Interval Linear Programming Research Articles

Uncertain multi‐objective programming approach for planning supplementary irrigation areas in rainfed agricultural regions

AbstractRainfed agriculture is crucial for ensuring global food and water security, and supplementary irrigation is an effective means of improving the economic benefits in rainfed agricultural regions. This study proposes a novel uncertain optimization approach to optimize supplementary irrigation areas in rainfed agricultural regions. The approach incorporates multi‐objective linear programming, interval linear programming, fuzzy goal programming and stochastic expected value programming. In the proposed interval multi‐fuzzy goal stochastic expected‐value programming, uncertainties are expressed in the form of discrete intervals, probability distributions and fuzzy goals. This approach, which considers the randomness of precipitation during the optimization process and allocates limited irrigation water resources to different subareas and crops, was applied to a case study of crop irrigation area planning in Guyuan City, Ningxia Hui Autonomous Region, northwestern China. The maximum economic benefits and the minimum sum of the Gini coefficients among the different subareas and crops were regarded as the planning objectives, and a series of optimal irrigation areas with different crops and subareas under different water levels were obtained. The optimization results revealed that vegetables and fruits consumed large amounts of irrigation water to increase their economic benefits. In addition, allocating a large amount of irrigation water to wheat is essential to decrease the Gini coefficient and meet food security constraints, particularly at extremely low water levels. Compared with current management, the optimized irrigation area decreased by 30%, the crop water deficit index increased by 9%, the economic benefits increased by 3% and the total Gini coefficient of crops decreased by 17%, indicating that the optimization approach could fairly and reasonably allocate irrigation water resources. Our research provides a mathematical approach for decision‐makers to plan supplementary irrigation areas in rainfed agricultural regions.

Some new results on rough interval linear programming problems and their application to scheduling and fixed-charge transportation problems

This paper focuses on linear programming problems in a rough interval environment. By introducing four linear programming problems, an attempt is being made to propose some results on optimal value of a linear programming problem with rough interval parameters. To obtain optimal solutions of a linear programming problem with rough interval data, constraints of the four proposed linear problems are applied. In this regard, firstly, the largest and the smallest feasible spaces for a linear constraint set with rough interval coefficients and parameters are introduced. Then, a rough interval for optimal value of such problems is obtained. Further, an upper approximation interval and a lower approximation interval as the optimal solutions of linear programming problems with rough interval parameters are achieved. Moreover, two solution concepts, surely and possibly solutions, are defined. Some numerical examples demonstrate the validity of the results. In particular, a scheduling problem and a fixed-charge transportation problem (FCTP) under rough interval uncertainty are investigated.

On the new solution to interval linear fractional programming problems

On the new solution to interval linear fractional programming problems

A NOVEL METHOD FOR SOLVING FULLY FUZZY SOLID TRANSPORTATIONS PROBLEMS

In this paper, we proposes a new method for solving solid transportation problem under uncertainty environments. The fully fuzzy solid transportation problem has been formulated. To reduce the model into crisp equivalent, we have used existing method for approximation of fuzzy numbers by interval numbers and its arithmatics. A simplex method and existing method for solving Interval Linear Programming problems are used for solving solid transportation problem with fuzzy parameters and decision variables. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy and reasonable.

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.

A study on the solution of interval linear fractional programming problem

Interval linear fractional programming problem (ILFPP) approaches uncertain-ties in real-world systems such as business, manufacturing, finance, and eco-nomics. In this study, we propose solving the interval linear fractional pro-gramming (ILFP) problem using interval arithmetic. Further, to construct the problem, a suitable variable transformation is used to form an equivalent ILP problem, and a new algorithm is depicted to obtain the optimal solution with-out converting the problem into its conventional form. This paper compares the range, solutions, and approaches of ILFP with fuzzy linear fractional pro-gramming (FLFP) in solving real-world optimization problems. The illustrated numerical examples show a better range of interval solutions on practical appli-cations of ILFPs and uncertain parameters.

Optimal design of low impact development at a community scale considering urban non-point source pollution management under uncertainty

Urban non-point source (NPS) pollution management is receiving increased attention because of the environmental and ecological consequences of urbanization. We developed a Storm Water Management Model (SWMM)-based interval double-sided chance-constrained programming (SWMM–IDCCP) model for the optimal design of low impact development (LID) during urban NPS pollution management at a community scale. Pollution control efficiency and economic costs related to LID practices were considered in the SWMM–IDCCP model, simultaneously addressing uncertainties in pollutant output and decision-making processes. Specifically, by combining SWMM, Monte Carlo simulation, interval linear programming, and double-sided chance-constrained programming, the SWMM–IDCCP model can evaluate export loads of urban NPS pollutants and pollution control efficiencies of LID practices and analyze the corresponding uncertainties. Concurrently, the proposed model can provide reliable and optimal implementation schemes for LID practices under different scenarios, dealing with uncertainties such as discrete intervals and double-sided random parameters. The proposed model was applied to the Dongguan University of Technology in Dongguan City, South China. The random distribution characteristics of pollution load were identified. The ranges of pollution load for total suspended solids, chemical oxygen demand, total nitrogen, and total phosphorus were 76,430–165,698, 44,366–74,155, 2118–3,830, and 97–177 kg in 2021, respectively. Probability distribution characteristics of pollution control efficiencies of three LID practices (i.e., bio-retention cell, green roof, and permeable pavement) for the four urban NPS pollutants were obtained. Using the SWMM-IDCCP model, LID configuration schemes were obtained under diverse pollution control targets and confidence levels, taking into account variations in the decision preference of different administrators. This research indicates that the SWMM-IDCCP model can be used to manage urban NPS pollution at a community scale under uncertainty.

Co-optimization of building energy systems with renewable generations combining active and passive energy-saving

Establishing clean and efficient building energy systems (BES) is an efficient path to promote the low-carbon energy transition to achieve the goal of carbon peak and carbon neutrality. To fully tap the energy saving potential of buildings, a co-optimization method combined active and passive energy-saving technologies is proposed for BES. The structures and characteristics of active and passive energy-saving means are modeled and analyzed. On this basis, a double-layer co-optimization model is built to optimize the planning and operation of BES separately. Furthermore, a carbon tax is introduced to establish the connection between active and passive means, and actual uncertainties of source-load are considered through the probabilistic scenario generation and interval linear programming approach. Case studies on the BES in Xiong’an New Area, China show that the proposed co-optimization method saves about 2.2%–3.4% costs than considering only the active energy-saving. Analysis of forecast errors for different scenarios reveals planning options and detailed costs under different levels of uncertainty.

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.

Three different optimization techniques for solving the fully rough interval multi-level linear programming problem

Due to the importance of the multi-level fully rough interval linear programming (MLFRILP) problem to address a wide range of management and optimization challenges in practical applications, such as policymaking, supply chain management, energy management, and so on, few researchers have specifically discussed this point. This paper presents an easy and systematic roadmap of studies of the currently available literature on rough multi-level programming problems and improvements related to group procedures in seven basic categories for future researchers and also introduces the concept of multi-level fully rough interval optimization. We start remodeling the problem into its sixteen crisp linear programming LP problems using the interval method and slice sum method. All crisp LPs can be reduced to four crisp LPs. In addition, three different optimization techniques were used to solve the complex multi-level linear programming issues. A numerical example is also provided to further clarify each strategy. Finally, we have a comparison of the methods used for solving the MLFRILP problem.

Parametric approach for multi-objective enhanced interval linear fractional programming problem

The design (decision) variables in the presented article of a multi-objective interval fractional optimization problem based on a linear function are assumed to take the form of a closed interval using the concept of the parametric form of an interval. The original problem is initially changed into equivalent multi-objective interval linear programming with the design variables as closed intervals. Further, it is made free from interval uncertainty by changing into a classical single-objective problem using the weighted-sum method. The solutions of the model are theoretically justified by its existence. Finally, a numerical example and a case study on the agricultural planting structure optimization problem with hypothetical data are presented to support the recommended technique for the model.

Development of a multi-regional factorial optimization model for supporting electric power system's low-carbon transition – A case study of Canada

In this study, a multi-regional factorial optimization model (MRFO) has been developed for supporting nation-wide transitions to low-carbon electric power systems (EPS) under the commitment to reduce greenhouse gas emissions. Through integrating non-deterministic optimization methods (interval linear, chance-constrained, and mixed-integer linear programming) with factorial analysis, MRFO can address multiple uncertainties stated as intervals and probability distribution in system parameters and objectives; it can also unveil the effects of multiple uncertain parameters and their interactions on system performance. A Canadian case study is provided to demonstrate the applicability of the proposed approach. Optimal schemes of electricity generation, capacity expansion, and inter-regional trades at different risk levels are examined with the objective of minimizing the total system costs. Results indicate that renewable energy (i.e., wind and solar) would play an important role in facilitating the transitions to low-carbon EPS, which would contribute to approximately 13% of the total national electricity generation by 2050. In addition, increasing the collaboration among regional EPS would have positive effects on the national penetration of low-carbon power generation in the light of the diversities in regional generation mixes. The findings can support the national efforts in formulating desired long-term power generation expansion plans and emission reduction policies.

Joint Optimization of Planning and Operation for Integrated Electrothermal and Hydrogen Coupling System

In view of the rough modeling of hydrogen energy system, insufficient analysis of hydrogen energy utilization mode, and the lack of consideration of the multiple uncertain factors, this paper proposes a joint optimization method of planning and operation for integrated electrothermal and hydrogen coupling system (IEHS) based on multiverse optimizer algorithm (MVO) and interval linear programming method. Firstly, the frame structure of IEHS is constructed, and multiple utilization modes of hydrogen energy are analyzed. Secondly, considering the hydrogen physical characteristics, the operation model of high-pressure hydrogen storage device based on Van Deemter equation is established, and the linear model of compressor power consumption is built considering dynamic compression ratio. On this basis, the bilevel optimization model of planning and operation for IEHS is established considering multiuncertainty and multiple hydrogen energy utilization modes. The upper model realizes the capacity configuration of equipment with the goal of minimizing the total construction and operation cost within the service life. The lower model optimizes the output of each equipment with the goal of minimizing the operation cost. The example results show that MVO and interval linear programming method for solving the bilevel optimization model have the better convergence speed and convergence ability. Further, the multiple utilization modes of hydrogen energy can effectively reduce the construction and operation cost of IEHS, and accurately depicting the physical characteristics of the hydrogen storage system can increase the coordination and optimization ability of IEHS. Considering multiple uncertain factors, the optimal configuration scheme and operation cost of IEHS are more stable.

Day-ahead interval scheduling strategy of power systems based on improved adaptive diffusion kernel density estimation

The increasing penetration of renewable energy causes high uncertainty, which further complicates the optimal scheduling operation of power systems. The uncertainty of renewable energy output is represented by empirical prediction intervals in the traditional interval optimization scheduling model, but the interval range is not precise enough. To address this problem, a novel improved interval optimization method is proposed in this paper. First, the improved adaptive diffusion kernel density estimation (IADKDE) is used to obtain more accurate intervals for the renewable energy output. Furthermore, a data-driven adaptive optimal bandwidth selection is adopted to select the optimal bandwidth instead of normal reference rules in IADKDE. In addition, the day-ahead scheduling optimization model is developed by IADKDE considering the driving requirements of electric vehicles (EVs) owners. The proposed model is described in details and solved by interval linear programming method. Finally, the effectiveness and accuracy of the proposed method are validated, and the comparative analysis with interval optimization by empirical prediction intervals, extreme learning machine(ELM) and stochastic optimization is given. It is demonstrated that the proposed method can obtain more accurate interval ranges of uncertain variables and has strong applicability.

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.

A Novel Interval Programming Method and Its Application in Power System Optimization Considering Uncertainties in Load Demands and Renewable Power Generation

This paper expresses the output power of renewable generators and load demand as interval data and develops the interval economic dispatch (IED), as well as interval reactive power optimization (IRPO) models. The two models are generalized into a specific type of linear interval programming (LIP) and nonlinear interval programming (NLIP), respectively. A security limits method (SLM) is proposed to solve LIP and NLIP problems. As for the LIP, the maximum radii of the interval variables are first calculated by the optimizing-scenarios method (OSM) for defining security limits, and the LIP is transformed into deterministic linear programming (LP), for which its constraints are the security limits, which can be solved by the simplex method. As for the NLIP, Monte Carlo simulations were used to obtain the maximum radii of the interval variables, and the average interval ratio of the interval variables is defined to compute the security limits for transforming the NLIP to deterministic nonlinear programming (NLP), which can be solved by using the interior point method. Finally, the IED and IRPO are used to verify the effectiveness and engineering of the proposed SLM.

Day-Ahead Optimal Scheduling of Integrated Energy System Based on Type-II Fuzzy Interval Chance-Constrained Programming

Renewable energy sources (RES) generation has huge environmental and social benefits, as a clean energy source with great potential. However, the difference in the uncertainty characteristics of RES and electric–thermal loads poses a significant challenge to the optimal schedule of an integrated energy system (IES). Therefore, for the different characteristics of the multiple uncertainties of IES, this paper proposes a type-II fuzzy interval chance-constrained programming (T2FICCP)-based optimization model to solve the above problem. In this model, type-II fuzzy sets are used to describe the uncertainty of RES in an IES, and interval numbers are used to describe the load uncertainty, thus constructing a T2FICCP-based IES day-ahead economic scheduling model. The model was resolved with a hybrid algorithm based on interval linear programming and T2FICCP. The simulations are conducted for a total of 20 randomly selected days to obtain the advance operation plan of each unit and the operation cost of the system. The research results show that the T2FICCP optimization model has less dependence on RES output power and load forecasting error, so can effectively improve the economy of IES, while ensuring the safe and stable operation of the system.

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.

A factorial interval chance-constrained diet model for dairy farms under climate change: A case study for the Province of Saskatchewan, Canada

A large amount of greenhouse gases is emitted from the dairy industry in Canada. Actions must be taken to achieve Canada's environmental goals of decreasing greenhouse gas (GHG) emissions by 30% below 2005 levels by 2030. Dietary management plays an important role in reducing such emission in dairy cattle production for current and future climate change conditions. In this study, a factorial interval chance-constrained diet model has been developed for Saskatchewan's dairy cow farms to control emissions by incorporating factorial analysis, interval linear programming, and chance-constrained programming. Multiple uncertainties (e.g., the price of the milk, the price of the feed, and the disease of animals) have been considered in this model. The chance-constrained programming method, which is effective to tackle uncertainties, has been first integrated in the animal nutrition optimization model. This model can provide the schemes of feed selection for diet formulation under trade-offs between farm profit and environmental impact. In addition, it could facilitate decision-makers to identify interactive relationships among various nutrient levels and manage methane emissions and nitrogen excretion from dairy cattle in dairy farms under various risks. Results show that canola meal, rolled barley grain, soybean meal, and whey would be contained in all optimized diets; the forage proportion would have the most significant effect on methane emissions; factors of forage and crude protein would have negative effects on methane emissions, while the neutral detergent fiber (NDF) would have a positive effect on methane emissions. Results indicate that reducing the number of cows is still an important strategy to mitigate carbon emissions under the condition of meeting the growing milk demand. Overall, this model could manipulate the formula of different types of dairy cattle to maximise the benefits of farm and alleviate its negative environmental impact. • A FICC diet model has been developed for the Province of Saskatchewan. • Uncertainties and complexities existed in the dairy farm diet have been considered. • Various diet schemes of dairy farms have been obtained under different risk levels. • Increasing the milk production in dairy cattle could reduce greenhouse gas emissions .

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.