Interval-parameter Two-stage Stochastic Programming Research Articles

A Two-Stage Stochastic Fuzzy Mixed-Integer Linear Programming Approach for Water Resource Allocation under Uncertainty in Ajabshir Qaleh Chay Dam

Due to the dry climate and unsuitable distribution of rainfall in Iran, sustainable agriculture depends on the proper use of water resources. In this study, the optimal allocation of water at Ajabshir Qaleh Chay Dam in agricultural sector is investigated using an interval parameter two-stage stochastic mixed-integer linear programming approach. Indeed, interval parameters two-stage stochastic programming (ITSP) with fuzzy variables is developed based on mixed-integer programming for a water resource allocation model to retrieve the water shortage of agricultural products and to achieve the optimal allocation of Ajabshir Qaleh Chay Dam water through its river canals between different products under uncertainty conditions. In this developed method, called extended ITSP (EITSP), a number of alternatives are used to compensate for the difference between the amount of promised water allocation targets and the actual allocated water in the optimal allocation of water. Then a new solving approach based on Huang Algorithm, fuzzy chance constrained programming and Zimmermann fuzzy programming will be presented to solve the problems. Furthermore, using a case study in this dam, the results are obtained for the developed approaches to clarify the described methods and to compare these results with each other. Finally, comparing the total system profits of the models shows that in the fuzzy model, the profit and system certainty increase simultaneously. Therefore, due to the lack of water resources in the agricultural sector and the uncertainty, the agricultural authorities of Ajabshir can decrease the unsustainability of water resources using the optimal model while increasing the cost-effectiveness of the farmers.

Optimal Model of Desalination Planning Under Uncertainties in a Water Supply System

A requirement for developing desalination efforts in coastal regions suffering water scarcity is proposed to address the increased water demand and limited traditional water supply. The determination of a plant capacity and water allocation scheme in a multiple-source water supply system, as the first problem in planning desalination under streamflow and water demand uncertainties, remains a challenge. To address this gap, an interval-parameter two-stage stochastic programming model is developed in this study. The first-stage problem is to determine a proper desalination plant capacity, and the second is the development of a water allocation scheme under the uncertainties of natural streamflows, water demands, benefits and economic losses. The objective function is to maximize the net benefit of the system, and the cost function of desalination, including capital and operational costs, implying environmental impact, is linearized within a range of plant capacities to solve the model. The proposed approach is applied to an urban area of Weihai in China to illustrate the validity of the model. The results suggest a capacity of 46 × 103 m3/d in 2030 and 55 × 103 m3/d in 2040. Sensitivity analyses of the parameters indicate that a decrease in the unit price of electricity leads to an increase in the utilization level of desalinated seawater. A complementary relationship was observed between reclaimed water and desalinated seawater, in that a decrease in the use rate of reclaimed water from 0.38 to 0.18 led to a 15% increase in desalinated plant capacity.

Optimization of Ecological Water Supplement Scheme for Improved Suitable Habitat Area for Rare Migratory Birds in Nature Reserves Using Interval-Parameter Fuzzy Two-Stage Stochastic Programming Model.

The optimization of ecological water supplement scheme in Momoge National Nature Reserve (MNNR), using an interval-parameter two-stage stochastic programming model (IPTSP), still experiences problems with fuzzy uncertainties and the wide scope of the obtained optimization schemes. These two limitations pose a high risk of system failure causing high decision risk for decision-makers and render it difficult to further undertake optimization schemes respectively. Therefore, an interval-parameter fuzzy two-stage stochastic programming (IPFTSP) model derived from an IPTSP model was constructed to address the random variable, the interval uncertainties and the fuzzy uncertainties in the water management system in the present study, to reduce decision risk and narrow down the scope of the optimization schemes. The constructed IPFTSP model was subsequently applied to the optimization of the ecological water supplement scheme of MNNR under different scenarios, to maximize the recovered habitat area and the carrying capacity for rare migratory water birds. As per the results of the IPFTSP model, the recovered habitat areas for rare migratory birds under low, medium and high flood flow scenarios were (14.06, 17.88) × 103, (14.92, 18.96) × 103 and (15.83, 19.43) × 103 ha, respectively, and the target value was (14.60, 18.47) × 103 ha with a fuzzy membership of (0.01, 0.83). Fuzzy membership reflects the possibility level that the model solutions satisfy the target value and the corresponding decision risk. We further observed that the habitat area recovered by the optimization schemes of the IPFTSP model was significantly increased compared to the recommended scheme, and the increases observed were (5.22%, 33.78%), (11.62%, 41.88%) and (18.44%, 45.39%). In addition, the interval widths of the recovered habitat areas in the IPFTSP model were reduced by 17.15%, 17.98% and 23.86%, in comparison to those from the IPTSP model. It was revealed that the IPFTSP model, besides generating the optimal decision schemes under different scenarios for decision-makers to select and providing decision space to adjust the decision schemes, also shortened the decision range, thereby reducing the decision risk and the difficulty of undertaking decision schemes. In addition, the fuzzy membership obtained from the IPFTSP model, reflecting the relationship among the possibility level, the target value, and the decision risk, assists the decision-makers in planning the ecological water supplement scheme with a preference for target value and decision risk.

Simulation of water resource allocation for sustainable urban development: An integrated optimization approach

The water conflicts between society and ecosystems pose great challenges to sustainable development in water-stressed areas. This study proposes a system dynamics model consisting of social, economic, water demand and water supply sectors to investigate the complex interactions within the water system in Huainan City. With the simulation results of water resources, an optimization model for water resource allocation based on interval-parameter two-stage stochastic programming was developed. Taking into account the value of ecosystem services, the optimal water resource allocation was simulated under various scenarios for the development of supplementary water sources. The simulation results indicate that relying only on the local water resources and the inflow of the Huaihe River would lead to the degradation of ecosystem functions, resulting in excessive system costs. The development of reused wastewater and harvested rainwater could be an effective way to guarantee both water supply security and ecosystem health, as well as reduce the cost of water allocation system. The integrated optimization model may be applicable to water resources management in other regions and could help stakeholders and decision-makers obtain comprehensive insights to establish and implement advantageous strategies for sustainable regional development.

Interval-Parameter Two-Stage Stochastic Programming (IPTSP) Model of Ecological Water Replenishment Scheme in the National Nature Reserve for Improved Suitable Habitat for Rare and Endangered Migrant Birds

In this work, an interval-parameter two-stage stochastic programming (IPTSP) model of water resources allocation was established for maximizing the restored habitat area of large, rare, and endangered water birds by adjusting the recommended scheme of water replenishment under different scenarios and constraints. The established model can efficiently deal with the uncertainties, such as the interval parameters and random variables, in the management system of water resources simultaneously. A case study was conducted in the Momoge National Nature Reserve (MNNR) in northeast China to maximize the restored habitat area of large, rare and endangered water birds based on limited water resources. According to the previous studies, a water area with a depth of 0–40 cm is a suitable habitat area in the MNNR for the Siberian crane, oriental stork, and red-crowned crane. The results of the present work show that the habitat area restored by water replenishment schemes under low, medium, and high flood flow scenarios after optimization increased in comparison to 13.36 × 103 ha of the recommended scheme, with an increase of [0.62, 5.23], [1.49, 6.42], and [2.43, 7.17] × 103 ha, respectively (the two numbers within each bracket represent the lower and upper bounds of the restored habitat areas). As a result, the carrying capacity of suitable habitat areas increased by [0.82, 6.88], [1.96, 8.45], and [3.21, 9.43] × 103 birds, correspondingly. The restored wetland area of the project recommendation scheme was 34.23 × 103 ha, and that of the optimal water replenishment schemes was [29.35, 41.01], [31.02, 44.13], and [33.88, 46.04] × 103 ha, respectively under the three flood flow scenarios. The results reveal that the model constructed in this work realizes the optimization and adjustment of the initial scheme to an increased restored wetland and habitat area with an increase in the flow level. Here, the upper bound of the interval value mentioned above is significantly higher than the lower bound value, which indicates that a feasible decision space was provided for decision makers to optimize and adjust the recommended scheme on the basis of the actual situation. The model-optimized schemes significantly improved the utilization of limited water resources. The results of this study can provide valuable theoretical support for the restoration and protection of rare and endangered water bird habitats and planning and management of water resources.

Basin Initial Water Rights Allocation under Multiple Uncertainties: a Trade-off Analysis

Economic losses and inequities caused by uncertainties in the availability of water intensify the competition between water sectors, making the allocation of water rights of vital importance for minimizing water conflicts. In this study, an Interval-parameter Two-stage Stochastic Programming (ITSP) model for water rights allocation is developed that contains an industrial allocation preference coefficient and involves the risk control of Conditional Value-at-Risk theory and Gini coefficient constraints (ITSP-CG). Using China’s Taihu Basin as a case study, it is shown that optimized water rights allocation schemes can reduce the risk of inequitable localized water deficits, a narrower confidence interval results in higher economic loss, and, when the confidence level is fixed, tighter control of water availability results in water efficient sectors having an increasing preference for allocation schemes. It is also shown that Basin Authorities need to trade-off the equitable allocation of water rights and economic returns over a particular planning period.

A hybrid system dynamics and optimization approach for supporting sustainable water resources planning in Zhengzhou City, China

Problems with water resources restrict the sustainable development of a city with water shortages. Based on system dynamics (SD) theory, a model of sustainable utilization of water resources using the STELLA software has been established. This model consists of four subsystems: population system, economic system, water supply system and water demand system. The boundaries of the four subsystems are vague, but they are closely related and interdependent. The model is applied to Zhengzhou City, China, which has a serious water shortage. The difference between the water supply and demand is very prominent in Zhengzhou City. The model was verified with data from 2009 to 2013. The results show that water demand of Zhengzhou City will reach 2.57 billion m3 in 2020. A water resources optimization model is developed based on interval-parameter two-stage stochastic programming. The objective of the model is to allocate water resources to each water sector and make the lowest cost under the minimum water demand. Using the simulation results, decision makers can easily weigh the costs of the system, the water allocation objectives, and the system risk. The hybrid system dynamics method and optimization model is a rational try to support water resources management in many cities, particularly for cities with potential water shortage and it is solidly supported with previous studies and collected data.

Identification of low-carbon strategies for post-earthquake development in the city of Guangyuan based on an inexact two stage stochastic programming approach

In this research, an interval-parameter two-stage stochastic programming approach was developed for supporting the identification of low-carbon strategies and the post-earthquake reconstruction in the city of Gugangyuan, Sichuan province of China. The approach can effectively deal with uncertainties that can be expressed as both probability density functions and interval numbers without any distribution information. Also, the approach can provide useful decision alternatives for supporting long-term corrections for any plans associated with carbon dioxide emission limits within a city context. The developed approach was applied in the city of Guangyuan. Three scenarios were established including the baseline (i.e., regional planning) as scenario 1, zero carbon emission as scenario 2, and carbon sink as scenario 3. Also, based on practical information and the collected data, three projection values (i.e., low, medium and high levels) for the carbon emission of the entire city were generated and analyzed under varying probability levels (i.e., 20, 60, and 20%). The results indicated that in order to achieve low-carbon post-earthquake development in the city, the priority areas and industries for carbon emission reduction would be Chaotian (the secondary industry) > Wangcang (the secondary industry) > Lizhou (the secondary industry) = Qingchuan (the tertiary industry) > Cangxi (the secondary industry) = Chaotian (the tertiary industry).

An Extended Two-stage Stochastic Programming Approach for Water Resources Management under Uncertainty

In this study, extended two-stage stochastic programming with fuzzy variables is developed for water resources management under uncertainty. First, the problem is formulated and solved by an extended interval-parameter two-stage stochastic programming (ITSP) approach for retrieving water shortages. To this end, some alternatives are considered to retrieve the difference between the quantities of promised water-allocation targets and the actual allocated water. An extended ITSP is then developed for the problem under the fuzzy uncertainty by using fuzzy variables and solved using fuzzy chance-constrained programming based on the idea of possibility theory. Furthermore, an illustrative example is also given to clarify the methods discussed in this paper.

An inventory-theory-based interval-parameter two-stage stochastic programming model for water resources management

In this study, an inventory-theory-based interval-parameter two-stage stochastic programming (IB-ITSP) model is proposed through integrating inventory theory into an interval-parameter two-stage stochastic optimization framework. This method can not only address system uncertainties with complex presentation but also reflect transferring batch (the transferring quantity at once) and period (the corresponding cycle time) in decision making problems. A case of water allocation problems in water resources management planning is studied to demonstrate the applicability of this method. Under different flow levels, different transferring measures are generated by this method when the promised water cannot be met. Moreover, interval solutions associated with different transferring costs also have been provided. They can be used for generating decision alternatives and thus help water resources managers to identify desired policies. Compared with the ITSP method, the IB-ITSP model can provide a positive measure for solving water shortage problems and afford useful information for decision makers under uncertainty.