Optimization of the industrial structure facing sustainable development in resource-based city subjected to water resources under uncertainty

An inexact stochastic fuzzy programming (ISFP) approach has been developed for the optimization of the industrial structure in resource-based city subjected to water resources under uncertainty in present study. The ISFP method incorporates the techniques of inexact stochastic programming and inexact fuzzy chance-constrained programming, where the uncertainties are expressed as interval, fuzzy sets, and probability distribution, respectively. Moreover, it can also examine the risk of violating fuzzy tolerance constraints. The developed method is subsequently employed in a realistic case for industrial development in the Jinchang city, Gansu province, China. The result can help to analyze whether the water resources carrying capacity of Jinchang can meet the need of local economic development plan under uncertainty and help decision maker to optimize the industry structure under water resource constraints to meet the maximum economic efficiency.     

Two-stage fuzzy chance-constrained programming: application to water resources management under dual uncertainties

In this study, a two-stage fuzzy chance-constrained programming (TFCCP) approach is developed for water resources management under dual uncertainties. The concept of distribution with fuzzy probability (DFP) is presented as an extended form for expressing uncertainties. It is expressed as dual uncertainties with both stochastic and fuzzy characteristics. As an improvement upon the conventional inexact linear programming for handling uncertainties in the objective function and constraints, TFCCP has advantages in uncertainty reflection and policy analysis, especially when the input parameters are provided as fuzzy sets, probability distributions and DFPs. TFCCP integrates the two-stage stochastic programming (TSP) and fuzzy chance-constrained programming within a general optimization framework. TFCCP incorporates the pre-regulated water resources management policies directly into its optimization process to analyze various policy scenarios; each scenario has different economic penalty when the promised amounts are not delivered. TFCCP is applied to a water resources management system with three users. Solutions from TFCCP provide desired water allocation patterns, which maximize both the system’s benefits and feasibility. The results indicate that reasonable solutions were generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of stream flows, α-cut levels and fuzzy dominance indices.     

Inexact stochastic dynamic programming method and application to water resources management in Shandong China under uncertainty

In this study, an interval parameter multistage joint-probability programming (IMJP) approach has been developed to deal with water resources allocation under uncertainty. The IMJP can be used not only to deal with uncertainties in terms of joint-probability and intervals, but also to examine the risk of violating joint probabilistic constraints in the context of multistage. The proposed model can handle the economic expenditure caused by regional water shortage and flood control. The model can also reflect the related dynamic changes in the multi-stage cases and the system safety under uncertainty. The developed method is applied to a case study of water resources allocation in Shandong, China, under multistage, multi-reservoir and multi-industry. The violating reservoir constraints are addressed in terms of joint-probability. Different risk levels of constraint lead to different planning. The obtained results can help water resources managers to identify desired system designs under various economic, environment and system reliability scenarios.     

Water resources management under uncertainty: factorial multi-stage stochastic program with chance constraints

Due to rapid growth of population and development of economy, water resources allocation problems have aroused wide concern. Therefore, optimization of water resources systems is complex and uncertain, which is a severe challenge faced by water managers. In this paper, a factorial multi-stage stochastic programming with chance constraints approach is developed to deal with the issues of water-resources allocation under uncertainty and risk as well as their interactions. It can deal with uncertainties described as both interval numbers and probability distributions, and can also support the risk assessment within a multistage context. The solutions associated with different risk levels of constraint violation can be obtained, which can help characterize the relationship between the economic objective and the system risk. The inherent interactions between factors at different levels and their effects on total net benefits can be revealed through the analysis of multi-parameter interactions.     

Development of an inexact two-stage stochastic model with downside risk control for water quality management and decision analysis under uncertainty

Water quality management along rivers involves making water-allocation plans, establishing water quality goals, and controlling pollutant discharges, which is complicated itself but further challenged by existence of uncertainties. In this study, an inexact two-stage stochastic downside risk-aversion programming (ITSDP) model is developed for supporting regional water resources allocation and water quality management problems under uncertainties. The ITSDP method is a hybrid of interval-parameter programming, two-stage stochastic programming, and downside risk measure to tackle uncertainties described in terms of interval values and probability distributions. A water quality simulation model was provided for reflecting the relationship between the water resources allocation, wastewater discharge, and environmental responses. The proposed approach was applied to a hypothetical case for a shared stream water quality management with one municipal, three industrial and two agricultural sectors. A number of scenarios corresponding to different river inflows and risk levels were examined. The results demonstrated that the model could effectively communicate the interval-format and random uncertainties, and risk-aversion into optimization process, and generate a trade-off between the system economy and stability. They could be helpful for seeking cost-effective management strategies under uncertainties, and gaining an in-depth insight into the water quality management system characteristics, and make cost-effective decisions.     

Risk aversion based interval stochastic programming approach for agricultural water management under uncertainty

In this study, a risk aversion based interval stochastic programming (RAIS) method is proposed through integrating interval multistage stochastic programming and conditional value at risk (CVaR) measure for tackling uncertainties expressed as probability distributions and intervals within a multistage context. The RAIS method can reflect dynamic features of the system conditions through transactions at discrete points in time over the planning horizon. Using the CVaR measure, RAIS can effectively reflect system risk resulted from random parameters. When random events are occurred, the adjustable alternatives can be achieved by setting desired targets according to the CVaR, which could make the revised decisions to minimize the economic penalties. Then, the RAIS method is applied to planning agricultural water management in the Zhangweinan River Basin that is plagued by drought due to serious water scarcity. A set of decision alternatives with different combinations of risk levels employed to the objective function and constraints are generated for planning water resources allocation. The results can not only help decision makers examine potential interactions between risks under uncertainty, but also help generate desired policies for agricultural water management with a maximized payoff and a minimized loss.     

A dual two-stage stochastic model for flood management with inexact-integer analysis under multiple uncertainties

This study introduces a hybrid optimization approach for flood management under multiple uncertainties. An inexact two-stage integer programming (ITIP) model and its dual formation are developed by integrating the concepts of mixed-integer and interval-parameter programming techniques into a general framework of two-stage stochastic programming. The proposed approach provides a linkage to pre-defined management policies, deals with capacity-expansion planning issues, and reflects various uncertainties expressed as probability distributions and discrete intervals for a flood management system. Penalties are imposed when the policies are violated. The marginal costs are determined based on dual formulation of the ITIP model, and their effects on the optimal solutions are investigated. The developed model is applied to a case study of flood management. The solutions of binary variables represent the decisions of flood-diversion–capacity expansion within a multi-region, multi-flow-level, and multi-option context. The solutions of continuous variables are related to decisions of flood diversion toward different regions. The solutions of dual variables indicate the decisions of marginal costs associated with the resources of regions’ capacity, water availability, and allowable diversions. The results show that the proposed approach could obtain reliable solutions and adequately support decision making in flood management.     

Agricultural effluent control under uncertainty: An inexact double-sided fuzzy chance-constrained model

An inexact double-sided fuzzy chance-constrained programming (IDFCCP) method was developed in this study and applied to an agricultural effluent control management problem. IDFCCP was formulated through incorporating interval linear programming (ILP) into a double-sided fuzzy chance-constrained programming (DFCCP) framework, and could be used to deal with uncertainties expressed as not only possibility distributions associated with both left- and right-hand-side components of constraints but also discrete intervals in the objective function. The study results indicated that IDFCCP allowed violation of system constraints at specified confidence levels, where each confidence level consisted of two reliability scenarios. This could lead to model solutions with high system benefits under acceptable risk magnitudes. Furthermore, the introduction of ILP allowed uncertain information presented as discrete intervals to be communicated into the optimization process, such that a variety of decision alternatives can be generated by adjusting the decision-variable values within their intervals. The proposed model could help decision makers establish various production patterns with cost-effective water quality management schemes under complex uncertainties, and gain in-depth insights into the trade-offs between system economy and reliability.     

An inexact fuzzy-chance-constrained two-stage mixed-integer linear programming approach for flood diversion planning under multiple uncertainties

In this study, an inexact fuzzy-chance-constrained two-stage mixed-integer linear programming (IFCTIP) approach is developed for flood diversion planning under multiple uncertainties. A concept of the distribution with fuzzy boundary interval probability is defined to address multiple uncertainties expressed as integration of intervals, fuzzy sets and probability distributions. IFCTIP integrates the inexact programming, two-stage stochastic programming, integer programming and fuzzy-stochastic programming within a general optimization framework. IFCTIP incorporates the pre-regulated water-diversion policies directly into its optimization process to analyze various policy scenarios; each scenario has different economic penalty when the promised targets are violated. More importantly, it can facilitate dynamic programming for decisions of capacity-expansion planning under fuzzy-stochastic conditions. IFCTIP is applied to a flood management system. Solutions from IFCTIP provide desired flood diversion plans with a minimized system cost and a maximized safety level. The results indicate that reasonable solutions are generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of flood flows.     

Development of a fuzzy-queue-based stochastic quadratic program for water resources planning

In this study, a fuzzy-queue (FQ)-based inexact stochastic quadratic programming (SQP) method is developed through coupling FQ technique with inexact SQP. FQ-SQP improves upon the existing stochastic programming methods by considering the effects of queuing phenomenon during the water resources allocation process. FQ-SQP cannot only handle uncertainties expressed as interval values, random variables, and fuzzy sets, but also tackle nonlinearity in the objective function; more importantly, it can reflect the effects of FQ on water resources allocation and system benefit. The FQ-SQP model is applied to a case study of planning water resources management, where FM/FM/1 (fuzzy exponential interarrival time, fuzzy exponential service time, and one server) queue is incorporated within the SQP modeling framework. Based on α-cut analysis technique, interval solutions with fuzzy arrival and service rates have been generated, which result in different water resources allocation patterns as well as changed waiting water amounts and system benefits. Results indicate that consideration of queuing problem impacts on water resources allocation can provide more useful information for decision makers and gain in-depth insights into the effects of queuing problems for water resources allocation.     

Factorial two-stage stochastic programming for water resources management

This study presents a factorial two-stage stochastic programming (FTSP) approach for supporting water resource management under uncertainty. FTSP is developed through the integration of factorial analysis and two-stage stochastic programming (TSP) methods into a general modeling framework. It can handle uncertainties expressed as probability distributions and interval numbers. This approach has two advantages in comparison to conventional inexact TSP methods. Firstly, FTSP inherits merits of conventional inexact two-stage optimization approaches. Secondly, it can provide detailed effects of uncertain parameters and their interactions on the system performance. The developed FTSP method is applied to a hypothetical case study of water resources systems analysis. The results indicate that significant factors and their interactions can be identified. They can be further analyzed for generating water allocation decision alternatives in municipal, industrial and agricultural sectors. Reasonable water allocation schemes can thus be formulated based on the resulting information of detailed effects from various impact factors and their interactions. Consequently, maximized net system benefit can be achieved.     

Planning hydroelectric resources with recourse-based multistage interval-stochastic programming

Optimization models play an important role in long-term hydroelectric resources planning. The effectiveness of an optimization model, however, depends on its capability of dealing with uncertainties. This study presents a multistage interval-stochastic programming model for long-term hydropower planning, in which uncertainties are reflected as randomness and intervals. The model is developed based on interval programming technique and recourse-based multistage stochastic programming and using the expected value of long-term hydroelectric profit as the objective function. A solution method of the developed model is also presented, which is based on a decomposition method by partitioning the multistage interval-stochastic program into two-stage stochastic programming sub-problems in each scenario-tree node. A hypothetical case study is used to demonstrate the developed model and its solution method. Modeling results demonstrates the computationally effectiveness of the solution method and reveal the applicability of the developed model for long term planning of hydroelectric resources.     

Fuzzy chance-constrained linear fractional programming approach for optimal water allocation

A fuzzy chance-constrained linear fractional programming method was developed for agricultural water resources management under multiple uncertainties. This approach improved upon the previous programming methods, and could reflect the ratio objective function and multiple uncertainties expressed as probability distributions, fuzzy sets, and their combinations. The proposed approach is applied to an agricultural water resources management system where many crops are considered under different precipitation years. Through the scenarios analyses, the multiple alternatives are presented. The solutions show that it is applicable to practical problems to address the crop water allocation under the precipitation variation and sustainable development with ratio objective function of the benefit and the irrigation amount. It also provides bases for identifying desired agriculture water resources management plans with reasonable benefit and irrigation schedules under crops.     

Multistage scenario-based interval-stochastic programming for planning water resources allocation

In this study, a multistage scenario-based interval-stochastic programming (MSISP) method is developed for water-resources allocation under uncertainty. MSISP improves upon the existing multistage optimization methods with advantages in uncertainty reflection, dynamics facilitation, and risk analysis. It can directly handle uncertainties presented as both interval numbers and probability distributions, and can support the assessment of the reliability of satisfying (or the risk of violating) system constraints within a multistage context. It can also reflect the dynamics of system uncertainties and decision processes under a representative set of scenarios. The developed MSISP method is then applied to a case of water resources management planning within a multi-reservoir system associated with joint probabilities. A range of violation levels for capacity and environment constraints are analyzed under uncertainty. Solutions associated different risk levels of constraint violation have been obtained. They can be used for generating decision alternatives and thus help water managers to identify desired policies under various economic, environmental and system-reliability conditions. Besides, sensitivity analyses demonstrate that the violation of the environmental constraint has a significant effect on the system benefit.     

A multistage simulation-based optimization model for water resources management in Tarim River Basin, China

In this study, a multistage simulation-based optimization model is developed for supporting water resources management under uncertainty. The system couples a lumped rainfall-runoff model with an inexact multistage stochastic program, where its random parameter is provided by the statistical analysis of simulation outcomes. Moreover, penalties are exercised with recourse against any infeasibility, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic consequences when the promised water-allocation targets are violated. The developed model can also reflect dynamic features of the system conditions through transactions at discrete points in time over a multistage context. The developed model is applied to a real case of planning water resources management in Tarim River Basin, which is one of the most serious water-shortage regions of China. A variety of policies associated with different water-allocation targets are analyzed. The results are helpful for decision makers identifying optimal water-allocation plans for mitigating the conflict among ecological protection, economic development, and regional sustainability.     

Planning water resources management systems using a fuzzy-boundary interval-stochastic programming method

In this study, a fuzzy-boundary interval-stochastic programming (FBISP) method is developed for planning water resources management systems under uncertainty. The developed FBISP method can deal with uncertainties expressed as probability distributions and fuzzy-boundary intervals. With the aid of an interactive algorithm woven with a vertex analysis, solutions for FBISP model under associated α-cut levels can be generated by solving a set of deterministic submodels. The related probability and possibility information can also be reflected in the solutions for the objective function value and decision variables. The developed FBISP is also applied to water resources management and planning within a multi-reservoir system. Various policy scenarios that are associated with different levels of economic consequences when the pre-regulated water-allocation targets are violated are analyzed. The results obtained are useful for generating a range of decision alternatives under various system conditions, and thus helping decision makers to identify desired water resources management policies under uncertainty.     

Analyzing urban water supply through an acceptability-index-based interval approach

Incorporation of uncertainties within an urban water supply management system has been a challenging topic for many years. In this study, an acceptability-index-based two-step interval programming (AITIP) model was developed for supporting urban water supply analysis under uncertainty. AITIP improved upon the traditional two-step interval programming (TIP) through incorporating the acceptability level of constraints violation into the optimization framework. A four-layer urban water supply system, including water sources, treatment facilities, reservoirs, and consuming zones, was used to demonstrate the applicability of proposed method. The results indicated that an AITIP model was valuable to help understand the effects of uncertainties related to cost, constraints and decision maker’s judgment in the water supply network, and capable of assisting urban water managers gain an in-depth insight into the tradeoffs between system cost and constraints-violation risk. Compared with TIP, the solutions from AITIP were of lower degree of uncertainty, making it more reliable to identify effective water supply patterns by adjusting decision variable values within their solution intervals. The study is useful in helping urban water managers to identify cost-effective management schemes in light of uncertainties in hydrology, environment, and decisions. The proposed optimization approach is expected to be applicable for a wide variety of water resources management problems.     

Total consumption controlled water allocation management for multiple sources and users with inexact fuzzy chance-constrained programming: a case study of Tianjin,China

Environmental and ecological issues caused by water resources crisis have brought enormous challenges to the sustainable development of water-deficient area. Water resources allocation management balancing the relationship between the social-economic development and the ecological environment has become a hot topic in recent years. In this paper, an inexact fuzzy chance-constrained programming (IFCCP) approach is proposed for regional water resource allocation optimization with the aim of promoting the harmonious development of the social economic and the ecological environment, improving water utilization efficiency, and realizing water resources consumption control under uncertainties. The method is incorporated with interval parameter programming, fuzzy programming, and chance-constrained programming, for handling system uncertainties and balancing the optimal objectives with the risk of violating system constraints. Under this framework, an IFCCP model for water resources allocation management was successfully formulated and applied to a typical water-deficit area, Tianjin, China, for obtaining a better water resources plan among multiple users under resources and environmental limitation. Different total water consumption control policies are designed for assessing regional water allocation schemes. The results indicated that the gap of supply and demand will only be solved by foreign water, the transferred water from Luan River and Changjiang River would still be the main supplier in planning horizon. Moreover, the strict total water consumption control policy would guarantee the water requirement of ecological environment, lead to changes in the structure of water supply, actively guide on water conservation, and promote the large-scale utilization of desalted water and recycle water.     

Water quality management in a wetland system using an inexact left-hand-side chance-constrained fuzzy multi-objective approach

Water quality management is a significant item in the sustainable development of wetland system, since the environmental influences from the economic development are becoming more and more obvious. In this study, an inexact left-hand-side chance-constrained fuzzy multi-objective programming (ILCFMOP) approach was proposed and applied to water quality management in a wetland system to analyze the tradeoffs among multiple objectives of total net benefit, water quality, water resource utilization and water treatment cost. The ILCFMOP integrates interval programming, left-hand-side chance-constrained programming, and fuzzy multi-objective programming within an optimization framework. It can both handle multiple objectives and quantify multiple uncertainties, including fuzziness (aspiration level of objectives), randomness (pollutant release limitation), and interval parameters (e.g. water resources, and wastewater treatment costs). A representative water pollution control case study in a wetland system is employed for demonstration. The optimal schemes were analyzed under scenarios at different probabilities (p _i , denotes the admissible probability of violating the constraint i). The optimal solutions indicated that, most of the objectives would decrease with increasing probability levels from scenarios 1 to 3, since a higher constraint satisfaction probability would lead to stricter decision scopes. This study is the first application of the ILCFMOP model to water quality management in a wetland system, which indicates that it is applicable to other environmental problems under uncertainties.     

Risk assessment of agricultural irrigation water under interval functions

In recent years, water shortages and unreliable water supplies have been considered as major barriers to agricultural irrigation water management in China, which are threatening human health, impairing prospects for agriculture and jeopardizing survival of ecosystems. Therefore, effective and efficient risk assessment of agricultural irrigation water management is desired. In this study, an inexact full-infinite two-stage stochastic programming (IFTSP) method is developed. It incorporates the concepts of interval-parameter programming and full-infinite programming within a two-stage stochastic programming framework. IFTSP can explicitly address uncertainties presented as crisp intervals, probability distributions and functional intervals. The developed model is then applied to Zhangweinan river basin for demonstrating its applicability. Results from the case study indicate that compromise solutions have been obtained. They provide the desired agricultural irrigation water-supply schemes, which are related to a variety of tradeoffs between conflicting economic benefits and associated penalties attributed to the violation of predefined policies. The solutions can be used for generating decision alternatives and thus help decision makers to identify desired agricultural irrigation targets with maximized system benefit and minimized system-failure risk. Decision makers can adjust the existing agricultural irrigation patterns, and coordinate the conflict interactions among economic benefit, system efficiency, and agricultural irrigation under uncertainty.