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# System prepended metadata

title: Selecting the Best Approach to Modeling the Performance of Water Supply System Using the Combination of Rough Set Theory with Multi‐Criteria Decision Making<br>結合粗糙集理論和多準則決策，選擇供水系統效能建模的最佳方法

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# Selecting the Best Approach to Modeling the Performance of Water Supply System Using the Combination of Rough Set Theory with Multi‐Criteria Decision Making<br>結合粗糙集理論和多準則決策，選擇供水系統效能建模的最佳方法

## Abstract 摘要
The purpose of this study is to select the best modeling approach (simulation or optimization) for operation the water supply system using multi-criteria decision-making method. For this purpose, the Geophysical Fluid Dynamics Laboratory-Earth System Models (GFDLESM2M) and the Model for Interdisciplinary Research on Climate-ESM (MIROC-ESM) models were selected to predict the changing trend of the climatic variables of rainfall and temperature, respectively.

本研究的目的是使用多準則決策方法為供水系統的運行選擇最佳建模方法（模擬或優化）。 為此，選擇了地球物理流體動力學實驗室-地球系統模型（GFDLESM2M）和氣候跨學科研究模型-ESM（MIROC-ESM）模型分別預測降雨和溫度氣候變量的變化趨勢。


*ESM : Earth System Model 地球系統模型*

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Then Artificial Neural Network (ANN) model and a decision support system tool named Cropwat were used to simulate water resources and consumption; 
然後使用人工神經網絡（ANN）模型和名為Cropwat的決策支持系統工具模擬水資源和消耗



*Cropwat是聯合國糧食與農業組織的土地與水開發部門，所開發的一個決策資源工具，它可以依據土壤、氣候、和農作物的資料，計算出作物需要的水量，以及灌溉的需求。*

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and to model the behavior of the water supply system, the MODified SYMyld (MODSIM) (as simulator) and the modeling language and optimizer LINGO 18 (as optimizer) were used in the future time period (2026–2039) and the results were compared with the baseline period (1987–2000) for the Idoghmush reservoir (Iran). 

為了模擬供水系統的行為，針對Idoghmush 水庫（伊朗）的資料，使用修改後的 SYMyld (MODSIM)（作為模擬器）與建模語言和優化器 LINGO 18（作為優化器）被用於未來時間段（2026-2039），並將結果與基線期（1987-2000）進行比較 。




Target : Idoghmush reservoir (Iran) 伊朗的一個水庫
Range : 2026–2039 VS 1987–2000
- model the behavior of the water supply system 對供水系統的行為建模
simulator : MODified SYMyld (MODSIM)
optimizer : LINGO 18

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<!--![](https://i.imgur.com/0OWI6j8.png)-->

The results of MODSIM simulation model show that the indexes of reliability, vulnerability, reseiliency and flexibility in the future time period under the RCP2.6 emission scenario compared to the baseline time period decreased by 9%, decreased by 22%, increased by 4%, and decreased by 2%, respectively.
MODSIM仿真模型結果表明，RCP2.6排放情景下未來時間段的可靠性、脆弱性、彈性和靈活性指標較基線時間段分別下降9%，下降22%，上升4% , 下降2%。

![](https://i.imgur.com/mZctVlY.png)

- *RCPs : Representative Concentration Pathways 代表性濃度途徑*
- *RCP2.6: Is a warming-reducing scenario (radiative forcing decreases in 2100)*
*是個暖化減緩的情境（輻射強迫力在2100年呈減少趨勢)*
- *RCP4.5 and RCP6.0 : Belongs to the stable scenario (the change in radiative forcing is relatively stable in 2100)*
*屬於穩定的情境（輻射強迫力的變化在2100年呈較為穩定狀態）*
- *RCP8.5: 溫室氣體高度排放的情境（輻射強迫力在2100年呈持續增加趨勢）*

*Positive radiative forcing means that the Earth receives more energy from the sun's radiation than it emits into space. This net gain in energy would result in warming of the Earth's climate. Conversely, negative radiative forcing means that the Earth radiates more energy into space than it receives from the Sun, causing the surface to cool
正輻射強迫意味著地球接收太陽輻射的能量多於它向太空釋放輻射的能量。這種能量的淨增益將導致地球氣候變暖。相反，負輻射強迫說明地球向太空輻出的能量多於它從太陽接收到的能量，從而導致地表冷卻*

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The results ofthe LINGO 18 optimization model show that the reliability, vulnerability, resiliency and flexibility indexes in the future time period under the RCP2.6 emission scenario compared to the baseline time period decreased by 13%, decreased by 17%, increased by 14% and increased by 3%, respectively.
LINGO 18優化模型結果顯示，RCP2.6排放情景下未來時間段的可靠性、脆弱性、彈性和靈活性指標較基線時間段分別下降13%，下降17%，上升14%，上升3%。
![](https://i.imgur.com/4TrQFCn.png)


Due to the different results obtained from optimization and simulation approaches for the study area, the Multi-Attributive Ideal-Real Comparative Analysis (MAIRCA) multi-criteria decision-making method was used to select a more appropriate approach.
由於研究對象的優化和模擬方法獲得的結果不同，採用多屬性理想-真實比較分析（MAIRCA）多準則決策方法來選擇更合適的方法。

The results show that for water resources management planning, the simulation approach is given priority over the optimization approach due to its characteristics.
研究結果表明，在水資源管理規劃中，由於其特點，模擬方法優先於優化方法。


## 1 introduction 簡介
The climate change phenomenon affects on the climatic parameters, especially temperature and precipitation directly and indirectly, which have been studied in different regions. For example,
氣候變化現象直接或間接影響氣候參數，尤其是溫度和降水，已在不同區域進行了研究。 例如，


- Marengo et al. (2012)
using the Intergovernmental Panel on Climate Change(IPCC) Fourth Assessment Report (AR4) and the application of the HadCM3 climate model, stated that temperature will rise between 4 and 6 °C due to extreme heat in the southern American continent.
利用政府間氣候變化專門委員會（IPCC）第四次評估報告（AR4）和HadCM3氣候模型的應用，指出由於南美洲大陸的極端高溫，溫度將上升4至6°C。

- Zarghami et al. (2016) 
studied the temperature, precipitation, and runoff changes of Yamchi Dam in the time period (2011–2030) using the HadCM3 climate model under the RCP2.6, RCP4.5 and RCP8.5 scenarios. The results showed that the temperature will increase by 0.77 °C and precipitation will decrease by 11 mm. 
利用HadCM3氣候模型研究了RCP2.6、RCP4.5和RCP8.5情景下鹽池壩在時間段（2011-2030）的溫度、降水和徑流變化。 結果表明，氣溫將升高0.77℃，降水量將減少11毫米。

- Rani and Sreekesh (2019) 
evaluated the flow responses under the climate change scenarios in the western Himalayan watershed using the Soil and Water Assessment Tool (SWAT) and cartosat Digital Evaluation Model (DEM). The results showed with the slow disappearance of snow cover as a result of rising temperature, the water availability will be reduced in the second half of this century. 
使用土壤和水評估工具 (SWAT) 和 cartosat 數字評估模型 (DEM) 評估了西喜馬拉雅流域氣候變化情景下的流量響應。 結果表明，隨著溫度升高，積雪逐漸消失，本世紀下半葉可用水量將減少。

- Nabeel and Athar (2020) 
estimated rainfall in wet and dry days for Pakistan using 25 climate models under the fifth IPCC report with RCP4.5 and RCP8.5 scenarios. Generaly, the results showed that under both emission scenarios, the average annual rainfall and the number of wet days in future time periods will decrease compared to the baseline time period, but rainfall volume per a wet day will increase.
根據第五次 IPCC 報告中的 RCP4.5 和 RCP8.5 情景，使用 25 個氣候模型估算了巴基斯坦乾濕天氣的降雨量。 總體而言，結果表明，在兩種排放情景下，未來時間段的年平均降雨量和濕天數與基線時間段相比將減少，但每濕天的降雨量將增加。

- Daba and You (2020) 
investigated the climate change effects on the Awash River with using the CSIRO-MK3-6-0 and MIROC-ESM-CHEM climate models under RCP4.5 and RCP8.5 scenarios. The results showed that temperature, precipitation amount and river discharge volume will decrease in the future years due to climate change.
在 RCP4.5 和 RCP8.5 情景下使用 CSIRO-MK3-6-0 和 MIROC-ESM-CHEM 氣候模型調查了氣候變化對阿瓦什河的影響。 結果表明，由於氣候變化，未來幾年氣溫、降水量和河流流量將減少。

- Mandal et al.(2021) 
investigated the effects of climate change on the hydrology and biomass yield of the tropical river basin under the HadGEM2-ES, MIROC-ESM, NCAR-CCSM4, CSIRO-MK3-6-0 and CESMI-CAM5 climate models.
在 HadGEM2-ES、MIROC-ESM、NCAR-CCSM4、CSIRO-MK3-6-0 和 CESMI-CAM5 氣候模型下調查了氣候變化對熱帶河流域水文和生物量產量的影響。

- Dau et al. (2021) 
assessed the water availability in the Huong river basin in Vietnam under the climate change effects and population growth. The results showed that the future time, temperature and annual rainfall will be increased by 0.2 to 3.5 °C and 1 to 8% respectively and water shortages wouldn’t exist without considering population projections in 2080s.
評估了氣候變化影響和人口增長下越南香江流域的可用水量。 結果表明，如果不考慮2080年代的人口預測，未來時間、溫度和年降雨量將分別增加0.2至3.5°C和1至8%，並且不會存在缺水問題。

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The trend of changing temperature and precipitation parameters in the coming years will change the supply and demand of water resources, which requires the use of management tools based on optimization or simulation approaches with the aim of optimal efficiency of available water resources. For instance:
未來幾年溫度和降水參數變化的趨勢將改變水資源的供需，這需要使用基於優化或模擬方法的管理工具，以實現可用水資源的最佳效率。 例如：

- Shenava and Shourian (2018) 
presented a model for reservoir optimal operation with the aim of enhancing downstream demands supply and flood damage mitigation by coupling MODSIM and Competitive Optimization Algorithm (ICA) for Gotvand dam in Iran.
提出了一個水庫優化運行模型，旨在通過耦合 MODSIM 和伊朗 Gotvand 大壩的競爭優化算法 (ICA) 來增強下游需求供應和減輕洪水損害。

- Sherafatpour et al. (2019) 
presented an integrated hydroeconomic model for allocating agricultural water by coupling the Positive Mathematical Programming (PMP) economic model and MODSIM water allocation planning model for Zayandeh Rood dam in Iran.
通過耦合正數學規劃 (PMP) 經濟模型和伊朗 Zayandeh Rood 大壩的 MODSIM 水資源分配規劃模型，提出了用於分配農業用水的綜合水文經濟模型。

- Fadaeizadeh and Shourian (2019) 
investigated the optimum water resources allocation through quantification of the agricultural demands by combining the MODSIM and the Particle Swarm Optimization (PSO) algorithm in the Atrak River Basin in Iran.
通過結合 MODSIM 和粒子群優化 (PSO) 算法對伊朗阿特拉克河流域的農業需求進行量化，研究了最佳水資源分配。

- Jamshidpey and Shoorian (2021) 
used the Gray Wolf Optimization (GWO) algorithm and the MODSIM to investigate the climate change effects on the Zayandehrud river and agricultural water demand in the Borkhar plain in Iran. The results showed that the cultivation area will decrease due to lack of water resources and agricultural water demand will increase due to rising the temperatures.
使用灰狼優化 (GWO) 算法和 MODSIM 調查氣候變化對伊朗 Borkhar 平原 Zayandehrud 河和農業用水需求的影響。 結果表明，由於水資源缺乏，耕地面積將減少，氣溫升高將增加農業需水量。

- Behboudian and Kerachian (2021) 
evaluated the resiliency of water supply system in Zarrinehrud river basin using MODSIM. The results showed that 40% reduction in agricultural water demand by 2023 will maximize the the water supply system resiliency index.
使用 MODSIM 評估 Zarrinehrud 河流域供水系統的恢復能力。 結果表明，到 2023 年農業用水需求減少 40% 將使供水系統彈性指數最大化。

- Ashrafi et al. (2022) 
evaluated the water and soil stability of Zarrinehrud basin using the SWAT and the MODSIM models under climate change. The results showed that the best management scenario was to allocate water to Lake Urmia from new sources, rehabilitate irrigation and drainage networks, and modify the cultivation pattern.
使用氣候變化下的 SWAT 和 MODSIM 模型評估了 Zarrinehrud 流域的水土穩定性。 結果表明，最佳管理方案是從新水源向烏爾米亞湖分配水資源，修復灌溉和排水網絡，並調整耕作方式。

- Pourmoghim et al. (2022) 
used the SWAT-MODSIM coupled model to evaluate the resiliency and improve the condition of Lake Urmia against droughts resulting from human activity.
使用 SWAT-MODSIM 耦合模型來評估 Urmia 湖的恢復能力並改善其應對人類活動造成的干旱的條件。


- Ashofteh et al. (2017)
used reservoir performance indexes to evaluate the performance of the Gharnaghu multipurpose dam with the aim of irrigating agricultural lands under climate change conditions. The results showed that the time reliability, vulnerability, resiliency and flexibility indexes in the future period will be decreased by 18%, increased by 150%, decreased by 33% and decreased by 47%, respectively compared to the baseline time period.
使用水庫性能指標來評估 Gharnaghu 多功能大壩的性能，目的是在氣候變化條件下灌溉農田。 結果顯示，未來一段時間的可靠性、脆弱性、彈性和靈活性指標與基線時間段相比分別下降18%、上升150%、下降33%和下降47%。


- Ashofteh et al. (2019)
evaluated the best adaptation scenario with climate change effects on the water resources and consumptions in Gharnaghu water supply system with eight reservoir performance indexes including time reliability, volumetric reliability, availability, supply to demand, and sum of squared deficits.
評估了氣候變化對 Gharnaghu 供水系統水資源和消耗影響的最佳適應情景，八個水庫性能指標包括時間可靠性、體積可靠性、可用性、供需關係和平方和赤字。


- Golfam et al. (2019b) 
used Analytic Hierarchy Process (AHP) and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) MCDM methods to determine the best adaptation scenario with climate change in the agricultural sector in Gharnaghu basin during the period (2040–2069). The results showed that the best alternative using the AHP model was reducing agricultural water by 25% with weight equal to 0.335, while the TOPSIS model showed that the best alternative was reducing demand by 15% with a distance of 0.20805 from the ideal alternative.
使用層次分析法 (AHP) 和通過與理想解決方案相似的偏好排序技術 (TOPSIS) MCDM 方法來確定 Gharnaghu 盆地農業部門在此期間 (2040-2069) 氣候變化的最佳適應情景。 結果表明，使用 AHP 模型的最佳替代方案是將農業用水減少 25%，權重等於 0.335，而 TOPSIS 模型顯示，最佳替代方案是將需求減少 15%，與理想替代方案的距離為 0.20805。

- Golfam et al. (2019a) 
evaluated the agricultural water supply under five climate change adaptation scenarios for the period (2040–2069) using two methods of multi-criteria decision making: the VIKOR (Vlse Kriterijumsk Optimizacija Kompromisno Resenje) and Fuzzy Order Weighted Average (FOWA). The results showed that using both VIKOR and FOWA methods, the fifth altenative (reducing water consumption in the agricultural sector by 25%), was the best scenario for adaptation to climate change.
使用兩種多標準決策方法：VIKOR (Vlse Kriterijumsk Optimizacija Kompromisno Resenje) 和模糊階加權平均數 (FOWA)，評估了 2040-2069 年期間五種氣候變化適應情景下的農業供水情況。 結果表明，同時使用 VIKOR 和 FOWA 方法的第五個備選方案（將農業部門的用水量減少 25%）是適應氣候變化的最佳方案。


- Ashofteh et al. (2020) 
presented a methodology in order to select the best alternative for river-water transferring based on the Interactive Multi-Criteria Decision-Making (TODIM) method in the Khodaafarin irrigation network, Iran. The results of AHP weighting method showed the investment costs was the most effective criterion and results of the TODIM method revealed that water transferring through earthen canal to a pumping lift station was the best option.
提出了一種方法，以便在伊朗 Khodaafarin 灌溉網絡中基於交互式多標準決策 (TODIM) 方法選擇河水調水的最佳替代方案。 AHP 加權方法的結果表明投資成本是最有效的標準，TODIM 方法的結果表明通過土渠將水輸送到抽水站是最佳選擇。


- Theochari et al.(2021) 
developed a methodology in Geographic Information System (GIS) using a MCDM approach with several spatial criteria to propose suitable locations for hydrometeorological and hydrometric station network in the Sarantapotamos river basin in the western part of the Attica Region, Greece.
使用具有多個空間標準的 MCDM 方法開發了地理信息系統 (GIS) 方法，為希臘阿提卡地區西部 Sarantapotamos 河流域的水文氣象和水文站網絡提出合適的位置。

- Abdi-Dehkordi et al. (2021)
evaluated the water resources systems in the Karun basin using seven indexes of quantitative water stress, water quality, environmental water stress, agricultural revenue, agricultural/industrial water productivity, irrigation system cost, and water transfer revenue.
(2021)，利用定量水分壓力、水質、環境水分壓力、農業收入、農業/工業用水生產率、灌溉系統成本和調水收入七項指標評估了卡倫流域的水資源系統。


- Velasquez and Hester (2013)
According to the multi-dimensional nature of water resources management due to the presence of decision-makers and policy-makers and their multiple considerations on one hand and presence of stakeholders with different goals on the other hand, selection a common strategy in which all groups are satisfied with maximum benefit is very difficult. One of the most widely used tools to overcome this challenge is multi-criteria decision-making methods. These methods have evolved to adapt to different types of programs and issues, so that small changes in these methods have created a variety of branches. A combination of different concepts along with methods that are in their original form, can be used more effectively.
根據水資源管理的多維性，一方面存在決策者和政策制定者及其多重考慮，另一方面存在目標不同的利益相關者，選擇一個共同的戰略，其中所有群體 對利益最大化感到滿意是很困難的。 克服這一挑戰的最廣泛使用的工具之一是多標準決策方法。 這些方法已經進化以適應不同類型的程序和問題，因此這些方法的微小變化創造了各種分支。 可以更有效地使用不同概念及其原始形式的方法的組合。

- Kuang et al. (2014)
Various MCDM methods have been used in water resources management, including evaluating and prioritizing water conservation strategies using the Preference Ranking Organization Method for Enrichment Evaluation (PROMITHEE II) method to assess alternatives based on gray numbers for Ontario, Canada.
各種 MCDM 方法已用於水資源管理，包括使用偏好排名組織方法進行濃縮評估 (PROMITHEE II) 方法評估和優先考慮水資源保護策略，以評估加拿大安大略省基於灰數的備選方案


- He et al. (2020)
Identification of optimal groundwater remediation strategies for a naphthalene-contaminated site by using the developed PROMETHEE-TOPSIS method 
使用開發的 PROMETHEE-TOPSIS 方法確定萘污染場地的最佳地下水修復策略
![](https://i.imgur.com/PNA2aXY.png)
![](https://i.imgur.com/ID4IHJ9.png)



- Hadian et al. (2022)
Investigation of flood susceptibility in temperate Mediterranean climate using MAIRCA method.
使用 MAIRCA 方法調查溫帶地中海氣候的洪水敏感性。

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The purpose of present study is to select the best approach between simulation and optimization approaches for water resources management in the future time interval using MAIRCA decision-making method. Given that the decision-making to choose the appropriate approach is for the future time period, first climatic processing of climatic parameters of temperature and precipitation must be done.
本研究的目的是使用 MAIRCA 決策方法在水資源管理模擬和優化方法之間選擇未來時間間隔內的最佳方法。 鑑於選擇合適方法的決策是針對未來時間段的，首先必須對溫度和降水等氣候參數進行氣候處理。


In this study, the proper climatic models for temperature and precipitation were GFDLESM2M and MIROC-ESM based on the results of their evaluation. In Sect. 2, the method and results of evaluating climate models will be described in detail.
在本研究中，根據評估結果，適合溫度和降水的氣候模型是 GFDLESM2M 和 MIROC-ESM。在第二章，將詳細描述評估氣候模型的方法和結果。


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## 2 Materials and Methods 材料和方法
Climate models were first evaluated from the Fifth Assessment Report (AR5) of the IPCC,and after selecting the best model, temperature and precipitation scenarios were created under the emission scenarios of RCP2.6 and RCP8.5. Then the discharge of the dam reservoir in the future years was estimated using the ANN conceptual model and the amount of irrigation water demand in the agricultural sector was estimated with the Cropwat model.
In the next step, the water supply system and demand status will be simulated using the MODSIM model, and optimized with LINGO 18 model. Reservoir performance indexes will be calculated using the results of both models. In the last step, the MAIRCA decisionmaking method will be used to determine the most appropriate approach for policy-making of reservoir operation based on the performance indexes of each of them. Also the AHP method developed with Rough Set Theory (RST) called Rough-AHP will be used to weight the criteria. The flowchart of the present study is shown in Fig. 1.
氣候模型首先根據IPCC第五次評估報告（AR5）進行評​​估，在選擇最佳模型後，在RCP2.6和RCP8.5排放情景下創建溫度和降水情景。然後使用 ANN 概念模型估算未來幾年大壩水庫的流量，並使用 Cropwat 模型估算農業部門的灌溉需水量。
下一步將使用MODSIM模型模擬供水系統和需求狀態，並使用LINGO 18模型進行優化。將使用兩個模型的結果計算水庫性能指標。在最後一步中，將使用 MAIRCA 決策方法，根據每個水庫的性能指標確定最合適的水庫運行決策方法。此外，使用粗糙集理論 (RST) 開發的 AHP 方法(稱為 Rough-AHP) 將用於衡量標準。本研究的流程圖如圖 1 所示。

![](https://i.imgur.com/zxZ39hm.png)

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###  2.1<br>Step 1: Selection the Climate Models, Climatic Scenario Generation for Temperature and Rainfall<br>第 1 步：選擇氣候模型，生成溫度和降雨的氣候情景
#### 2.1.1<br>Evaluation of Climate Models<br>氣候模型評估
Suitable climatic model was selected for the study area among the 28 models in the fifth IPCC report under two scenarios RCP2.6 and RCP8.5 using statistical criteria including correlation coefficients(r\), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Nash–Sutcliffe Efficiency (NSE) coefficient.The results of coefficients are presented in Table 1.

在IPCC第五次報告的28個模型中為研究區域選擇了在RCP2.6和RCP8.5兩種情景下合適的氣候模型。，其中使用了包括相關係數（r）、均方根誤差（RMSE）、平均絕對誤差（MAE)和 纳什效率係數 (NSE) 在內的統計標準，係數結果如表 1 所示。

![](https://i.imgur.com/wWXQ29j.png)


As shown in Table 1, the GFDL-ESM2M climate model with r, RMSE, MAE, and NSE
equal to 91%, 8.1 mm, 6.0 mm, and 0.8, respectively have the best performance among other climate models for rainfall model. Also, MIROC-ESM climate model with r, RMSE, MAE, and NSE equal to 98%, 2.9 °C, 2.1 °C, and 0.9 has the best performance among other climate models for temperature.

如表 1 所示，r、RMSE、MAE 和 NSE 分別等於 91%、8.1 mm、6.0 mm 和 0.8 的 GFDL-ESM2M 氣候模型在降雨模型的其他氣候模型中表現最好。 此外，r、RMSE、MAE 和 NSE 等於 98%、2.9 °C、2.1 °C 和 0.9 的 MIROC-ESM 氣候模型在其他氣候模型中的溫度性能最好。

![](https://i.imgur.com/uv3KmY4.png)

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#### 2.1.2<br>Generating Climatic Scenarios of Temperature and Rainfall<br>生成溫度和降雨的氣候情景
After selecting the appropriate climatic model, rainfall and temperature values are extracted for historical data under two RCP2.6 and RCP8.5 climatic scenarios and climatic data are extracted and downscaled using Geographic Information System (ARCGIS) software. The results are shown in Fig. 2. According to Fig. 2a, the amount of rainfall in the future (2026–2039) will increase by 17 and 25% compared to the base (1987–2000) under the RCP2.6 and RCP8.5 scenarios, respectively. This increase is especially in spring and autumn and in April and October, when the rainfall is usually higher than other months. As shown in Fig. 2b, the temperature will increase by 1.5 and 1.3 °C relative to the base in the future under the RCP2.6 and RCP8.5 scenarios. The results of climate change scenarios are used to calculate reservoir discharge and water consumption for the agricultural sector in the next step.
選擇合適的氣候模型後，提取RCP2.6和RCP8.5兩種氣候情景下降雨量和溫度值的歷史數據，並使用地理信息系統（ARCGIS）軟件提取和縮小氣候數據。 結果如圖2所示。根據圖2a，在RCP2.6和RCP8下，未來（2026-2039）的降雨量將比基準（1987-2000）分別增加17和25％。 這種增加尤其在春季和秋季以及 4 月和 10 月，這時的降雨量通常高於其他月份。 如圖2b所示，RCP2.6和RCP8.5情景下，未來溫度相對基準分別升高1.5和1.3℃。 氣候變化情景的結果用於計算下一步農業部門的水庫流量和用水量。
![](https://i.imgur.com/ahvPZu9.png)
![](https://i.imgur.com/hlIQgdP.png)

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### 2.2<br>Step 2: Calculation of Reservoir Discharge by ANN and Agricultural Water Consumption by Cropwat<br>第 2 步：通過 ANN 計算水庫流量和通過 Cropwat 計算農業耗水量
#### 2.2.1<br>Calculate the Resevoir Discharge<br>計算水庫流量
In this research, the ANN conceptual model was used to estimate the inflow to the reservoir in the future time. For this purpose, long-term time series of temperature and precipitation were included in the ANN model. First, two middle layers and one output layer with five hidden neurons were considered. The function type for the middle and output layers are both hyperbolic nonlinear. In this study, MATLAB (2022) was used to build the ANN model. The features of the ANN model are described below. The 75% of the observed data was used to train the model. Number of layers is equal to 4 and number of neurons in the middle layer is 5–20. The middle and output layer transfer function arr Sigmoid Function- Hyperbolic Functions. Number of Repetitions per Run is 1000.
ANN performance in calibration and validation with RMSE and MAE equal to 7.9 and 5.2 m3/s and NSE equal to 0.8 and r equal to 90%, respectively, showed that this model can be reliable for simulating reservoir discharge in the future. Because RMSE is 7.9 m3 / s and MAE is 5.2 m3 / s, which shows that the data simulated by the model are slightly different from the amount of each input data, and also the NASH coefficient of 0.8 indicates the good performance of the model in the simulation. Next, the future runoff time series for the RCP2.6 and RCP8.5 scenarios are obtained, as shown in Fig. 3. According to Fig. 3, the inflow to the reservoir will increase by 12 and 13% in the future compared to the base under the RCP2.6 and RCP8.5, respectively. The reason for this increase in runoff can be considered a significant increase in precipitation obtained from the GFDL-ESM2M model in the future period under both emission scenarios in October compared to the base period, so that the average precipitation under both emission scenarios is about 2.7 times the precipitation in October of the base time period.
在這項研究中，ANN 概念模型被用來估計未來時間到水庫的流入量。為此，ANN 模型中包含溫度和降水的長期時間序列。首先，考慮了兩個中間層和一個具有五個隱藏神經元的輸出層。中間層和輸出層的函數類型都是雙曲非線性的。在本研究中，使用 MATLAB (2022) 構建 ANN 模型。 ANN 模型的特點如下所述。 75% 的觀測數據用於訓練模型。層數等於 4，中間層的神經元數為 5-20。中間層和輸出層傳遞函數為 Sigmoid Function- Hyperbolic Functions。每次運行的重複次數為 1000。
RMSE 和 MAE 分別等於 7.9 和 5.2 m3/s，NSE 等於 0.8 和 r 等於 90% 時，ANN 在校準和驗證中的性能表明該模型可以可靠地模擬未來的水庫流量。因為RMSE為7.9 m3/s，MAE為5.2 m3/s，說明模型模擬的數據與每次輸入數據量略有不同，同時NASH係數為0.8說明模型在模擬。接下來，得到RCP2.6和RCP8.5情景下的未來徑流時間序列，如圖3所示。根據圖3，未來入庫流量將比上年增加12%和13%。 RCP2.6和RCP8.5下的base分別。徑流增加的原因可以認為是 GFDL-ESM2M 模型得到的 10 月份兩種排放情景下的未來時期降水量較基期顯著增加，因此兩種排放情景下的平均降水量約為 基準時間段 10 月降水量的2.7倍。
![](https://i.imgur.com/itOYEl3.png)


#### 2.2.2<br>Estimate the Amount of Water Consumption in the Agricultural Sector<br>估算農業用水量
The Cropwat model (developed by FAO (2009)) was used to calculate the water consumption of the agricultural sector in the future. Temperature and rainfall time series were used as input to the Cropwat model for the future time period. The rate of evapotranspiration of the reference crop ( ET0 ) was calculated by the Monteith-Penman-FAO equation based on climatic data including minimum and maximum temperature, relative humidity, radiation duration, and wind speed.
Changes in water demand in the future period under emission scenarios compared to the base period are presented in Fig. 4. According to Fig. 4, irrigation water demand in the agricultural sector will increase by about 15% in the future over the base period under both RCP2.6 and RCP8.5 scenarios.
Cropwat 模型（由糧農組織開發（2009 年））用於計算未來農業部門的用水量。 未來時間段的溫度和降雨時間序列被用作 Cropwat 模型的輸入。 參考作物的蒸散率 (ET0) 是根據氣候數據（包括最低和最高溫度、相對濕度、輻射持續時間和風速）通過 Monteith-Penman-FAO 方程計算得出的。
排放情景下未來時期需水量與基期相比的變化如圖4所示。根據圖4，在RCP2.6 和 RCP8.5 場景之下，未來農業灌溉需水量將比基期增加15%左右。
![](https://i.imgur.com/8fIen8h.png)
The results of ANN and Cropwat models are used to simulate and optimize the water supplying system.
ANN 和 Cropwat 模型的結果用於模擬和優化供水系統。

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### 2.3<br>Step 3: Simulation and Optimization of Water Supplying System Using MODSIM and LINGO 18 Respectively, and Calculation the Water Supply System Performance Indexes<br>分別使用MODSIM和LINGO 18對供水系統進行仿真優化，計算供水系統性能指標
#### 2.3.1<br>MODSIM Simulation Model<br>MODSIM 仿真模型
The MODSIM model is a decision support system for operation planning and water resource management developed by Colorado State University. This model is designed in such a way that managers and decision-makers are able to fully understand the complex and coordinated actions required in river management, assess the effects of hydrology, economics, environment, etc. In addition, they will be able to take the necessary steps for long-term executive planning, drought planning, agricultural and environment water rights analysis (Labadie 2006). The newer versions of MODSIM, developed with the NET Framework, also allow the user to customize MODSIM for each component, including input and output reports, and also it provides access another models that run concurrently with MODSIM without need to change the original source.

MODSIM模型是科羅拉多州立大學開發的用於運行規劃和水資源管理的決策支持系統。 該模型的設計方式使管理者和決策者能夠充分理解河流管理所需的複雜和協調行動，評估水文、經濟、環境等方面的影響。此外，他們將能夠採取長期執行規劃、乾旱規劃、農業和環境水權分析的必要步驟（Labadie 2006）。 使用 NET Framework 開發的較新版本的 MODSIM 還允許用戶為每個組件自定義 MODSIM，包括輸入和輸出報告，並且它還提供訪問與 MODSIM 同時運行的其他模型，而無需更改原始碼。

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The Network Flow Programming (NFP) method is used to determine the amount of water allocation between the demand nodes and water resources in MODSIM, and the water allocation between the consumption nodes is solved repeatedly at each time step to minimize the cost of the network flow. For this purpose, the problem is solved by Lagrangian relaxation algorithm (Bertsekas and Tseng 1994) and the optimization parameters of Ult, Llt, bit, and network flow are repeated until convergence is achieved. The general equation in the NFP method is obtained at each time step by sequentially solving the network flow optimization from Eqs. (1) to (3) (Labadie 2006):

在MODSIM中採用網絡流規劃（NFP）方法確定需求節點與水資源之間的水量分配，並在每個時間步重複求解消耗節點之間的水量分配，以最小化網絡流量成本 . 為此，通過拉格朗日鬆弛算法（Bertsekas 和 Tseng 1994）解決問題，並重複 Ult、Llt、bit 和網絡流量的優化參數，直到實現收斂。 NFP 方法中的一般方程是在每個時間步通過從 Eqs 順序求解網絡流優化而獲得的。 (1) 至 (3)（Labadie 2006）：

![](https://i.imgur.com/CM9jVSz.png)
![](https://i.imgur.com/TU69SnL.png)

where, A = total network links; N = set of network nodes; Oi = set of output links from node i; Ii = all input flow links to node i; ql = flow at link l; Cl = flow cost coefficient in link l;Llt = low flow limit at junction l; Ult = high flow limit at junction l, and bit = profit (positive) and loss (negative) of node i at time t.
其中，A = 網絡鏈接總數； N = 一組網絡節點； Oi = 來自節點 i 的一組輸出鏈接； Ii = 到節點 i 的所有輸入流鏈接； ql = 鏈路 l 處的流量； Cl = 鏈路 l 中的流量成本係數；Llt = 連接點 l 的低流量限制； Ult = 連接點 l 的高流量限制，和 bit = 節點 i 在時間 t 的利潤（正）和損失（負）。


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#### 2.3.2<br>LINGO 18 Optimization Model<br>LINGO 18 優化模型
The LINGO 18 optimization model is a powerful tool for solving linear and nonlinear models. This model has various capabilities, including very advanced solvers with high model speed, simple expression of the mathematical model in the software environment, high power of model analysis, having various mathematical, statistical and probabilistic functions. In the present study, the objective function in the LINGO 18 optimization model is to minimize deficiencies. The objective function is to minimize the relative deficit in water allocation to the agricultural sector per month. The objective function and the corresponding constraints are considered according to Eqs. (4) to (8):

LINGO 18 優化模型是求解線性和非線性模型的強大工具。 該模型具有多種功能，包括具有高模型速度的非常先進的求解器，軟件環境中數學模型的簡單表達，強大的模型分析能力，具有各種數學、統計和概率函數。 在本研究中，LINGO 18 優化模型中的目標函數是最小化缺陷。 目標函數是最小化每月分配給農業部門的相對缺水。 根據方程式考慮目標函數和相應的約束條件。 (4)至(8)：
![](https://i.imgur.com/R5pJ0BB.png)

which, Def = objective function; Dt = water demand per month; Davet = average water demand per month; Ret = monthly release rate; St = the amount of reservoir storage at the beginning and end of period t; St + 1 = the amount of reservoir storage at the beginning and end of period t + 1; Qt = volume of inflow to the dam reservoir per month; EVt = evaporation rate from dam lake; Smax = total reservoir capacity; Smin = dead volume of the dam; SPt = amount of spill volume from the dam; n = length of operation period; a and b = constants obtained from the surface-volume curve of the reservoir, such that A = a St + b (A is the reservoir lake surface) (a and b are 0.0568 and 0.7855, respectively).
其中，Def = 目標函數； Dt = 每月需水量； Davet = 每月平均需水量； Ret = 月釋放率； St = t期初、期末水庫蓄水量； St+1=t+1期初、期末水庫蓄水量； Qt = 每月流入大壩水庫的流量； EVt = 大壩湖的蒸發率； Smax = 總水庫容量； Smin = 大壩的死體積； SPt = 來自大壩的溢流量； n = 運營期的長度； a、b=從水庫表容曲線求得的常數，A=aSt+b（A為水庫湖面）（a、b分別為0.0568、0.7855）。

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#### 2.3.3<br>Water Supply Performance Indexes<br>供水績效指標
In order to evaluate the efficiency of water supply system, it is necessary to measure it by reservoir performance indexes. By monitoring the performance of the reservoir through the performance index, the efficiency of the reservoir in the face of climate change can be investigated. In this study, four indexes of reliability, resiliency, vulnerability and flexibility will be used, the first three of them were first proposed by Hashimoto et al. (1982) to evaluate the performance of reservoirs. Due to the lack of a similar trend in the changes of the first three indexes, Loucks (1997) presented the flexibility index, which is a combination of the above three indexes. The characteristics of the four indexes are given in Table 2. In this Table, lack of water supply means failure system, and water supply indicates the success of the system. As can be seen from the concept of Table 2, high reliability, high resiliency, low vulnerability, and consequently high flexibility is considered desirability for any water supply system project.
為了評價供水系統的效率，需要用水庫效能指標來衡量。通過效能指標監測水庫的效能，可以研究水庫在面對氣候變化時的效率。在這項研究中，將使用可靠性、彈性、脆弱性和靈活性四個指標，其中前三個指標首先由 Hashimoto (1982)等人提出，評估水庫的性能。由於前三個指標的變化趨勢不相似，Loucks（1997）提出了彈性指標，它是上述三個指標的組合。表2給出了四個指標的特徵。在該表中，供水不足表示系統失敗，供水表示系統成功。從表 2 的概念可以看出，高可靠性、高彈性、低脆弱性以及由此帶來的高靈活性被認為是任何供水系統項目的可取之處。

![](https://i.imgur.com/wlVC9YZ.png)
where, REL = temporal reliability (%); RES = resiliency index (%); VUL = vulnerability index (%); FEL = flexibility index; N = the number of months in which the release from the reservoir is equal to or greater than the demand at the downstream of the reservoir; N’ = number of months in which release from reservoir is less than demand at the downstream of the reservoir; T = operation interval; Dt And Dt+1 = the volume of demand at the downstream of the reservoir at time t and t + 1, respectively; Davet = average amount of demand in the whole operation period; Ret and Ret+1 = release volume from the reservoir at time t and t + 1, respectively; Ct = counting function (such that it considers releases equal to or greater than demand over the entire operating period); Ct’ = counting function (in such a way that it considers the release of less than demand in the whole period of operation); N’’ = number of failures after successes; deficitt = amount of water shortage
其中，REL = 時間可靠性 (%)； RES = 彈性指數 (%)； VUL = 脆弱性指數（%）； FEL = 柔韌性指數； N = 水庫洩水量等於或大於水庫下游需求量的月數； N’ = 水庫洩水量小於水庫下游需求量的月數； T = 操作間隔； Dt 和 Dt+1 = 時間 t 和 t + 1 時水庫下游的需求量； Davet = 整個運營期的平均需求量； Ret 和 Ret+1 = 分別在時間 t 和 t + 1 從儲層釋放體積； Ct = 計數函數（這樣它認為釋放量等於或大於整個運行期間的需求量）； Ct’ = 計數功能（考慮在整個運行期間釋放未滿足需求的功能）； N'' = 成功後的失敗次數； deficitt = 缺水量

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### 2.4 Step 4: Selection the Best Modeling Approach Using MAIRCA MCDM Method<br>使用 MAIRCA MCDM 方法選擇最佳建模方法
#### 2.4.1 Multi‐Criteria Decision‐Making Method<br>多準則決策方法
In order to adopt a proper operation policy and allocation of water resources in accordance with the status of the water supply system in the future, the most effective approach should be selected from simulation and optimization models according to different criteria.
為了根據未來供水系統的狀況採取適當的運行策略和水資源分配，應根據不同的標準從模擬和優化模型中選擇最有效的方法。

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- (a) Weighing the criteria set using Rough-AHP method
The first step in multi-criteria decision-making methods is to determine the appropriate criteria to the problem being decided and to choose the weighting method. In the present study, the Rough-AHP method is used to weight the criteria. 
AHP weighting and ranking method is one of the most well-known multi-criteria decision-making methods that has been used by experts in various fields (Saaty 1980). The steps of the AHP method include the following steps:
(a)使用 Rough-AHP 方法對標準集進行權衡
多準則決策方法的第一步是確定適合被決策問題的準則並選擇加權方法。 在本研究中，使用 Rough-AHP 方法對標准進行加權。
AHP 加權和排序方法是最著名的多標準決策方法之一，已被各個領域的專家使用（Saaty 1980）。 AHP方法的步驟包括以下步驟：

Step 1: The first step is to identify the problem accurately and model the goal, criteria, and alternatives levels.
第 1 步：第一步是準確識別問題並對目標、標準和備選方案級別進行建模。

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Step 2: In this step, pairwise comparison matrixes are provided for water resources management experts to determine the preference of the two criterias. Pairwise comparison is done according to the preference of each criterion over the other criterion by assigning a number from 1 (indicates equal importance) to 9 (indicating absolute importance). The expression of the preference of the criteria is given in the form of numbers in Table 2.


步驟2：在此步驟中，為水資源管理專家提供成對比較矩陣以確定兩個標準的偏好。 成對比較是根據每個標準相對於另一個標準的偏好進行的，方法是分配一個從 1（表示同等重要性）到 9（表示絕對重要性）的數字。 表2以數字形式給出了準則偏好的表達。

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When the problem has m alternatives and n criterias, a n × n pairwise comparison matrix must be created and n matrixes of m × m pairwise comparison must be formed. The pairwise comparison matrix is constructed according to Eq. (9):


當問題有m個備選方案和n個標準時，必須創建一個n×n的兩兩比較矩陣，並組成n個m×m兩兩比較的矩陣。 成對比較矩陣是根據等式構建的。 (9):
![](https://i.imgur.com/7TmekVD.png)
where aij = the preference of element i over element j.
其中 aij = 元素 i 對元素 j 的偏好。

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The main diameter of the pairwise comparison matrix is 1, because the preference of each criterion over itself is equal to 1. Upper-triangular elements of the pairwise comparison matrix are scored by the experts, and lower-triangular elements are obtained using the reciprocal principle.
The numberical expression for preferences presented in Table 3.
成對比較矩陣的主倍率為1，因為每個標準對自身的偏好度等於1。成對比較矩陣的上三角元素由專家打分，下三角元素使用倒數原理得到。表 3 中顯示了偏好的數字表達式。
![](https://i.imgur.com/dWIcywk.png)

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Step 3: In this step, the relative weight of each alternative regarding to each criterion is calculated using various methods such as geometric mean, arithmetic mean, linear mean. To calculate the weight vector in AHP method, the geometric mean method is used in which, first the geometric mean of the elements of each row is calculated and then the obtained vector is normalized and the weight vector is obtained.

步驟3：在這一步中，使用幾何平均數、算術平均數、線性平均數等多種方法計算每個備選方案相對於每個標準的相對權重。層次分析法計算權向量採用幾何平均法，首先計算每行元素的幾何平均，然後對得到的向量進行歸一化，得到加權向量。

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Step 4: In this step, the Decision Cosistency Rate (DCR) is calculated, which indicates the confidence of experts in judging the preference of the criteria. The decision cosistency rate is the result of dividing the Decision Consistency Index (DCI) by Decision Random Incomatibility Index (DRI) and is obtained according to Eqs. (10) and (11).


第四步：在這一步中，計算決策一致性率（DCR），它表示專家判斷標準偏好的信心。決策一致性率是決策一致性指數 (DCI) 除以決策隨機不相容指數 (DRI) 的結果，並根據等式獲得。 (10) 和 (11)。

![](https://i.imgur.com/ddlohan.png)

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(b) Rough set theory
(b) 粗糙集理論
Rough set theory, proposed by Pawlak (1982), is a mathematical tool that is able to deal with subjective and imprecise concepts. The most important feature of RST is to check the mental information of experts without the need for any additional hypotheses and information. In other words, in this theory the ambiguity in the data is not expressed through the membership function, but is expressed using a boundary region of the set. The boundary region is the difference between the upper approximation and the lower approximation, which can neither be rejected as a member of the goal nor can it be rejected. Figure 5 shows the concept of RST.

粗糙集理論由 Pawlak (1982) 提出，是一種能夠處理主觀和不精確概念的數學工具。 RST 最重要的特點是在不需要任何額外假設和信息的情況下檢查專家的心理信息。 也就是說，在該理論中，數據中的模糊性不是通過隸屬函數來表達的，而是通過集合的邊界區域來表達的。 邊界區域是upper approximation和lower approximation之間的差異，既不能被拒絕為目標的成員，也不能被拒絕。 圖 5 顯示了 RST 的概念。
![](https://i.imgur.com/NOUxtt5.png)



Another important feature of RST is the calculation of the degree of satisfaction from the judgment of the experts, which also indicates the need for change in judgments. The following are the steps of the Rough-AHP weighting method:

RST的另一個重要特點是從專家的判斷中計算出滿意程度，這也說明了判斷需要改變。 以下是Rough-AHP加權法的步驟：

---
• Creating a hierarchical analysis model
創建層次分析模型

---
• Performing pairwise comparisons between criteria according to the goal level
根據目標級別在標準之間進行成對比較

---
• Controling the DRI
控制 DRI

---
• Aggrigation the expert’s opinions and converting them to the rough set at the group level through calculating upper and lower approximations
聚合專家的意見並通過計算上下近似值將其轉換為組級別的粗糙集

In this step, to calculate the upper and lower approximations, first the geometric mean of the same elements is calculated from the pairwise comparison matrices and then the upper and lower approximation sets are determined based on Eqs. (12) and (13) respectively.


在這一步中，為了計算上下近似值，首先從兩兩比較矩陣中計算出相同元素的幾何平均數，然後根據式（1）確定上下近似集。 (12) 和 (13) 分別。
![](https://i.imgur.com/SM13Q8Z.png)
![](https://i.imgur.com/t1lE1xl.png)


The most important reason for using geometric mean to aggregate expert opinions is that it preserves the reciprocal property of the pairwise comparison matrix without violating the Pareto principle (Forman and Peniwati 1998).
使用幾何平均數來聚合專家意見的最重要原因是它保留了成對比較矩陣的倒數性質，而不違反帕累托原則（Forman 和 Peniwati 1998）。

---
- Calculate the upper and lower limits and the rough number of each element 
The upper limit, the lower limit, and the rough number of each element are calculated based on Eqs. (14) to (16), respectively:
計算每個元素的上下限和粗略量
每個元素的上限、下限和粗略數是根據Eqs(14)至(16)計算的，分別為：
![](https://i.imgur.com/M45aywa.png)

---
- Create an aggrigated pairwise comparison matrix based on rough numbers
基於粗略數創建聚合成對比較矩陣

---
- Calculate the degree of satisfaction from judgments
根據判斷計算滿意度

In this step, the degree of satisfaction with the judgments is calculated using the convex linear composition (Ayağ and Özdemir 2009) according to Eq. (17):
在此步驟中，使用凸線性組合（Ayağ 和 Özdemir 2009）根據等式計算對判斷的滿意度。 (17):
![](https://i.imgur.com/Naz8fS1.png)

---
Criteria for the present study include: (1) model preparation cost, (2) operation policy approach, (3) the quality of the solutions of each model and (4) the time of construction and execution of models. The criterion of model preparation cost is negative and other criteria are positive. A negative criterion is a criterion that is better to be the minimum, while a positive criterion is better to be the maximum.
本研究的準則包括：(1) 模型準備成本，(2) 操作策略方法，(3) 每個模型解決方案的質量以及 (4) 模型構建和執行的時間。 模型製備成本的標準為負，其他標準為正。 負標準是最好是最小值的標準，而正標準是最好是最大值的標準。

---
\(c\) MAIRCA MCDM Method
\(c\) MAIRCA MCDM 方法
In the present study, the MAIRCA ranking method is used to determine the priority of the appropriate approach to water resources management. The MAIRCA method was first developed by Pamacur et al. (2014). The main idea of this method is to determine the gap between the ideal and empirical weights, and finally the sum of the gaps for each criterion determines the final gap for each alternative. In this method, the alternative with the best rating will be the one that has the minimum distance from the final gap. In other words, the alternative with the lowest value for the final gap has the value closest to the criterion ideal weight. The following are the steps of the MAIRCA method:

在本研究中，MAIRCA 排序方法用於確定適當的水資源管理方法的優先級。 MAIRCA 方法首先由 Pamacur 等人開發。 (2014)。 該方法的主要思想是確定理想權重和經驗權重之間的差距，最後每個標準的差距之和決定了每個備選方案的最終差距。 在這種方法中，具有最佳評級的備選方案將是與最終差距距離最小的備選方案。 換句話說，最終差距值最低的備選方案具有最接近標準理想權重的值。 以下是 MAIRCA 方法的步驟：

---
- (I) Forming the initial decision matrix
In this step, the initial decision matrix is formed based on the aggregation of expert opinions to evaluate the performance of each alternative based on each criteria assuming the existence of m alternatives and n criterias ccording to Eq. (18).
(I) 形成初始決策矩陣
在這一步中，初始決策矩陣是基於專家意見的聚合形成的，以根據每個標准假設存在 m 個備選方案和 n 個標準，根據等式評估每個備選方案的性能。 (18).
![](https://i.imgur.com/O8ZlUVN.png)

---
- (II) Determine the preference according to the choice of alternatives
The basic assumption of the MAIRCA method is that at the beginning of the decision-making process, the experts are neutral in choosing the alternatives and therefore the probability of choosing the alternatives is equal. The preference of selecting an alternative from the m alternative is calculated based on Eq. (19):
(II) 根據備選方案的選擇確定偏好
MAIRCA 方法的基本假設是在決策過程開始時，專家們在選擇備選方案時是中立的，因此選擇備選方案的概率是相等的。 從 m 個備選方案中選擇一個備選方案的偏好是基於等式計算的。 (19):
![](https://i.imgur.com/VvusmDV.png)

---
- (III) Calculation of theoretical evaluation matrix
In this step, the theoretical evaluation matrix is obtained by multiplying the preference of the alternatives by the weight of the criteria and in accordance with Eq. (20):
（III）理想矩陣的計算
在這一步中，通過將備選方案的偏好乘以標準的權重並根據等式獲得理想矩陣。 (20):
![](https://i.imgur.com/mwLzlcC.png)
Given that the preference of all alternatives is equal, the TP matrix can be represented as Eq. (21):
鑑於所有備選方案的偏好相等，TP 矩陣可以表示為等式。 (21):
![](https://i.imgur.com/PPJfQip.png)

---
- (IV) Formation of a real assessment matrix
In this step, the real assessment matrix is calculated based on the product of the initial decision matrix elements in the theoretical evaluation matrix elements. The values of the real assessment matrix are calculated based on the profit or cost of the criterion, respectively, based on Eq. (22).
（IV）現實矩陣的形成
在該步驟中，現實矩陣是根據初始決策矩陣元素與理想矩陣元素的乘積來計算的。 真實矩陣的值分別基於標準的利潤或成本計算，基於等式。 (22).
![](https://i.imgur.com/w0rQ9tg.png)
![](https://i.imgur.com/19uS2Wb.png)

---
- (V) Calculate the total gap matrix
In this step, the total gap matrix is calculated based on the difference between the elements of the theoretical evaluation matrix and the real assessment matrix according to Eq. (23).
(V) 計算總差距矩陣
在這一步中，總差距矩陣是根據公式（1）將理想矩陣與現實矩陣之間的元素相減計算的。 (23).
![](https://i.imgur.com/ArwpcZ6.png)

---
- (VI) Calculate the final values of criteria functions for alternatives
In this step, the value of the criteria functions is calculated by summing the rows of the gap matrix for each alternative according to Eq. (24):
(VI) 計算備選標準函數的最終值
在此步驟中，標準函數的值是通過根據等式對每個備選方案的間隙矩陣的行求和來計算的。 (24):
![](https://i.imgur.com/eFQ8fm1.png)


---
## 3 case study 案例研究
Study area in this research is Aidoghmush basin that with an area of 1802 square kilometers located in East Azerbaijan province in Iran. This basin is adjacent to Gharnaghuchai basin in the north and Ajichai basin in the south. The 80-km-long Idoghmush River originates from the Ghur-Ghur heights and flows into the Ghezel Ozan River. The height of this basin varies from 1100 to 2500 m. Also, the annual discharge of this dam is about 170 × 106 m3 and the average annual rainfall is 378 mm. The most important objective of Idoghmush dam construction is to supply water needed for the agricultural sector of the region (Ashofteh et al. 2013).
本研究的研究區域是位於伊朗東阿塞拜疆省的Aidoghmush盆地，面積1802平方公里。 該盆地北鄰Gharnaghuchai盆地，南鄰Ajichai盆地。 全長 80 公里的 Idoghmush 河發源於 Ghur-Ghur 高地，流入 Ghezel Ozan 河。 這個盆地的高度從 1100 米到 2500 米不等。 此外，該壩的年流量約為 170×106 立方米，年平均降雨量為 378 毫米。 Idoghmush 大壩建設的最重要目標是為該地區的農業部門提供所需的水（Ashofteh 等人，2013 年）。
![](https://i.imgur.com/MFi0j9w.png)


---
## 4 Results 結果
### 4.1<br>Result of MODSIM Simulation Model<br>MODSIM仿真模型結果
The results of system simulation with MODSIM model for base and future periods under RCP 2.6 and RCP 8.5 were obtained and the results are presented as reservoir performance indexes in Table 4. According to Table 4, changes in reservoir performance indexes in the future over the base have decreased in some cases and increased in others. Lack of the same trend in changes in reservoir performance indexes shows that for planning of water supply systems in the future years should not only consider changes in reservoir inflow volume during operation period, but also examine the reservoir inflow changes in each month is very important in determining the amount of indexes and finally developing a suitable model for releasing the reservoir. According to Table 4, the reliability rate under the RCP2.6 scenario decreased by 9% compared to the base period and the vulnerability index decreased by 17%. The significant reduction in vulnerability indicates that demand shortages occurred in the months when the system was more vulnerable. Also, the resiliency has increased by 4% under the RCP2.6 scenario compared to the base period, which indicates that the ability of the system to return to the desired state will be in a better position than the base period and the flexibility index under the RCP2.6 scenario will not change much compared to the base period. According to Table 4, the reliability index and vulnerability index under the RCP8.5 scenario will decrease by 5 and 30%, respectively, compared to the base period, and the resiliency index will increase by 9% that indicates the ability of the system to return to the desired state, which will lead to a 10% increase in the flexibility index. This indicates that the system will be more flexible in dealing with the adverse effects of climate change in the future period under the RCP8.5 scenario than the base period and also better than the future period under the RCP2.6 scenario.
得到RCP 2.6和RCP 8.5下基期和未來期MODSIM模型系統模擬結果，結果以水庫性能指標形式呈現在表4中。根據表4，未來水庫性能指標相對於基期的變化在某些情況下減少，在其他情況下增加。水庫性能指標變化趨勢不一致，說明未來幾年的供水系統規劃不僅要考慮水庫運行期間水量的變化，為了確定指標的數量並最終開發出合適的釋放儲存模型，考察水庫各月間水量的變化也很重要。由表4可知，RCP2.6情景下可靠性率較基期下降9%，脆弱性指數下降17%。脆弱性的顯著降低表明需求短缺發生在系統更脆弱的月份。另外，RCP2.6情景下的彈性較基期提高了4%，表明系統恢復到理想狀態的能力將比基期處於更好的位置，彈性指標在RCP2.6情景與基期相比變化不大。由表4可知，RCP8.5情景下的可靠性指標和脆弱性指標較基期分別下降5%和30%，彈性指標提高9%，表明系統能夠恢復到理想狀態，這將導致柔韌性指數提高10%。這表明RCP8.5情景下未來時期系統在應對氣候變化不利影響方面比基期更加靈活，也優於RCP2.6情景下未來時期。
![](https://i.imgur.com/JDLOQzd.png)

---
### 4.2<br>Result of LINGO 18 Optimization Model<br>LINGO 18優化模型的結果
The results of reservoir performance indexes from the optimization model with LINGO 18 model for the base and future period (under RCP 2.6 and RCP 8.5) are given in Table 5. According to Table 5, the changes in reservoir performance indexes in the future period compared to the base period as in the reservoir simulation indexes have decreased in some cases and increased in others. For example, the reliability and vulnerability indexes in the future period under the RCP2.6 scenario have decreased by 13 and 17% compared to the base period, respectively. But on the other hand, the resiliency index has increased by 15% in the future period under the RCP2.6 compared to the base period. This means that the damage to the system has been such that the system has been able to return to the desired state. Overall, the flexibility index has grown by 4%, which indicates that the system, despite the reduction in reliability, will have more flexibility to the destructive effects of climate change in the future period under the RCP2.6 than the base period. Also, in the future period under RCP8.5 scenario, compared to the base period, the reliability and vulnerability indexes will decrease by 6 and 19%, respectively. On the other hand, the resiliency index will increase by 15% in the future period under the RCP8.5 compared to the base period. Finally, the flexibility index will increase by 13%.
LINGO 18 優化模型得到的基期和未來時期（RCP 2.6 和 RCP 8.5 下）水庫性能指標結果如表 5 所示。根據表 5，未來時期水庫性能指標變化與基期相比，水庫模擬指標有的下降，有的上升。例如，RCP2.6情景下未來時期的可靠性和脆弱性指標較基期分別下降了13%和17%。但另一方面，在RCP2.6下，未來時期的彈性指數較基期上升了15%。這意味著對系統的損壞已經使系統能夠恢復到所需的狀態。總體而言，靈活性指數增長了4%，這表明儘管可靠性有所降低，但RCP2.6下未來時期系統對氣候變化破壞性影響的靈活性將高於基期。另外，在RCP8.5情景下的未來時期，與基期相比，可靠性和脆弱性指標將分別下降6%和19%。另一方面，在RCP8.5下，未來時期的彈性指數將比基期提高15%。最後，靈活性指數將增加13%。
![](https://i.imgur.com/pO4iLuq.png)

---
### 4.3<br>Results of Rough‐AHP Weighting Method<br>Rough-AHP加權法結果
---
- (a) Hierarchical modeling
In the first step, after determining the goal level, criteria level, and alternatives level, the hierarchical model was modeled as Fig. 6.
第一步，在確定了目標層級、準則層級和備選層級後，構建層次模型如圖 6 所示。
![](https://i.imgur.com/vvbf54f.png)

---
- (b) Criteria pair comparisons and controling the inconsistency rate
After determining the criteria set and completing the pairwise comparison matrixes by five experts, the inconsistency rate of the experts’ judgments was examined using Expert Choice 11 software, which is designed based on the AHP method. Table 6 shows the pairwise comparison matrixes of the first experts and their inconsistency rate. Table 6 is a pairwise comparison matrix of the criteria, the upper half elements of which was completed by the first expert, and the lower half elements were calculated based on the reciprocal principle relative to the original diameter. After entering the above matrix in the Expert Choice model, the inconsistency rate was calculated, the value of which is less than 0.1, which indicates that the expert judgment is consistent.
(b) 準則對比較和控制不一致率
在確定標準集並由五位專家完成兩兩比較矩陣後，利用基於層次分析法設計的Expert Choice 11軟件對專家判斷的不一致率進行檢驗。 表 6 顯示了第一位專家的兩兩比較矩陣及其不一致率。 表6是標準的兩兩比較矩陣，上半部分元素由第一位專家完成，下半部分元素根據相對於原始直徑的倒數原則計算。 將上述矩陣輸入專家選擇模型後，計算出不一致率，其值小於0.1，說明專家判斷一致。
![](https://i.imgur.com/KcNJjoG.png)
<font color= #0000FF>上面這張表格可能有錯 The above form may be wrong</font>
![](https://i.imgur.com/XMRRFUu.png)

![](https://i.imgur.com/O2m2yhV.png)


---
- \(c\) Calculate the geometric mean of the same elements in the pairwise comparison matrix 
The same elements in the pairwise comparisons matrix and their geometric mean are shown in Table 7. As shown in Table 7, the lower approximation, the set of numbers smaller than the geometric mean, and the higher approximation, the numbers greater than the geometric mean, are the same elements in the pairwise comparison matrix.
\(c\) 計算成對比較矩陣中相同元素的幾何平均數
表7顯示了成對比較矩陣中的相同元素及其幾何平均值。如表7所示，較低近似值（小於幾何平均值的一組數字）和較高近似值（大於幾何平均值）是成對比較矩陣的相同元素。
![](https://i.imgur.com/0CtHPHy.png)
![](https://i.imgur.com/ULqRL9a.png)
![](https://i.imgur.com/lN1iWJ2.png)

---
- (d) Calculate the upper limit and the lower limit
After determining the set of high and low approximations in the previous step, the same elements and their geometric mean were arranged and the upper and lower limits were calculated, the results of which are given in Table 8.
(d) 計算上限和下限
在確定了上一步的高低近似值集合後，將相同的元素及其幾何平均數進行排列，計算上下限，結果如表8所示。
![](https://i.imgur.com/0GbpRYb.png)
![](https://i.imgur.com/XlUJknX.png)

---
- (e) Formation of cumulative pairwise comparisons based on rough numbers
The cumulative pairwise comparison matrix was then formed based on rough numbers. To form this matrix, after calculating and placing the upper-half elements, the lower-half elements were calculated based on the reciprocal principle in AHP method. The cumulative pairwise comparison matrix is given in Table 9.
(e) 基於粗略數形成累積的成對比較矩陣
然後基於粗略數形成累積的成對比較矩陣。 為了形成這個矩陣，在計算和放置上半部分元素之後，根據層次分析法中的倒數原理計算下半部分元素。 在表 9 中給出累積成對比較矩陣。
![](https://i.imgur.com/rl5braD.png)
![](https://i.imgur.com/37zTLSo.png)


---
- (f) Calculate the degree of satisfaction from the judgment of experts
At this stage, the degree of satisfaction with the scoring of experts was calculated. The results are presented in Table 10.
(f) 根據專家的判斷計算滿意度
本階段計算專家打分的滿意度。 結果如表 10 所示。
![](https://i.imgur.com/uUVgAck.png)
![](https://i.imgur.com/pH3BvzC.png)


---
- (g) Calculate the final weight of the criteria
The criterion final weight of the model preparation cost was 0.437, the criterion of the operating policy approach was 0.273, the criterion of the quality of the solutions was 0.223 and the criterion of construction and execution time of the model was 0.0656. According to the results of the weighting method, the most important criterion in evaluating the alternatives is to consider the model preparation cost, because it can not be used until the model can be prepared.
(g) 計算標準的最終權重
模型準備成本(c1)的準則最終權重為0.437，操作策略方法(c2)的準則為0.273，解決方案的質量準則(c3)為0.223，模型的構建和執行時間的準則(c4)為0.0656。 根據加權法的結果，評估備選方案最重要的標準是考慮模型準備成本，因為只有模型準備好才能使用。

After that, the criterion of the reservoir operation policy approach is in the second place, because the reservoir operation policy approach has a direct effect on the monthly release from the reservoir and causes significant changes in the process of changing reservoir performance indexes as a basis for future planning.
其後，水庫調度策略方法(c2)的判據排在第二位，因為水庫調度策略方法直接影響水庫的月釋放量，並在改變水庫性能指標的過程中引起顯著變化，作為未來規劃的判斷依據。

The criterion of construction and execution time of the model is in the last rank, which shows that according to experts, the construction and execution time of the model has no significant effect on the accuracy of the results.
模型的構建和執行時間的標準排在最後，這表明根據專家的說法，模型的構建和執行時間對結果的準確性沒有顯著影響。

---
### 4.4<br>Result of MAIRCA Decision‐Making Method<br>MAIRCA決策方法的結果

---
- (a) Formation of matrices
In the first step of the MAIRCA decision ranking method, the initial decision matrix was completed by four experts and their opinions were aggregated. Then the theoretical evaluation matrix is formed in the second step. In the third step, the real assessment matrix was calculated. Next, the general gap matrix is formed in the fourth step. The results are presented in Table 11.
(a) 矩陣的形成
在 MAIRCA 決策排序方法的第一步中，初始決策矩陣由四位專家完成並彙總他們的意見。 然後在第二步形成理論評價矩陣。 第三步，計算真實的評估矩陣。 接下來，在第四步中形成通用間隙矩陣。 結果如表 11 所示。
![](https://i.imgur.com/98pvlRI.png)
![](https://i.imgur.com/T3QKGgg.png)



![](https://i.imgur.com/vcrRqOE.png)

---
- (b) Calculate values of criteria functions
In the last step, the final values of the criteria functions for each alternative were calculated, which are presented in Table 12. The results of Table 12 show that the simulation approach with MODSIM model with lower criterion function is superior to the optimization approach with LINGO 18 model. The simulation approach with the MODSIM model as a more appropriate approach shows that experts consider the simulation results by simulators widely used in different regions to be more appropriate than the optimization with the LINGO 18 model whose objective function and constraints are determined by policy-makers. Because policymakers may not use the objective function appropriate to the situation of the study area and based on unrealistic results, the status of the water supply system will be more endangered. Also, the provision cost of the LINGO 18 optimization model has led experts to consider free access to the MODSIM simulation model as one of its advantages. The possibility of customization in the MODSIM model has caused it to have a lot of flexibility for modeling a wide range of water systems with different characteristics. Also, the ability of MODSIM model to provide more quality criteria for solutions based on performance indexes of water supply system, which aims to provide more stakeholder satisfaction, has played an important role in choosing a simulation approach with MODSIM model.
- (b) 計算準則函數的值
在最後一步中，計算了每個備選方案的標準函數的最終值，如表 12 所示。表 12 的結果表明，使用具有較低標準函數的 MODSIM 模型的仿真方法優於使用 LINGO 18 的優化方法。 MODSIM模型作為更合適的模擬方法，表明專家認為不同地區廣泛使用的模擬器的模擬結果比LINGO 18模型的優化更合適，因為LINGO 18模型的目標函數和約束由決策者確定。由於政策制定者可能沒有使用適合研究區域情況的目標函數和基於不切實際的結果，供水系統的現狀將更加危險。此外，LINGO 18 優化模型的提供成本讓專家們將免費存取 MODSIM 仿真模型視為其優勢之一。 MODSIM 模型中定制的可能性使其具有很大的靈活性，可以對范圍廣泛的具有不同特徵的水系統進行建模。此外，MODSIM 模型能夠根據供水系統的性能指標為解決方案提供更多質量標準，旨在提供更多的利益相關者滿意度，這在選擇 MODSIM 模型的仿真方法方面發揮了重要作用。
![](https://i.imgur.com/eA2PzgL.png)
![](https://i.imgur.com/8Aetj9C.png)



The results of the present study show that choosing the proper approach for managing available water resources is very important and effective. The results of a choice can be cited when it is based on the suitable criteria that reflect the actual decision space. The use of multi-criteria decision-making methods has the ability to consider different criteria to select the best alternative and can include all the opinions of experts and their desired criteria. Also, the general process of this study can be generalized to other case studies.
本研究的結果表明，選擇適當的方法來管理可用水資源非常重要且有效。 當基於反映實際決策空間的合適標準時，可以引用選擇的結果。 使用多準則決策方法能夠考慮不同的準則來選擇最佳方案，並且可以包括專家的所有意見及其期望的準則。 此外，本研究的一般過程可以推廣到其他案例研究。

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## 5 concluding remarks 結論
Ensuring the security of water resources, especially in the agricultural sector as one of the infrastructures of sustainable development is of special importance. Preservation of current water resources and planning to deal with factors that could jeopardize the quantity and quality of water resources is essential. The phenomenon of climate change is considered as one of the natural critical factors in the occurrence of water stress and it is necessary to carefully study its effects on climate parameters to minimize its adverse effects.

對於農業部門來說，水資源是作為持續發展所必要的重要基礎設施。保存當前的水資源並且計畫如何去處理可能危及水源的因素是相當重要的。對於缺水的情況，氣候變化的各種現象被認為是一個自然產生的危機因素。有必要對其深度研究以求最小化它的負面影響

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In addition to identify the effects of any type of threat to water resources, it is necessary that the policy-making approach be considered in accordance with the situation in the study region, because the choice of any approach plays a key role in how decisionmakers respond to factors threatening the sustainability of water resources. Choosing the right approach for realistic evaluation of results is influenced by various factors, and one of the best tools for aggregating various opinions of different groups is to use multi-criteria decision-making methods.

除了確定任何類型的威脅對水資源的影響之外，有必要針對不同的研究區域去考慮不同的決策方法。因為任何一種決策方法都對決策者如何應對各種水資源危機起到關鍵性的影響。選擇正確的結果評估方式受到很多因素影響，整合不同群體的各種意見的最佳工具之一是使用多準則決策方法

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On the other hand, in any decision-making process, the inherent uncertainties arising from the human knowledge of the group of experts who are in the first step of each process always challenge the sustainability of the results obtained. So far, various theories have been proposed to reduce the uncertainties. One of the most important is RST theory based on mathematical concepts, which due to its flexibility can be combined with weighting and ranking methods of multi-criteria decisions and reduce uncertainties in the decisionmaking process.

另一方面，在任何決策過程中，處於每個過程第一步的專家團隊的人為知識所產生的內在不確定性總是挑戰所獲得結果的可持續性。到目前為止，已經提出各種理論來減少不確定性。其中最重要的是基於數學概念的RST理論，由於其靈活性，可以與多準則決策的加權和排序方法結合，減少決策過程的不確定性。

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In the present study, first the effects of climate change were investigated on water supply system in the future period. The results of climatic models showed that in the future time period, the climatic parameter of temperature under RCP2.6 and RCP8.5 scenarios will increase by 1.5 and 1.3%, respectively, compared to the base time period, and the climatic parameter of precipitation will increase by 17 and 25% in the future time period, under RCP2.6 and RCP8.5 emission scenarios respectively, over the base time period.

在本研究中，首先調查了氣候變化對未來供水系統的影響。氣候模型結果表明，未來時間段，RCP2.6和RCP8.5情景下溫度氣候參數將分別增加1.5%和1.3%，降水氣候參數則是分別增加17%和25%。

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In the next step, the effects of increasing temperature and rainfall on water required for agricultural irrigation and inflow to reservoir were investigated. The results showed that the water required by the agricultural sector under both scenarios will increase by 15% and the inflow to reservoir under the scenarios RCP2.6 and RCP8.5 will increase by 12 and 13%, respectively. Then the water supply system was modeled with two approaches of simulation with MODSIM model and the optimization with LINGO 18 model and four indexes of reliability, resiliency, vulnerability and flexibility for both approaches were extracted separately. Due to the same amount and trend of performance indexes of water supply system in simulation and optimization approaches, MAIRCA decision-making method was used to select a more appropriate approach.

下一步，調查溫度上升和降雨量增加對農業灌溉所需水量和水庫流入量的影響。結果表明，兩種情景下農業部門的用水需求都將增加15%，而流入水庫的水量在兩個情景下分別增加12%和13%，然後採用MODSIM模型仿真和LINGO 18模型優化兩種方法對供水系統進行建模，分別提取兩種方法的可靠性、彈性、脆弱性和靈活性4個指標。由於模擬和優化方法中供水系統性能指標的數量和趨勢相同，因此採用MAIRCA決策方法來選擇更合適的方法。

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In this study, Rough-AHP weighting method was used to weight the criteria. In this way, the mental information of the experts is expressed definitively, but with the RST tool they are splited to upper and lower limits and calculated the degree of satisfaction with the judgments. In fact, this method is a response to uncertainties with definite numbers. The results of Rough-AHP method showed that the model provision cost with a weight of 0.475 was the most important criteria and the criterion construction and execution time of the model with a value of 0.065 was the least important criteria for experts. Finally, the criteria of operation policy approach and the criterion of quality of solutions with weights of 0.273 and 0.223, respectively, were ranked second and third. In the last step, with MAIRCA decision-making method, modeling approaches were prioritized and the simulation approach with a criteria function value of 0.467 was selected as the best approach for allocating water resources in the future.

本研究採用Rough-AHP加權法對標准進行加權。在這樣的情況下，專家的心理信息表達明確，但用RST工具將他們拆分為上下限，計算出對判斷的滿意程度。實際上，這種方法是對不確定性做出確定性的數字回應。Rough-AHP方法的結果表明，<模型提供成本>權重為0.475是最重要的標準，而<模型構建和執行時間>權重為0.065，是專家最不重視的標準。最後，<操作策略方法>權重為 0.273 ，<解決方案質量>權重為 0.223，分別是第二和第三名。最後一步，基於 MAIRCA 決策的結果，模擬方法被優先考慮，並選擇標準函數值為 0.467 的模擬方法作為未來分配水資源的最佳方法。