSIS

In this study, an interval linear fractional bi-level programming (ILFBP) model was developed for managing irrigation-water resources under uncertainty. The ILFBP can fully address system fairness, uncertainties, and the leader–follower relationship of decision makers in the optimization process, which can better reflect the complexity of real decision-making process and help formulate reasonable water policies. An interactive fuzzy coordination algorithm based on satisfaction degree was introduced to solve the ILFBP model. In order to evaluate the applicability of optimization schemes, the interval analytic hierarchy process (IAHP) and the interval technique for order preference by similarity to an ideal solution (TOPSIS) method were integrated as IAHP-TOPSIS. To verify its validity, the developed optimization-evaluation framework was applied to an irrigation water management case study in the middle reaches of the Shiyang River Basin, located in the northwest China. The ILFBP model results show that the total water allocation is [6.73, 7.37] × 108 m3, saving nearly 0.9 × 108 m3 more than the current situation. The benefit per unit of water is [2.38, 2.95] yuan/m3, nearly 0.4 yuan/m3 more than the status quo, and the Gini coefficient is within a reasonable range of [0.35, 0.38]. The ILFBP model can well balance economic benefits and system fairness. Through the evaluation bases on IAHP-TOPSIS, the results of ILFBP show better water allocation effects and applicability than the other two models in this study area. Furthermore, due to various characteristics such as geographical location, population and area, there are three irrigation districts, Xiying, Donghe, and Qinghe, showing higher importance than others when considering regional water allocation. These findings can provide useful information for limited water resource managers and help decision makers determine effective alternatives of water resource planning under uncertainty.

Solving methods for interval linear programming problem: a review and an improved method

Interval linear programming is used for tackling interval uncertainties in real-world systems. An arbitrary point is a feasible point to the interval linear programming model if it lies in the largest feasible region of the interval linear programming model, and it is optimal if it is an optimal solution to a characteristic model. The optimal solution set to the interval linear programming is the union of all solutions that are optimal for a characteristic model. In this paper, we review some existing methods for solving interval linear programming problems. Using these methods the interval linear programming model is transformed into two sub-models. The optimal solutions of these sub-models form the solution space of these solving methods. A part of the solution space of some of these methods may be infeasible. To eliminate the infeasible part of the solution space of above methods, several methods have been proposed. The solution space of these modified methods may contain non-optimal solutions. Two improvement methods have been proposed to remove the non-optimal solutions of the solution space of above modified methods. Finally, we introduce an improved method and its sub-models. The solution space of our method is absolutely both feasible and optimal.

An interval multi-objective programming model for irrigation water allocation under uncertainty

An interval linear multi-objective programming (ILMP) model for irrigation water allocation was developed, considering conflicting objectives and uncertainties. Based on the generation of interval numbers through statistical simulation, the ILMP model was solved using a fuzzy programming method. The model balances contradictions among economic net benefit, crop yield and water-saving in irrigation systems incorporating uncertainties in both objective functions and constraints that are based on the conjunctive use of surface water and groundwater. The model was applied to Hulan River irrigation district, northeast China. Tradeoffs between various crops in different subareas under different frequencies were analyzed, and scenarios with different objectives were considered to evaluate the changing trend of irrigation water allocation.Results indicated that the ILMP model provided effective linkages between revenue/output promotion and water saving, and offers insights into tradeoffs for irrigation water management under uncertainty.

Solving the interval linear programming problem: A new algorithm for a general case

Based on the binding constraint indices of the optimal solution to the linear programming (LP) model, a feasible system of linear equations can be formed. Because an interval linear programming (ILP) model is the union of numerous LP models, an interval linear equations system (ILES) can be formed, which is the union of these conventional systems. Hence, a new algorithm is introduced in which an arbitrary characteristic model of the ILP model is chosen and solved. The set of indices of its binding constraints is then obtained. This set is used to form and solve an ILES using the enclosure method. If all the components of the interval solutions to this system are strictly non-negative, the optimal solution set (OSS) of the ILP model is determined as the subscription of the zone created by reversing the signs of the binding constraints of the worst model and the binding constraints of the best model. The solutions to several problems obtained by the new algorithm and a Monte Carlo simulation are compared. The proposed algorithm is applicable to large-scale problems. To this end, an ILP model with 270 constraints and 270 variables is solved.

An improved risk-explicit interval linear programming model for pollution load allocation for watershed management.

Although the risk-explicit interval linear programming (REILP) model has solved the problem of having interval solutions, it has an equity problem, which can lead to unbalanced allocation between different decision variables. Therefore, an improved REILP model is proposed. This model adds an equity objective function and three constraint conditions to overcome this equity problem. In this case, pollution reduction is in proportion to pollutant load, which supports balanced development between different regional economies. The model is used to solve the problem of pollution load allocation in a small transboundary watershed. Compared with the REILP original model result, our model achieves equity between the upstream and downstream pollutant loads; it also overcomes the problem of greatest pollution reduction, where sources are nearest to the control section. The model provides a better solution to the problem of pollution load allocation than previous versions.

An efficient irrigation water allocation model under uncertainty

An interval linear fractional irrigation water allocation (ILFIWA) model is developed in response to the complexity of errors in estimating crop yields, fluctuating hydrological elements as well as varying economic profits in an irrigation system. The model is capable of quantitatively solving multi-objective problems, i.e. to obtain the maximum system net benefit and the minimum irrigation water use. Particularly, it can handle multi-objective functions expressed as ratios, such as irrigation water productivity. Moreover, it can reflect the uncertainties of the variables/parameters and functions involved. The potential of the developed model is shown by applying to a case study in northwest China. Results of the model can help make irrigation water allocation decisions for different time periods under varying flow levels. Comparison between ILFIWA model and ordinary interval linear programming model shows that the developed model is conducive to improving irrigation water productivity and saving irrigation water, and helping decision makers formulate desired irrigation water resources management policies under uncertainty.

Uncertain and multi-objective programming models for crop planting structure optimization

Crop planting structure optimization is a significant way to increase agricultural economic benefits and improve agricultural water management. The complexities of fluctuating stream conditions, varying economic profits, and uncertainties and errors in estimated modeling parameters, as well as the complexities among economic, social, natural resources and environmental aspects, have led to the necessity of developing optimization models for crop planting structure which consider uncertainty and multi-objectives elements. In this study, three single-objective programming models under uncertainty for crop planting structure optimization were developed, including an interval linear programming model, an inexact fuzzy chance-constrained programming (IFCCP) model and an inexact fuzzy linear programming (IFLP) model. Each of the three models takes grayness into account. Moreover, the IFCCP model considers fuzzy uncertainty of parameters/variables and stochastic characteristics of constraints, while the IFLP model takes into account the fuzzy uncertainty of both constraints and objective functions. To satisfy the sustainable development of crop planting structure planning, a fuzzy-optimizationtheory-based fuzzy linear multi-objective programming model was developed, which is capable of reflecting both uncertainties and multi-objective. In addition, a multiobjective fractional programming model for crop structure optimization was also developed to quantitatively express the multi-objective in one optimization model with the numerator representing maximum economic benefits and the denominator representing minimum crop planting area allocation. These models better reflect actual situations, considering the uncertainties and multi-objectives of crop planting structure optimization systems. The five models developed were then applied to a real case study in Minqin County, north-west China. The advantages, the applicable conditions and the solution methods of each model are expounded. Detailed analysis of results of each model and their comparisons demonstrate the feasibility and applicability of the models developed, therefore decision makers can choose the appropriate model when making decisions.

Convex contractive interval linear programming for resources and environmental systems management

It is likely that the most reliable estimation of system uncertainty in resources and environmental systems management (RESM) is a value range with an unknown distribution. Stochastic programming would be challenged by distortion of the original uncertain information through fabricating an inexistent probabilistic distribution function. Instead, interval linear programming (ILP), i.e. a synthesis of interval-set coefficients and the conventional linear programming, has been employed to identify the desired schemes for a number of RESM problems under interval uncertainty. However, its effectiveness is disabled by constraint violation which may lead to severe penalties on socio-economic or eco-environmental development. To mitigate such a challenge, a convex contractive interval linear programming (CCILP) approach is proposed in this study. It mainly consists of six modules: parameterizing an RESM problem as an ILP model, initializing a hyperrectangle decision space by two linear programming sub-models, revealing causes of constraint violation given a criterion, inferring feasibilities of potential solutions, finalizing a feasible hyperrectangle decision space by another linear programming sub-model, and supporting RESM of various complexities through alternative variants. A simple ILP model for RESM is introduced to demonstrate the procedures of CCILP and verify its advantages over existing ILP methods. The result indicates that CCILP is capable of robustly incorporating interval uncertainties into the optimization process, avoiding heavy computation burdens on complicated sub-models, eliminating occurrence of constraint violation, enabling provision of a hyperrectangle decision space, adapting to diverse system requirements, and increasing reliability of decision support for interval linear RESM problems.