Modeling of Water Resources Allocation and Water Quality Management for Supporting Regional Sustainability under Uncertainty in an Arid Region

Abstract

In this study, a scenario-based interval-stochastic fraticle optimization with Laplace criterion (SISFL) method is developed for sustainable water resources allocation and water quality management (WAQM) under multiple uncertainties. SISFL can tackle uncertainties presented as interval parameters and probability distributions; meanwhile, it can also quantify artificial fuzziness such as risk-averse attitude in a decision-making issue. Besides, it can reflect random scenario occurrence under the supposition of no data available. The developed method is applied to a real case of water resources allocation and water quality management in the Kaidu-kongque River Basin, where encounter serve water deficit and water quality degradation simultaneously in Northwest China. Results of water allocation pattern, pollution mitigation scheme, and system benefit under various scenarios are analyzed. The tradeoff between economic activity and water-environment protection with interval necessity levels and Laplace criterions can support policymakers generating an effective and robust manner associated with risk control for WAQM under multiple uncertainties. These discoveries avail local policymakers gain insight into the capacity planning of water-environment to satisfy the basin’s integrity of socio-economic development and eco-environmental sustainability.

1 Introduction

With the pace of accelerated industrialization and elevated urbanization, water deemed as the prerequisite natural resource has been countered severe anthropogenic influences, leading aggravated global water crisis associated with shrinking water availability and deteriorating water quality (Zeng et al. 2015). Particular in some developing countries, adverse impacts of water crisis have threatened to human-public health, improvement of life standards and socio-economic development, which would be perceived the most significant issue that policymaker has to face today (Ternes et al. 1999). Effective water planning and management index of water quantity and quality can provide guidance to the preservation of valuable water resources and total pollutant control simultaneously, which can alter the industrial types and economic patterns to attain sustainable development of regional eco-environment and socio-economy (Brouwer et al. 2008). However, a water resources allocation and water quality management (WAQM) plan is complicated with numbers of uncertainties incorporating with imprecise economic data, changed water policies, random water flows and fuzzy pollutant emission, where their interactions can fortify the conflict laden issues of water planning. The above issues require policymakers not only coordinating regional human activity and environment protection, but also considering numbers of influence factors (including economic, social and environmental), uncertainties and their interactions in a WAQM plan.

Previously, various mathematical programming approaches were developed for supporting WAQM under uncertainty (Huang and Loucks 2000; Li et al. 2006; Zeng et al. 2015; Dutta et al. 2016; Nematian 2016). Among them, interval-parameter programming (IPP) can handle indeterminacies fluctuating within a certain region, that cannot be quantified as membership or distribution functions; however, it has difficulties in resolving indeterminacies expressed as random variables and is lack of linkage to economic consequences of violated policies pre-regulated by authorities through taking recourse actions in order to correct any infeasibility (Li and Huang 2008). Therefore, two-stage stochastic programming (TSP) (i.e., one type of stochastic mathematical programming (SMP) approach) is introduced to tackle WAQM issues whose input data are represented as probabilities (Zeng et al. 2014). TSP can solve decision-making problems associated with randomness, which can rectify initial (first-stage) decision with probabilistic event; however, it has difficulties in incorporating other formats of uncertainties presented as possibilistic distributions within the optimization framework. In comparison, fuzzy programming (FP) has an advantage to tackle decision issues with imprecise information (Inuiguchi and Ramík 2000; Li and Huang 2009). Fractile optimization (FO) approach deemed as an effective solution algorithm of FP can handle necessity degree of event occurrence expressed as the possibility distributions, which can enable the flexible estimation of results with high satisfaction degree (Inuiguchi and Ramík 2000).

Although the above optimization methods can deal with various uncertainties in a WAQM planning system, they have difficulties to obtain reliable and rational input data such as promised water targets by policymakers. Especially when the targets are affected by risk attitudes/preferences of policymakers in response to decision makers’ experiences and personality traits, conventional mathematic programming has fall into dilemma. Therefore, scenario analysis (SA) is introduced to provide an effective manner to explore approaching dubious informations regarding the interaction between many factors (including risk attitudes) and decision outcomes (Peterson et al. 2003). In general, SA can create a ‘possibility space’ to reflect the complex and uncertain system, which may be influenced by process of risk attitudes and preferences (Swarta et al. 2004). However, limited data has difficulty in reflecting the probability of the scenario occurrence, which can be dealt with Laplace’s criterion. SA with Laplace’s criterion is effective in handling the probability of scenario occurrence under the limited data availability through supposing its occurrence equal (Laplace 1951; Aldea and Draghici 2014; Green and Weatherhead 2014).

Therefore, the objective of this study is to develop a scenario-based interval-stochastic fraticle optimization with Laplace criterion (SISFL) method for WAQM under multiple uncertainties. SISFL can not only tackle uncertainties presented as interval parameters and probability distributions, but also tackle random scenario occurrence under limited data availability; meanwhile, risk attitudes and preferences of decision makers have been reflected into the Laplace “possibility space’ to support generating more reliable decision in a WAQM plan. The SISFL method will be applied to a practical water management problem on integration of water quantity and quality in Kaidu-kongque River Basin, China, which is encountering atrophic available water and deteriorative water quality in response to high speedy economic development and population growth. Results with various η -level can be used for supporting the adjustment of the current water allocation patterns, as well as pollution discharge schemes. Results can be beneficial to generate an optimized water-environment policy to alleviate the conflict among economic growth, water-supply and pollutant control issue at the watershed level.

2 Methodology

In many real-world problems, when the parameter of a model fluctuates within a certain region, not stating a probability distribution, interval-parameter programming (IPP) can be imported to deal with such uncertainties expressed as interval as follows:

$$ \operatorname{Max}\ {f}^{\pm }={C}^{\pm }{x}^{\pm } $$

(1a)

subject to:

$$ {A}^{\pm }{x}^{\pm}\le {B}^{\pm } $$

(1b)

$$ {x}^{\pm}\ge 0 $$

(1c)

In model (1), an interval number x ^± can be expressed as a region (i.e., interval) with known lower- and upper-bound (Huang and Loucks 2000). TSP is introduced into model (1) to handle uncertain information presented as random variables, leading to an interval-stochastic programming (ISP) model as follows:

$$ \operatorname{Max}\ f=\sum_{i=1}^{I_1}{a}_i^{\pm }{x}_i^{\pm }-\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{v}_i^{\pm }{y}_{i h}^{\pm } $$

(2a)

subject to

$$ \sum_{i=1}^{I_1}{c}_{r i}^{\pm }{x}_i^{\pm}\le {e}_r^{\pm },\kern0.75em r=1,2,\dots, R $$

(2b)

$$ \sum_{i=1}^{I_1}{c}_{si}^{\pm }{x}_i^{\pm }+\sum_{i=1}^{I_2}{c}_{si}^{\pm }{x}_{si}^{\pm}\le {w}_h^{\pm },\kern0.75em s=1,2,\dots, S;\kern0.75em h=1,2,\dots, H $$

(2c)

$$ {x}_i^{\pm}\ge 0,\kern0.75em \mathrm{i}=1,2,\dots {,\mathrm{I}}_1 $$

(2d)

$$ {\mathrm{y}}_{ih}^{\pm}\ge 0,\kern0.75em \mathrm{i}=1,2,\dots {,\mathrm{I}}_2;\kern0.75em h=1,2,\dots, H $$

(2e)

where the initial decision (${x}_i^{\pm }$) is made before the occurrence of the random variables; when random event ω _h occur, a recourse action ${y}_{ih}^{\pm }$ can be rectify the initial decision benefit (i.e., $\sum_{i=1}^{I_1}{a}_i^{\pm }{x}_i^{\pm }$) (Li et al. 2006). However, parameters in objectives and constraints may be provided as indistinct manners, which can be influenced by subjective judgments of policymakers, leading ISP programming model infeasible (Tanaka et al. 2000). Therefore, a fractile optimization (FO) is introduced to reflect ambiguity in decision-making issues presented as possibilistic distribution. In general, an interval-stochastic fraticle optimization (ISFO) model can be formulated as follows (Inuiguchi and Ramík 2000):

$$ \operatorname{Max}\ f=\sum_{i=1}^{I_1}{\tilde{a}}_i^{\pm }{x}_i^{\pm }-\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{\tilde{v}}_i^{\pm }{y}_{i h}^{\pm } $$

(3a)

subject to

$$ \sum_{i=1}^{I_1}{c}_{r i}^{\pm }{x}_i^{\pm}\le {\tilde{e}}_r^{\pm },\kern0.75em r=1,2,\dots, R $$

(3b)

$$ \sum_{i=1}^{I_1}{c}_{si}^{\pm }{x}_i^{\pm }+\sum_{i=1}^{I_2}{c}_{si}^{\pm }{x}_{si}^{\pm}\le {w}_h^{\pm },\kern0.75em s=1,2,\dots, S;\kern0.75em h=1,2,\dots, H $$

(3b)

$$ {x}_i^{\pm}\ge 0,\kern0.75em \mathrm{i}=1,2,\dots {,\mathrm{I}}_1 $$

(3d)

$$ {\mathrm{y}}_{ih}^{\pm}\ge 0,\kern0.75em \mathrm{i}=1,2,\dots {,\mathrm{I}}_2;\kern0.75em h=1,2,\dots, H $$

(3e)

where ${\tilde{a}}_i^{\pm }$, ${\tilde{c}}_i^{\pm }$ and ${\tilde{\mathrm{e}}}_i^{\pm }$ are expressed as fuzzy coefficient in the objective and constraints, which can be restricted by fuzzy triangular numbers with possibility distribution. In general, possibility distribution can be regarded as fuzzy membership function, and possibility degree can be considered as the membership value (Liu and Liu 2002). Thus, Eq. (3a) can be transformed as follows:

$$ \begin{array}{l}\operatorname{Max}\ \tilde{f}=\sum_{i=1}^{I_1}{\tilde{a}}_i^{\pm }{x}_i^{\pm }-\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{\tilde{v}}_i^{\pm }{y}_{i h}^{\pm}\\ {}\kern3em =\left(\sum_{i=1}^{I_1}{a}_i^{\pm }{x}_i^{\pm }-\sum_{i=1}^{I_1}{\eta}_i^a{\omega}^a\left|{x}_i^{\pm}\right|\right)-\left(\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{v}_i^{\pm }{y}_{i h}^{\pm }-\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{\eta}_i^v{\omega}^v\left|{y}_{i h}^{\pm}\right|\right)\end{array} $$

(4)

Based on the possibility theory, necessity measure can be formulated as follows (Inuiguchi and Ramík 2000):

$$ Nec\left\{{\xi}_a\right\}= \inf \max \left(1-{\mu}_a(r),{\mu}_{\xi_a}(r)\right) $$

(5a)

$$ Nec\left\{{\xi}_v\right\}= \inf \min \left(1-{\mu}_v(r),{\mu}_{\xi_v}(r)\right) $$

(5b)

where ${\mu}_{\xi_c}$ is the membership function of the fuzzy set c; ${\tilde{\mu}}_c(r)$ denotes the certainty (or necessity) degree of the event that fuzzy possibilistic variable ${\tilde{v}}_i^{\pm }$restricted by the possibility distribution μ _c is in the fuzzy set c. Let ξ _c be a fuzzy variable with membership functionμ _c , and let u and r be real numbers. The necessity of a fuzzy event, characterized byξ ≤ r, is defined by $Nec\left\{{\xi}_c\le r\right\}=1-\underset{u> r}{ \sup }{\mu}_c(u)$ (Trumbo and McComa 2003; Zeng et al. 2014). Similarly, $Nec\left\{{\xi}_e\le r\right\}=1-\underset{u> r}{ \sup }{\mu}_e(u)$. According to the definition of necessity measure, ISFO model can be transferred as follows:

$$ \operatorname{Max}\ f=\left(\sum_{i=1}^{I_1}{a}_i^{\pm }{x}_i^{\pm }-\sum_{i=1}^{I_1}{\eta}_i^a{\omega}^a\left|{x}_i^{\pm}\right|\right)-\left(\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{v}_i^{\pm }{y}_{i h}^{\pm }-\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{\eta}_i^v{\omega}^v\left|{y}_{i h}^{\pm}\right|\right) $$

(6a)

Subject to:

$$ Nec\left\{\left({a}_i^{a\pm }{x}_i^{\pm },\omega \left|{x}_i^{\pm}\right|\right)\ge {\mu}_a\right\}\ge {\eta}_i^a $$

(6b)

$$ Nec\left\{\left({v}_i^{v\pm }{y}_i^{\pm },\omega \left|{y}_{i h}^{\pm}\right|\right)\le {\mu}_v\right\}\ge {\eta}_i^v $$

(6c)

$$ Nec\left\{\sum_{i=1}^{I_1}{c}_{r i}^{\pm }{x}_i^{\pm}\le {e}_r^{e\pm },\omega \left|{e}_r^{e\pm}\right|\right\}\ge {\eta}_i^e,\kern0.75em r=1,2,\dots, R $$

(6d)

$$ \sum_{i=1}^{I_1}{c}_{si}^{\pm }{x}_i^{\pm }+\sum_{i=1}^{I_2}{c}_{si}^{\pm }{x}_{si}^{\pm}\le {w}_h^{\pm },\kern0.75em s=1,2,\dots, S;\kern0.75em h=1,2,\dots, H $$

(6e)

$$ {x}_i^{\pm}\ge 0,\kern0.75em \mathrm{i}=1,2,\dots {,\mathrm{I}}_1 $$

(6f)

$$ {\mathrm{y}}_{ih}^{\pm}\ge 0,\kern0.75em \mathrm{i}=1,2,\dots {,\mathrm{I}}_2;\kern0.75em h=1,2,\dots, H $$

(6g)

However, in a practical decision-making issue, the input of initial variable is often impacted by various factors. Scenario analysis (SA) is introduced to provide an effective manner to explore approaching dubious informations associated with the interaction between many factors (including risk attitudes) and decision outcomes (Swarta et al. 2004). In practical decision issues, risk attitudes of policymakers deemed as an important component would influence scenario generation. Laplace’s criterion can handle probability of each scenario occurrence under the supposition of no data available on the probabilities of the various outcomes; the probabilities of each scenario appear reasonable to suppose that these are equal (Aldea and Draghici 2014). Thus, a scenario analysis with Laplace’s criterion (SAL) can be expressed as follows (Pažek and Rozman 2009):

$$ LS\left({A}_{m n}\right)=\frac{1}{m_2}\cdot \left[\left(\begin{array}{cccc}\hfill {PS}_{11}\hfill & \hfill {PS}_{12}\hfill & \hfill \dots \hfill & \hfill {PS}_{1 n}\hfill \\ {}\hfill {PS}_{21}\hfill & \hfill {PS}_{22}\hfill & \hfill \dots \hfill & \hfill {PS}_{2 n}\hfill \\ {}\hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill {PS}_{m_11}\hfill & \hfill {PS}_{m_12}\hfill & \hfill \dots \hfill & \hfill {PS}_{m_1 n}\hfill \end{array}\right)+\left(\begin{array}{cccc}\hfill {CS}_{\left({m}_1+1\right)1}\hfill & \hfill {CS}_{m_12}\hfill & \hfill \dots \hfill & \hfill {CS}_{\left({m}_1+1\right) n}\hfill \\ {}\hfill {CS}_{\left({m}_1+2\right)1}\hfill & \hfill {CS}_{\left({m}_1+2\right)2}\hfill & \hfill \dots \hfill & \hfill {CS}_{\left({m}_1+2\right) n}\hfill \\ {}\hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill {CS}_{m_21}\hfill & \hfill {CS}_{m_22}\hfill & \hfill \dots \hfill & \hfill {CS}_{m_2 n}\hfill \end{array}\right)\right] $$

(7d)

Thus, a scenario-based interval-stochastic fraticle optimization with Laplace’s criterion (SISFL) can be modified as follows:

$$ \begin{array}{l}\operatorname{Max}\ {\tilde{f}}_{LAP}^{\pm }=\Big\{\frac{1}{m_2}\cdot \left[\left(\begin{array}{cccc}\hfill {PS}_{11}\hfill & \hfill {PS}_{12}\hfill & \hfill \dots \hfill & \hfill {PS}_{1 n}\hfill \\ {}\hfill {PS}_{21}\hfill & \hfill {PS}_{22}\hfill & \hfill \dots \hfill & \hfill {PS}_{2 n}\hfill \\ {}\hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill {PS}_{m_11}\hfill & \hfill {PS}_{m_12}\hfill & \hfill \dots \hfill & \hfill {PS}_{m_1 n}\hfill \end{array}\right)+\left(\begin{array}{cccc}\hfill {CS}_{\left({m}_1+1\right)1}\hfill & \hfill {CS}_{m_12}\hfill & \hfill \dots \hfill & \hfill {CS}_{\left({m}_1+1\right) n}\hfill \\ {}\hfill {CS}_{\left({m}_1+2\right)1}\hfill & \hfill {CS}_{\left({m}_1+2\right)2}\hfill & \hfill \dots \hfill & \hfill {CS}_{\left({m}_1+2\right) n}\hfill \\ {}\hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill {CS}_{m_21}\hfill & \hfill {CS}_{m_22}\hfill & \hfill \dots \hfill & \hfill {CS}_{m_2 n}\hfill \end{array}\right)\right]\hfill \\ {}\kern3.359999em \cdot \left(\sum_{i=1}^{I_1}{a}_i^{\pm }{x}_i^{\pm }-\sum_{i=1}^{I_1}{\eta}_i^a{\omega}^a\left|{x}_i^{\pm}\right|\right)\Big\}-\left(\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{v}_i^{\pm }{y}_{i h}^{\pm }-\sum_{i=1}^{I_2}\sum_{h=1}^H{p}_h{\eta}_i^v{\omega}^v\left|{y}_{i h}^{\pm}\right|\right)\hfill \end{array} $$

(8a)

Subject to:

$$ Nec\left\{\left({a}_i^{a\pm }{x}_i^{\pm },\omega \left|{x}_i^{\pm}\right|\right)\ge {\mu}_a\right\}\ge {\eta}_i^a $$

(8b)

$$ Nec\left\{\left({v}_i^{v\pm }{y}_{i h}^{\pm },\omega \left|{y}_{i h}^{\pm}\right|\right)\le {\mu}_v\right\}\ge {\eta}_i^v $$

(8c)

$$ Nec\left\{\sum_{i=1}^{I_1}{c}_{r i}^{\pm }{x}_i^{\pm}\le {e}_r^{e\pm },\omega \left|{e}_r^{e\pm}\right|\right\}\ge {\eta}_i^e,\kern0.5em r=1,2,\dots, R $$

(8d)

$$ \sum_{i=1}^{I_1}{c}_{si}^{\pm }{x}_i^{\pm }+\sum_{i=1}^{I_2}{c}_{si}^{\pm }{x}_{si}^{\pm}\le {w}_h^{\pm },\kern0.75em s=1,2,\dots, S;\kern0.75em h=1,2,\dots, H $$

(8e)

$$ \begin{array}{cc}\hfill {x}_i^{\pm}\ge 0,\hfill & \hfill \mathrm{i}=1,2,\dots, {\mathrm{I}}_1\hfill \end{array} $$

(8f)

$$ \begin{array}{ccc}\hfill {\mathrm{y}}_{ih}^{\pm}\ge 0,\hfill & \hfill \mathrm{i}=1,2,\dots, {\mathrm{I}}_2;\hfill & \hfill h=1,2,\dots, H\hfill \end{array} $$

(8e)

In this study, a two-step solution process is developed to solve such SISFL model, which can transform SISFL model into two deterministic submodels associated with the lower and upper bounds of the desired objective-function values.

3 Application

Kaidu and Kongque River deemed as branches of Tarim River are responsible for water supply for Kuerle, Yanqi, Hejing, Heshuo, Bohu and Yuli counties, which situated in the middle reach of the Tarim River Basin, and with lengths of 610 km and 785 km (SYXUAR 2011). Kaidu-kongque River Basin (71°39″E-93°45″E, 34°20″N-43°39″N) has a total area of 62 × 10³ km² with a subtropical and continental arid monsoon climate. It has an average annual precipitation of 273 mm / year, and with an average year temperature of 11 to 12 °C (Huang et al. 2012). Kaidu-kongque River Basin is one of the most important bases of cotton and grain in the Tarim River Basin and the northwest of China. Accompanied by the development of economic crops planting, agricultural products processing and manufacturing have been accelerated; meanwhile, the national western development policy have promote the speed of the development of industrial activities (e.g., chemical industry, fossil oil industry, textiles, papermaking, electric power and transportation) (Zeng et al. 2015). In recent years, the population of Kaidu-kongque River Basin has exceed 1.5 million, the increment speed of GDP have reach 8.3% per year. The high-speed development of agriculture and industry can improve the human life greatly, however, numbers of problems have been raised as follows: (a) with increased population growth and high-speed economic development, enormous increasing of water demands reached the limits of what the natural system can provide. (b) excessive non-point source pollutant discharge due to over-fertilization can destroy the capacity of environmental self-purification, leading water environment disequilibrium. The water environmental degeneration due to long-term destruction would aggravate the water shrinking, leading chronic severe shortages. (c) rapid urbanization and over exploitation can destroy the original balance structure of land utilization, which can aggravate the land deterioration. (d) the deterioration of ecological environment has aggravated water crisis, (such as water shortages and pollution issues), which would be deemed as major obstacles to socioeconomic development in this region. Under these situations, an effective WAQM planning should to be advocated, which was consideration of balance the contradictions between development of economy and sustainability of environment.

Figure 1 presents the framework of SISFL method and its application to the Kaidu-kongque River Basin. Under risk of water deficits and pollution discharges, how to develop an optimal policy maximizing system benefit and minimizing risk associated with water-environment requirements and risk attitudes of policymakers are important issues to be handled in study region. However, since there are temporal variations in water availability and variability in pollution problems, the optimal schemes for WAQM can also vary correspondingly. Therefore, a scenario-based interval-stochastic fraticle optimization with Laplace’s criterion (SISFL) model can be embedded to resolve following issues: (i) what are the cost-efficient water allocation and pollution emission schemes; (ii) how to generate an optimized plan associate with water-allocation and pollution-control under various policy adjustment; (iii) what do uncertainties presented as probability / possibility distribution influence water-environment structure and system in WAQM of Kaidu-kongque Basin; (iv) how to balance the relationship between the risk attitude and system benefits.

Fig. 1

Framework of a scenario-based interval-stochastic fraticle optimization with Laplace’s criterion (SISFL) and its application in Kaidu-kongque River Basin

3.1 Objective Function

In the study region, water resources need to be allocated to multiple human activities (e.g., residential, industrial, agricultural, and ecological activities) of six zones. The objective is to maximize the system benefit subject to a set of constraints for relationships between decision variables and water-environment restrictions as follows:

$$ \mathit{\operatorname{Max}}\kern0.5em {\tilde{f}}_{LAP}=(1)+(2)-(3)-(4)-(5) $$

(9a)

• (1) Income from water allocation for municipal, agricultural, industrial and ecological activities (${\tilde{IT}}^{\pm }$):

$$ \begin{array}{l}\frac{1}{m_2}\cdot \left[\left(\begin{array}{cccc}\hfill {PS}_{11}\hfill & \hfill {PS}_{12}\hfill & \hfill \dots \hfill & \hfill {PS}_{1 n}\hfill \\ {}\hfill {PS}_{21}\hfill & \hfill {PS}_{22}\hfill & \hfill \dots \hfill & \hfill {PS}_{2 n}\hfill \\ {}\hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill {PS}_{m_11}\hfill & \hfill {PS}_{m_12}\hfill & \hfill \dots \hfill & \hfill {PS}_{m_1 n}\hfill \end{array}\right)+\left(\begin{array}{cccc}\hfill {CS}_{\left({m}_1+1\right)1}\hfill & \hfill {CS}_{m_12}\hfill & \hfill \dots \hfill & \hfill {CS}_{\left({m}_1+1\right) n}\hfill \\ {}\hfill {CS}_{\left({m}_1+2\right)1}\hfill & \hfill {CS}_{\left({m}_1+2\right)2}\hfill & \hfill \dots \hfill & \hfill {CS}_{\left({m}_1+2\right) n}\hfill \\ {}\hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill {CS}_{m_21}\hfill & \hfill {CS}_{m_22}\hfill & \hfill \dots \hfill & \hfill {CS}_{m_2 n}\hfill \end{array}\right)\right]\cdot \hfill \\ {}\begin{array}{l}\Big\{\left[\sum_{i=1}^3\sum_{j=1}^6{BI}_{i j}^{\pm}\cdot \left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)-\sum_{i=1}^3\sum_{j=1}^6\left({BI}_{i j}^{\pm },{\eta}_{n ec}^{BI}\omega \left|{BI}_{i j}^{\pm}\right|\right)\cdot \left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)\ \right]\hfill \\ {}+\Big[\sum_{m=1}^2\sum_{j=6}^6{BM}_{m j}^{\pm}\cdot \left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)-\sum_{m=1}^2\sum_{j=6}^6\left({BM}_{m j}^{\pm },{\eta}_{n ec}^{BM}\omega \left|{BM}_{m j}^{\pm}\right|\right)\cdot \left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)\hfill \\ {}+\left[\sum_{n=1}^5\sum_{j=1}^6{BA}_{n j}^{\pm}\cdot \left({WLA}_{n j}^{-}+\varDelta {WLA}_{n j}\cdot z\right)\cdot {ua}_{n j}-\sum_{n=1}^5\sum_{j=1}^6\left({BA}_{n j}^{\pm },{\eta}_{n ec}^{BA}\omega \left|{BA}_j^{\pm}\right|\right)\cdot \left({WLA}_{n j}^{-}+\varDelta {WLA}_{n j}\cdot z\right)\cdot {ua}_{n j}\right]\hfill \\ {}+\left[\sum_{k=1}^2\sum_{j=1}^6{BE}_{k j}^{\pm}\cdot \left({WLE}_{k j}^{-}+\varDelta {WLE}_{k j}\cdot z\right)-\sum_{k=1}^2\sum_{j=1}^6\left({BE}_{k j}^{\pm },{\eta}_{n ec}^{BE}\omega \left|{BE}_{k j}^{\pm}\right|\right)\cdot \left({WLE}_{k j}^{-}+\varDelta {WLE}_{k j}\cdot z\right)\right]\Big\}\hfill \end{array}\hfill \end{array} $$

(9b)

• (2) Loss from water shortage for municipal, agricultural, industrial and ecological activities (${\tilde{LWS}}^{\pm }$):

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\left\{\right[\left(\sum_{m=1}^2\sum_{j=1}^6{LM}_{m j}^{\pm }-\sum_{m=1}^2\sum_{j=6}^6\left({LM}_{m j}^{\pm },{\eta}_{n ec}^{LM}\omega \left|{LM}_{m j}^{\pm}\right|\right)\cdot {SLM}_{m j h}^{\pm}\right]+\hfill \\ {}\left[\sum_{n=1}^5\sum_{j=1}^6{LA}_{n j}^{\pm }-\sum_{n=1}^5\sum_{j=1}^6\left({LA}_{n j}^{\pm },{\eta}_{n ec}^{LA}\omega \left|{LA}_j^{\pm}\right|\right)\right]\cdot {SLA}_{n j}^{\pm}\cdot {ua}_{n j}+\Big[\sum_{i=1}^3\sum_{j=1}^6{LI}_{i j}^{\pm }-\hfill \\ {}+\sum_{i=1}^3\sum_{j=1}^6\left({LI}_{i j}^{\pm },{\eta}_{n ec}^{LI}\omega \left|{LI}_{i j}^{\pm}\right|\right)\left]\cdot {SLI}_{i j}^{\pm }+\left[\sum_{k=1}^2\sum_{j=1}^6{LE}_{k j}^{\pm }-\sum_{k=1}^2\sum_{j=1}^6\left({LE}_{k j}^{\pm },{\eta}_{n ec}\omega \left|{LE}_{k j}^{\pm}\right|\right)\cdot {SLE}_{k j}^{\pm}\right]\ \right\}\hfill \end{array} $$

(9c)

• (3) Cost for environmental retreatment from municipal activities (${\tilde{CMM}}^{\pm }$):

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\Big\{\sum_{m=1}^2\sum_{j=6}^6{CM}_{m j}^{\pm}\cdot \left[\left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)-{SLM}_{m j}^{\pm}\right]-\hfill \\ {}\sum_{m=1}^2\sum_{j=6}^6\left({CM}_{m j}^{\pm },{\eta}_{nec}^{CM}\omega \left|{CM}_{m j}^{\pm}\right|\right)\cdot \left[\left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)-{SLM}_{m j}^{\pm}\right]\cdot {POM}_{m j}\cdot \left(1-\alpha \right)\Big\}\hfill \end{array} $$

(9d)

• (4) Cost for environmental retreatment from agricultural activities (${\tilde{CMA}}^{\pm }$):

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\Big\{\sum_{n=1}^5\sum_{j=1}^6{CA}_{n j}^{\pm}\cdot \left[\left({WLA}_{n j}^{-}+\varDelta {WLA}_{n j}\cdot z\right)-{SLA}_{n j}^{\pm}\right]\cdot {ua}_{n j}\cdot {IRA}_{n j}\hfill \\ {}-\sum_{n=1}^5\sum_{j=1}^6\left({CA}_{n j}^{\pm },{\eta}_{n ec}^{CA}\omega \left|{CA}_{Nj}^{\pm}\right|\right)\cdot \left[\left({WLA}_{n j}^{-}+\varDelta {WLA}_{n j}\cdot z\right)-{SLA}_{n j}^{\pm}\right]\cdot {ua}_{n j}\cdot {IRA}_{n j}\Big\}\hfill \end{array} $$

(9e)

• (5) Cost for environmental retreatment from industrial activities (${\tilde{CMI}}^{\pm }$):

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\Big\{\sum_{i=1}^3\sum_{j=6}^6{CM}_{mj}^{\pm}\cdot \left[\left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)-{SLI}_{i j}^{\pm}\right]\cdot {IRI}_{i j}\cdot \left(1-\eta \right)-\hfill \\ {}\sum_{i=1}^3\sum_{j=6}^6\left({CM}_{mj}^{\pm },{\eta}_{nec}^{CM}\omega \left|{CM}_{mj}^{\pm}\right|\right)\cdot \left[\left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)-{SLI}_{i j}^{\pm}\right]\cdot {IRI}_{i j}\cdot \left(1-\eta \right)\Big\}\hfill \end{array} $$

(9f)

where ${\tilde{f}}_{LAP}^{\pm }$ is the net system benefit with Laplace criterion, which equals to the total returns minus the total costs. The detailed nomenclatures for the variables and parameters are provided in Appendix. In this study, five main point sources (including two municipal and three industrial plants) scatter along the river; meanwhile farming irrigation deemed as non-point sources plants are considered, where five crops (i.e., cearl, cotton, oil plants, vegetable, and fruit) irrigating in agricultural zones would generate amount of pesticide / chemical fertilizer residue washed away into the water body. A three-period planning horizon is selected, and each period concludes three water availability levels (i.e., low, medium and high levels).

3.2 Constraints

• (1) Water availability constraints:

$$ Nec\left\{\left({QK}_{hj}^{\pm },{\eta}^{QK}\omega \left|{QK}_{hj}^{\pm}\right|\right)=\left({QR}_{hj}^{\pm },\omega \left|{QR}_{hj}^{\pm}\right|\right)-{E}_{jh}^{\pm }-{H}_{jh}^{\pm }-{QU}_{jh}^{\pm}\ge \mu \right\}\ge {\eta}^{QK} $$

(10a)

• (2) Water quantity constraints:

$$ \begin{array}{l} Nec\left\{\sum_{m=1}^2\sum_{j=6}^6\sum_{h=1}^3{p}_{h j}\left[\left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)-{SLM}_{m j}^{\pm}\right]+\sum_{i=1}^3\sum_{j=1}^6\sum_{h=1}^3{p}_{h j}\right[\left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)\hfill \\ {}-{SLI}_{i j}^{\pm}\Big]+\sum_{n=1}^5\sum_{j=1}^6\sum_{h=1}^3{p}_{h j}\left[\left({WLA}_{n j}^{-}-\varDelta {WLA}_{n j}^{-}\cdot z\right)-{SLA}_{n j}^{\pm}\right]\cdot {ua}_{n j}+\hfill \\ {}\sum_{k=1}^2\sum_{j=1}^6\sum_{h=1}^3{p}_{h j}\left[\left({WLE}_{k j}^{-}+\varDelta {WLE}_{k j}\cdot z\right)-{SLE}_{k j}^{\pm}\Big)\right]\le \left({QK}_{h j}^{\pm },{\eta}^{QK}\omega \left|{QK}_{h j}^{\pm}\right|\right)\Big\}\ge {\eta}^{QK}\hfill \end{array} $$

(10b)

• (3) Wastewater treatment capacity constraints:

$$ \sum_{m=1}^2\sum_{j=6}^6\left[\left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)-{SLM}_{m j}^{\pm}\right]\cdot {POM}_{m j}\cdot \alpha \le {CTM}_{m j}^{\pm } $$

(10c)

$$ \sum_{i=1}^3\sum_{j=6}^6\left[\left({WLI}_{i j}^{-}+\varDelta {WLM}_{mj}\cdot z\right)-{SLI}_{i j}^{\pm}\right]\cdot {IRI}_{i j}\cdot \eta \le {CTI}_{i j}^{\pm } $$

(10d)

• (4) COD discharge constraints:

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\Big\{\sum_{m=1}^2\sum_{j=6}^6\left[\left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)-{SLM}_{m j}^{\pm}\right]\cdot {POM}_{m j}\cdot \left(1-\alpha \right)\cdot {dm}_{COD}\hfill \\ {}+\sum_{i=1}^3\sum_{j=6}^6\left[\left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)-{SLI}_{i j}^{\pm}\right]\cdot {IRI}_{i j}\cdot \left(1-\eta \right)\cdot {di}_{COD}\Big\}\le {DCM}_{m j h}^{\pm }+{DCI}_{i j h}^{\pm}\hfill \end{array} $$

(10e)

• (5) Nitrogen discharge constraints:

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\left\{\sum_{m=1}^2\sum_{j=6}^6\right[\left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)-{SLM}_{m j}^{\pm}\Big)\cdot {POM}_{m j}\cdot \left(1-\alpha \right)\cdot {dm}_{TN}+\\ {}\sum_{i=1}^3\sum_{j=6}^6\left[\left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)-{SLI}_{i j}^{\pm}\right]\cdot {IRI}_{i j}\cdot \left(1-\eta \right)\cdot {di}_{TN}\Big\}\le {DNM}_{m j h}^{\pm }+{DNI}_{i j h}^{\pm}\end{array} $$

(10f)

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\Big[\sum_{n=1}^5\sum_{j=1}^6\left[\left({WLA}_{n j}^{-}-\varDelta {WLA}_{n j}^{-}\cdot z\right)-{SLA}_{n j}^{\pm}\right]\cdot {ua}_{n j}\cdot {IRA}_{n j}\cdot {dar}_{TN}+\hfill \\ {}\sum_{n=1}^5\sum_{j=1}^6\left[\left({WLA}_{n j}^{-}-\varDelta {WLA}_{n j}^{-}\cdot z\right)-{SLA}_{n j}^{\pm}\right]\cdot {SA}_{n j}\cdot {das}_{TN}\Big]\le {DNA}_{n j h}^{\pm}\hfill \end{array} $$

(10g)

• (6) Phosphorus discharge constraints:

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\left\{\sum_{m=1}^2\sum_{j=6}^6\right[\left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)-{SLM}_{m j}^{\pm}\Big)\cdot {POM}_{m j}\cdot \left(1-\alpha \right)\cdot {dm}_{TP}+\hfill \\ {}\sum_{i=1}^3\sum_{j=6}^6\left[\left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)-{SLI}_{i j}^{\pm}\right]\cdot {IRI}_{i j}\cdot \left(1-\eta \right)\cdot {di}_{TP}\Big\}\le {DPM}_{m j h}^{\pm }+{DPI}_{i j h}^{\pm}\hfill \end{array} $$

(10h)

$$ \begin{array}{l}\sum_{h=1}^3{p}_{h j}\Big[\sum_{n=1}^5\sum_{j=1}^6\left[\left({WLA}_{n j}^{-}-\varDelta {WLA}_{n j}^{-}\cdot z\right)-{SLA}_{n j}^{\pm}\right]\cdot {ua}_{n j}\cdot {IRA}_{n j}\cdot {dar}_{TP}+\hfill \\ {}\sum_{n=1}^5\sum_{j=1}^6\left[\left({WLA}_{n j}^{-}-\varDelta {WLA}_{n j}^{-}\cdot z\right)-{SLA}_{n j}^{\pm}\right]\cdot {SA}_{n j}\cdot {das}_{TP}\Big]\le {DPA}_{n j h}^{\pm}\hfill \end{array} $$

(10i)

• (7) Soil loss constraints:

$$ \sum_{n=1}^5\sum_{j=1}^6\left[\left({WLA}_{n j}^{-}-\varDelta {WLA}_{n j}^{-}\cdot z\right)-{SLA}_{n j}^{\pm}\right]\cdot {SA}_{n j}\le {DPA}_{n j}^{\pm } $$

(10j)

• (8) Irrigation area constraints:

$$ {WLA}_{nj \min}^{\pm}\le \left({WLA}_{nj}^{-}-\varDelta {WLA}_{nj}^{-}\cdot z\right)\le {WLA}_{nj \max}^{\pm } $$

(10k)

• (9) Water demand of industrial production constraints:

$$ {WLI}_{ij \min}^{\pm}\le \left({WLI}_{ij}^{-}-\varDelta {WLI}_{ij}^{-}\cdot z\right)\le {WLI}_{ij \max}^{\pm } $$

(10l)

• (10) Water demand of municipality constraints:

$$ {WLM}_{mj \min}^{\pm}\le \left({WLM}_{mj}^{-}-\varDelta {WLM}_{mj}^{-}\cdot z\right)\le {WLM}_{mj \max}^{\pm } $$

(10m)

• (11) Fractile constraints:

$$ \begin{array}{l} Nec\left\{\right[\sum_{m=1}^2\sum_{j=6}^6\left({BM}_{m j}^{\pm },{\eta}_{n ec}^{BM}\omega \left|{BM}_{m j}^{\pm}\right|\right)\cdot \left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z\right)+\sum_{n=1}^5\sum_{j=1}^6\left({BA}_{n j}^{\pm },{\eta}_{n ec}^{AB}\omega \left|{BA}_j^{\pm}\right|\right)\hfill \\ {}\cdot \left({WLA}_{n j}^{-}+\varDelta {WLA}_{n j}\cdot z\right)\cdot {ua}_{n j}+\sum_{i=1}^3\sum_{j=1}^6\left({BI}_{i j}^{\pm },{\eta}_{n ec}^{BI}\omega \left|{BI}_{i j}^{\pm}\right|\right)\cdot \left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z\right)+\hfill \\ {}\sum_{k=1}^2\sum_{j=1}^6\left({BE}_{k j}^{\pm },{\eta}_{n ec}^{BE}\omega \left|{BE}_{k j}^{\pm}\right|\right)\cdot \left({WLE}_{k j}^{-}+\varDelta {WLE}_{k j}\cdot z\right)\left]\ge \mu \right\}\ge {\eta}_{n ec}\hfill \end{array} $$

(10n)

$$ \begin{array}{l} Nec\Big\{\sum_{m=1}^2\sum_{j=6}^6\sum_{h=1}^3{p}_{h j}\left({CM}_{m j}^{\pm },{\eta}_{n ec}^{CM}\omega \left|{CM}_{m j}^{\pm}\right|\right)\cdot \left({WLM}_{m j}^{-}+\varDelta {WLM}_{m j}\cdot z-{SLM}_{m j}^{\pm}\right)\cdot {POM}_{m j}\cdot \left(1-\alpha \right)+\hfill \\ {}\sum_{n=1}^5\sum_{j=1}^6\sum_{h=1}^3{p}_{h j}\left({CA}_{n j}^{\pm },{\eta}_{n ec}^{CA}\omega \left|{CA}_{Nj}^{\pm}\right|\right)\cdot \left({WLA}_{n j}^{-}+\varDelta {WLA}_{n j}\cdot z-{SLA}_{n j}^{\pm}\right)\cdot {ua}_{n j}\cdot {IRA}_{n j}+\hfill \\ {}\sum_{i=1}^3\sum_{j=6}^6\sum_{h=1}^3{p}_{h j}\left({CM}_{m j}^{\pm },{\eta}_{n ec}^{CM}\omega \left|{CM}_{m j}^{\pm}\right|\right)\cdot \left({WLI}_{i j}^{-}+\varDelta {WLI}_{i j}\cdot z-{SLI}_{i j}^{\pm}\right)\cdot {IRI}_{i j}\cdot \left(1-\eta \right)\left]\ge \mu \right\}\ge {\eta}_{n ec}\hfill \end{array} $$

(10o)

• (12) Non-negative constraints:

$$ \begin{array}{l}\begin{array}{cc}\hfill {SLM}_{mj}^{\pm },{SLA}_{nj}^{\pm },{SLI}_{ij}^{\pm },{SLE}_{kj}^{\pm}\ge 0\hfill & \hfill {BM}_{mj}^{\pm },{BA}_{nj b}^{\pm },{BI}_{ij}^{\pm },{BE}_{kj}^{\pm}\ge 0\hfill \end{array}\hfill \\ {}{CM}_{mj}^{\pm },{CA}_{nj}^{\pm },{CI}_{ij}^{\pm },{CE}_{kj}^{\pm}\ge 0\hfill \end{array} $$

(10p)

Constraint (2) presents that water deficits would be occur when water demands are not met, thus losses of water deficits among these competitive water consumers. Constraint (3) shows that the wastewater treatment capacity for point pollution sources. Constraint (4) presents the limited COD discharge into river from industry, which requires not to be exceeding the allowable discharge level after retreatment by sewage facilities. Constraint (5) to (6) shows the restrictions of nitrogen and phosphorus discharges from agricultural, municipal and industrial plants, which would be regulated by allowance thresholds. Constraint (7) presents that soil erosion and surface runoff due to its special geography in study region, which result in pollutant discharge carried by catchment process. The soil loss should be regulated by imposed allowance sediment loads permit. Constraint (8) presents restriction of irrigation developing scale in study region, which means that planning crop area should be restricted by allowable land resources; meanwhile it should maintain the minimum area to support food safety. Constraint (9) shows that the speed of industrial plants which should be maintain a rational pace according to regional development.

3.3 Data Acquirement

The parameters in this study are estimated according to government report, statistical yearbook, and related research works (Huang et al. 2012). Table 1 presents the various economic data (e.g., net benefit of water being delivered, loss of water not being delivered and retreatment cost) estimated by regional statistical yearbook with consideration of socioeconomic development (SYXUAR, from 2000 to 2013). In this study, available water resource levels are conducted through statistical analyses with the results of annual stream flow of the Kaidu-kongque River (2005–2012). The total available water can be divided into low, medium and high levels and their corresponding probabilities are 0.5, 0.3 and 0.2, where values of available water are presented as dual intervals (i.e., [(712, 725, 734), (824, 846, 856)], [(854, 863, 889), (910, 967, 990)] and [(998, 1001, 1012), (1101, 1178, 1235)] × 10⁶ m³). Meanwhile, allowance pollution discharges can be divided as low, medium and high levels and their probabilities are 0.583, 0.342 and 0.185. And the allowance amount of COD, TN and TP can be [7.05, 7.63], [1.02, 1.45] and [0.21, 0.92] at low level, they can be [7.89, 9.01], [1.45, 2.21] and [0.46, 1.11] at high level. Besides, eight scenarios are designed to compare diverse strategies associate with different risk attitudes between economic development and environmental protection as follows: (a) basic scenario (i.e., S1) present that water demand target would equal to current plan; (b) conservative scenarios (i.e., S2, S3 and S4) show that policymakers would slow down the speed of economic development to alleviate the pressure of water crisis. Under this situation, water demand targets are lower than the demand of economic development (i.e., 5%, 10% and 15%), which can also lessen pollution damages; (c) progressive scenarios (i.e., S5, S6 and S7) present that policymakers would focus on soc.-economy development instead of environmental protection. The increasing rate of water demand would be 5%, 8% and 10% according to the growing rate of economic development. (d) Laplace scenario (i.e., S8) is estimated by S1 to S7 by Laplace’s criterion.

Table 1 Economic data

4 Result Analysis

4.1 System Benefit

Figure 2 presents the system benefits of different scenarios under various necessity levels (i.e., η- necessity levels) through the SISFL model. The results indicate that progressive and conservative water plans (e.g., S6 and S3) corresponding to too high and low targets and expected values would all result in lower system benefits. Laplace scenario water plan (i.e., S8) can generate the highest system benefit; meanwhile it would product a robust and optimal result with consideration of balance of risk-attitude. Meanwhile, the results show that various η- levels can lead different varied system benefits, which would reflect expected system benefit preference and risk-averse attitude of policymakers. For instance, the system benefit under S1 would be diminished from $ 1.24 × 10⁹ to $ 1.01 × 10⁹, with incremental p-necessity levels from 0.60 to 0.995, respectively. It implies that increased indeterminacies in fuzzy objective corresponding to lower η--necessity level would result in higher benefits; vice versa.

Fig. 2

System benefits under S1, S3, S6 and S8 when η- levels are varied

4.2 Water Resources Allocation

Figure 3 showts the solutions for optimal targets and total water allocations for different activities under various scenarios (e.g., S1, S3, S6, and S8) when η-level is 1. Results indicate that water shortage would occurred if available water can not satisfy optimal target, thus, the actual water allocation would equal to pre-regulated target minus probabilistic shortage. For example, under S1, when inflow was high, optimized targets of municipal and agricultural users would be 37.30 × 10⁶ m³ and 1151.81 × 10⁶ m³; correspondingly, the actual allocations would be [33.54, 36.89] × 10⁶ m³ and [1016.67, 1118.06] × 10⁶ m³. Meanwhile, the results present that various scenarios would lead to different optimal targets and water allocations. The highest optimal targets are obtained in progressive scenario due to higher expected target values, while they corresponding to higher water shortages and allocations. The smallest shortages would occur under S8, which can generate more robust result based on synthesizing the risks of S6, S3 and S1. In comparison of total water allocation of four water users, the results illustrate that agriculture is encountering severe water deficit, although it obtain most available water. On the contrary, the municipal water demand would be satisfied greatly due to support of drinking safety policy.

Fig. 3

Optimal targets and total water allocations for various activities under various scenarios when η- level is 1

Figure 4 presents that water shortages for municipality under various scenario when η- levels are 0.7 and 1. The results demonstrate as follows: (a) although water shortages for municipality would be impacted by the random water availabilities, the values of deficit are relative small particular in wet season. (b) shortages are influenced by various η-levels in the constraint of water availability. For instance, water shortage of municipal users in Kuerla county (j = 1) would be 2.74 × 10⁶ m³ at low level, 2.15 × 10⁶ m³ at medium level, and 1.57 × 10⁶ m³ at high level when η- level is 0.7; while they would be 2.48 × 10⁶ m³, 1.92 × 10⁶ m³ l and 1.40 × 10⁶ m³ when η-level is 0.1. (c) the results indicate that shortages would be varied under various scenarios, S6 would lead to highest shortages, while S3 resulting in the lowest one.

Fig. 4

Water shortages for municipality under S1, S3, S6 and S8 when η- levels are 0.7 and 1 (“1” denoted as Kuerle county, “2” denoted as Yanqi county, “3” denoted as Hejing county, “4” denoted as Heshuo county, “5” denoted as Bohu county and “6” denoted as Yuli county)

4.3 Water Quality Management

From the results calculated from distributions of pollution discharge for various activities under S8 when η- level is 1. The results show that agricultural activities is main source of TN and TP pollutants, its discharge amounts have reach [65.4, 68.7] and [64.5, 72.3] % at highest; meanwhile, COD are produced from industry, which have reach [55.4, 57.9] % at highest. Figure 5 presents TP discharges for municipality among various zones under different scenarios when η-level is 1, which indicates that different scenarios would generate varied discharges. The results demonstrate that higher growth rate of economy would bring about higher TP discharge. For example, the excess TP discharge under S6 are higher than those under S1, S3 and S8. Figure 6 shows the excessive COD discharges from industry under diverse scenarios when η- level is 1. Due to various industrial layouts and productivities, the amount of excessive COD discharges is Kuelar > Yanqi > Hejing > Bohu > Yuli > Heshuo.

Fig. 5

Excess TP discharges for municipality among six zones under S1, S3, S6 and S8 when η- level is 1(“Zone 1” denoted as Kuerle county, “Zone 2” denoted as Yanqi county, “Zone 3” denoted as Hejing county, “Zone 4” denoted as Heshuo county, “Zone 5” denoted as Bohu county and “Zone 6” denoted as Yuli county)

Fig. 6

Excess COD discharges for industry under different scenarios and η- levels when water availability is low

4.4 Discussion

Through analysis of results, numbers of discoveries are found as follows: (a) severe water shortage in study region has posed negative effects on drinking security, food safety, and economic development. The reasons of sever deficit conclude objective (e.g., climate changes) and subjective factors (e.g., irrational water plans). Secondly, unreasonable industrial construction and extensive production mode have aggravated the water shortages and pollution issues. Meanwhile, overexploitation for agriculture can increase the water demand, which has exceeded what natural can afford. Besides, inefficiency water usage and backward recycling / retreatment techniques would lead to aggravating pollution issues. Thirdly, the backward fertilization management and irrigation regime can bring about degradation of land resources, leading heavy soil erosion and surface runoff. All above would aggravated nonpoint source pollution (nitrogen and phosphorus nutrients) emission, which may result in environment deterioration. Fourthly, unscientifically risk choice with uncertain importations can affect WAQM plans, which is not beneficial to generate neither adventurous nor conservative decisions. Correspondingly, the suggestions can be given as follows: (a) analysis of various reasons for water crisis in study region would be the first step for decision-making, which is beneficial to identify current policy associated with economy and environment. (b) adjustment of location of industry and transformation of extensive mode can improve efficiencies of water consumption and lessen pollution impacts; meanwhile, treatment technologies should be improved to improve the efficiency of pollutant removal; (c) improvement of irrigation regime and fertilizer regulation can would be developed into a specification, which is beneficial to promote efficiency of agricultural water-usage and lessen discharge. (d) the tradeoff between risk and system benefit should be taken into policy making process, which can support to fortify the robustness of WAQM plans.

5 Conclusions

In this study, a scenario-based interval-stochastic fraticle optimization with Laplace’s criterion (SISFL) method has been proposed for planning WAQM under uncertainties. SISFL has advantages as follows: (a) it can provide linkage between pre-regulated decision and recourse actions when random events occurs, (b) it can tackle imprecise information in objective function / constrain; (c) Laplace’s criterion can reflect the probability of scenario appearance under limited data availability. This study is an attempt to plan a WAQM through the SISFL method, which can effectively handle uncertainties expressed as multiple formats. However, multiple indeterminacies expressed as hybrid presentations would generate more complex systematic relationship in decision making processes; thus, a more robust optimization techniques would be required to handle such challenges, as well as promoting of its practical application to WAQM planning. Moreover, problems involving the efficient allocation of limited water resources and improper discharge of deteriorating water quality have challenged the local authority, which would require more effective methods such as market-trading mechanism to overcome them.

Appendix

1.1 Subscript

j:

District: j = 1 Kuerle, j = 2 Yanqi, j = 3 Hejing, j = 4 Heshuo, j = 5 Bohu and j = 6 Yuli;

m:

Municipal sector: m = 1 Residential use, m = 2 Municipal services;

n:

Agriculture sector: n = 1 Cearl, n = 2 Cotton, n = 3 Oil plants, n = 4 Vegetable, n = 5 Fruit;

i:

Industrial sector: i = 1 Agricultural processing industry, i = 2 oil industry, i = 3 Chemical Industry;

k:

Ecological sector: k = 1 Forest, k = 2 Safe water level of river and reservoir.

h:

Water level: h = 1 Low, h = 2 Medium, h = 3 High;

1.2 Notation

f ₁ :

system benefit without restricted policy (US $).

BM _mj ,BA _nj ,BI _ij ,BE _kj :

net benefit for municipality / agriculture / industry / ecology in district j per volume of water being delivered (US $/ m³).

WLM _mj ,WLA _nj ,WLI _ij ,WLE _kj :

water demand target for municipality / agriculture / industry / ecology in district j (m³).

WLM _mjmax,WLA _njmax,WLI _ijmax :

the maximum water demand target for municipality / agriculture / industry in district j (m³).

POM _mj :

the coefficient of waste water discharge per volume of water being used for municipality in district j.

ua _nj :

The coefficient of water consumption per unit area for agriculture in district j (m³/ ha).

IRA _nj :

the coefficient of pollution discharge of per volume of water being used for agriculture in district j.

IRI _ij :

the coefficient of waste water discharge per volume of water being used for industry in district j.

p _hj :

probability of random water availability ${QR}_{ij}^{\pm }$ under level h(%).

α, η :

recycling ratio of municipality / industry in district j.

SLM _mj ,SLA _nj ,SLI _ij ,SLE _kj :

water shortage for municipality / agriculture / industry / ecology in district j (m³).

CM _mj ,CA _nj ,CI _ij :

recycling cost for municipality / agriculture / industry in district j (US $/ m³).

QK _jh :

water availability from river and underground of district j under probabilityp _hj (m³).

QR _jh :

water flow from river of district j in period t under probabilityp _hj (m³).

E _j :

evaporation and infiltration loss of water from river of district j (m³).

H _j :

normal water requirement of watercourse of district j (m³).

QU _j :

water availability from underground water of district j (m³).

CTM _mj ,CTI _ij :

maximum capacity of recycling for municipality / industry in district j.

dm _COD ,dm _TN ,dm _TP :

the content of COD / TN / TP per volume of waste water for municipality in district j.

di _COD ,di _TN ,di _TP :

the content of COD / TN / TP per volume of waste water for industry in district j.

dar _TP ,dar _TN :

dissolved TN /TP content of runoff corresponding to agricultural activity i in district j (%).

das _TN ,das _TP :

TN / TP content of soil corresponding to agricultural activity i in zone j (kg/t).

${SA}_{nj}^{\pm }$ :

soil loss from agricultural activity i in zone j (t/km2).

DCM _mjh ,DNM _mjh ,DPM _mjh :

maximum allowable COD / TN / TP discharge for municipality in district j with probability p _hj of occurrence under scenario h (ton).

DCI _ijh ,DNI _ijh ,DPI _ijh :

maximum allowable COD / TN / TP discharge for industry in district j with probability p _hj of occurrence under scenario h (ton).

DCA _njh ,DNA _njh ,DPA _njh :

maximum allowable COD / TN / TP discharge for in district j with probability p _hj of occurrence under scenario h (ton).