# A Stochastic Multi-Objective Chance-Constrained Programming Model for Water Supply Management in Xiaoqing River Watershed

^{*}

## Abstract

**:**

## 1. Introduction

^{3}and the annual average water deficit is 50 billion m

^{3}[1]. Moreover, the average agricultural water utilization factor is 0.47, which is significantly lower than the global average level (0.7–0.8); the water consumption rate of ten thousand Yuan GDP is about 300 m

^{3}and two times higher than the global average [2]. Therefore, effective utilization of limited water resources is an important task for local administrators. However, as a complex and huge system, a large amount of system factors and their intricate relationships lead to the fact that the watershed exhibits a variety of characteristics, such as integrality, dynamics, multidimensionality, nonlinearity and uncertainty. In order to realize sustainable water resource utilization, an appropriate water supply management model at a watershed scale is desired.

## 2. Methodology

#### 2.1. Multi-Objective SCCP Model with Normal Probability Distribution

_{1}and f

_{2}represent two objective functions; X is a vector of the decision variable; B(t) and D(t) are two sets with random factors defined on a probability space T, $t\in T$, which are described as B~N (m

_{B}, δ

_{B}

^{2}) and D~N (m

_{D}, δ

_{D}

^{2}), where m

_{B}and m

_{D}denotes the mean value, respectively; δ

_{B}and δ

_{D}denotes the standard deviation, respectively; A, C

_{1}and C

_{2}are fixed vectors of auxiliary variables. To solve model (1), the constraints (1c) and (1d) are converted into their deterministic equivalents through using the SCCP approach with a series of predefined constraints-violation levels q

_{i}. Meanwhile, two objective functions are combined into one objective through designing various weight coefficients (i.e., w

_{1}and w

_{2}). The constraint (1e) ensures the non-negativity of decision vectors and nonzero of auxiliary variables. The model (1) is reformulated as follows:

_{i}. The constraint (2d) regulated that the summation of two weight coefficients is equal to one. Finally, a variety of solutions (i.e., f

_{1, opt}, f

_{2, opt}and X

_{opt}) are obtained through adjusting q

_{i}, w

_{1}and w

_{2}values, respectively.

#### 2.2. Multi-Objective SCCP Model with Log-Normal Probability Distribution

_{B}, δ

_{B}

^{2}) and ln(D(t)) ~N (m

_{D}, δ

_{D}

^{2}), respectively; variables B and D are expressed as $B\wedge \left({U}_{B},\text{\hspace{0.17em}}{V}_{B}^{2}\right)$ and $D\wedge \left({U}_{D},\text{\hspace{0.17em}}{V}_{D}^{2}\right)$, respectively. The following four equations are established to reflect the interactive relationships among four feature parameters (i.e., m

_{B}, δ

_{B}

^{2}, U

_{B}and V

_{B}

^{2}) [30].

_{i}levels and weight coefficients (w

_{1}and w

_{2}). The commercial software LINGO (LINGO 12.0, Lindo System Inc., Chicago, IL, USA) is used to code and solve the SMOCCP model, because it is capable of providing the user-friendly edit interface, embedding a series of valuable equations and functions and solving the optimization model with unlimited variables and constraints. The short computation time of solving this model (just a few seconds) is convenient to generate a variety of solutions under specific combinations of weighted coefficients and constraints-violation levels. Figure 1 shows the procedures of formulating and solving the proposed SMOCCP model, which can be summarized as follows:

- Step 1:
- Gain in-depth insights into the targeted watershed system, identify all uncertain variables and design major system objectives and constraints;
- Step 2:
- Formulate a SMOCCP model;
- Step 3:
- Determine two solution algorithm rules associated with the multi-objective functions and the parameters presented as log-normal probability distributions;
- Step 4:
- Combine two objective functions into an integrated one and convert stochastic constraints to their respective crisp equivalents;
- Step 5:
- Obtain final solutions of f
_{1, opt}, f_{2, opt}and X_{opt}under various probability levels and weight coefficients, respectively.

## 3. Case Study

#### 3.1. Introduction and Problem Description of Xiaoqing River Watershed

^{2}and reaches about 1/15 of total area in the Shandong Province. As shown in Figure 2, the main stream of the Xiaoqing River is sourced from four streamflows of the city Jinan with a total length of 237 km. It flows through ten regions (including towns and districts) of Jinan, Zibo, Binzhou, Dongying and Weifang from the west to the east, gathering the water from eighteen counties and finally falling into the bay Laizhou. As an important drainage channel, Xiaoqing River watershed is mainly responsible for agricultural irrigation and river transportation, and plays an important role in socio-economic development of the Shandong Province [31].

- (i)
- Severe water resource shortage and even flow cutoff in some tributaries: For example, the Jinan section of the Xiaoqing River is located in the mid-latitude zone in Northern China, where the rainfall distribution exhibits uneven characteristics and focuses on June to September, leading to frequent occurrences of drought and flooding disasters. The multi-year average surface runoff in this section is about 352.79 million m
^{3}, which is far below the required water demands. - (ii)
- Poor water quality: the Xiaoqing River receives industrial, agricultural and household wastewater sourced from eighteen counties, resulting in significant degradation of water quality. As stated in the “Report on the Water Quality of Critical Water Function Areas in the Shandong Province”, the total length of evaluated river is roughly 1682.6 km. Among them, the river length for meeting the water quality requirement is only 590.2 km, while the polluted river length reaches 1092 km.
- (iii)
- Imperfect infrastructure and management regime of this watershed: The overly high leakage loss of the water-transportation pipeline leads to a reduced amount of available water resources. A separate management mechanism is applied to this watershed for the time being, leading to unclear definitions in rights, responsibilities and obligations for the watershed management.

#### 3.2. Generalization of Xiaoqing River Watershed

#### 3.3. System Parameters and Model Formulation

_{1}= total system cost (RMB); k (k = 1, 2, …, K) = the index of time periods (i.e., months) where K is total number of time period; j (j = 1, 2, ..., J) = the index of water sources, where J is total number of water sources; t (t = 1, 2, …, T) = the index of treatment plants, where T = the total number of treatment plants; r (r = 1, 2, …, R) = the index of reservoirs, where R is total number of reservoirs; z (z = 1, 2, …, Z) is the index of tributaries, where Z is total number of tributaries; XJT

_{jtk}= the decision variables denoting water amounts transferred from water source to treatment plant, (×10

^{3}m

^{3}); XTR

_{trk}= the decision variables denoting water amounts allocated from treatment plant to reservoir, (×10

^{3}m

^{3}); XRZ

_{rzk}= the decision variables representing water amounts transferred from reservoir to tributary, (×10

^{3}m

^{3}); PR

_{jk}= water purchase cost from water source per month (RMB/ × 10

^{3}m

^{3}); CJT

_{jt}= water transfer cost from source to treatment plant (RMB/ ×10

^{3}m

^{3}); CTR

_{tr}= water transfer cost from treatment plant to reservoir (RMB/ × 10

^{3}m

^{3}); CRZ

_{tr}= water transfer cost from reservoir to tributary (RMB/ × 10

^{3}m

^{3}); f

_{2}= total leakage loss (×10

^{3}m

^{3}); LXJ

_{jt}= leakage loss in network from source to treatment plant, (%); LXT

_{tr}= leakage loss in network from treatment plant to reservoir, (%); LXZ

_{rz}= leakage loss in network from reservoir to tributary, (%). The objective Function (5a) is to minimize total system costs, which is calculated through the summation of water purchase and transportation costs. The decision variables are transferred water amounts among water sources, treatment plants, reservoirs and tributaries, respectively. The objective Function (5b) aims to achieve the minimization of total leakage loss in the entire transportation process.

- (1)
- Water consumption constraints:$$\sum _{r=1}^{R}(1-LX{Z}_{rz})}*XR{Z}_{rzk}\ge {D}_{zk}\left(\omega \right),\text{\hspace{0.17em}\hspace{1em}}\forall k,\text{\hspace{0.17em}}z$$$$XR{Z}_{rzk}\le ZR{Z}_{rz}*{U}_{RZ}\text{\hspace{0.17em}\hspace{1em}}\forall r,\text{\hspace{0.17em}}z,\text{\hspace{0.17em}}k$$
_{zk}(ω) = required water amounts of tributary in each month (×10^{3}m^{3}), which follows the log-normal distribution, i.e., ln(D_{zk}(ω)) ~N (m_{D}, δ_{D}^{2}); ZRZ_{rz}= binary variable (0 or 1) used to define paths from reservoir to tributary; U_{RZ}= the maximum capacity of prescribed paths from reservoir to tributary (×10^{3}m^{3}). The constraint (5c) is used to regulate allocated water amounts to tributaries that are higher than their required amounts. The constraint (5d) is used to ensure that water is transferred in prescribed paths where the paths are available while binary variable ZRZ_{rz}is “1”; otherwise, it will be “0”. - (2)
- Reservoir constraints:$$I{R}_{r1}=IR{O}_{r}+{\displaystyle \sum _{t=1}^{T}(1-LX{T}_{tr})}*XT{R}_{tr1}-{\displaystyle \sum _{z=1}^{Z}XR{Z}_{rz1}},\text{\hspace{0.17em}\hspace{1em}}\forall r$$$$I{R}_{rk}=I{R}_{r,k-1}+{\displaystyle \sum _{t=1}^{T}(1-LX{T}_{tr})}*XT{R}_{trk}-{\displaystyle \sum _{z=1}^{Z}XR{Z}_{rzk}},\text{\hspace{0.17em}\hspace{1em}}\forall r,k=2,\cdots ,K$$$$XT{R}_{trk}\le ZT{R}_{tr}*{U}_{TR}\text{\hspace{0.17em}\hspace{1em}}\forall t,r,k$$$$I{R}_{rk}\le V{R}_{rk}\text{\hspace{0.17em}\hspace{1em}}\forall r,k$$
_{r}= inventory amounts of reservoir at the first planning phase, (×10^{3}m^{3}); IR_{rk}= inventory amounts of reservoir at the end of month, (×10^{3}m^{3}); ZTR_{tr}= binary variables (0 or 1) used to regulate the paths from treatment plant to reservoir; U_{TR}= the maximum capacities of prescribed paths from treatment plant to reservoir, (×10^{3}m^{3}); VR_{rk}= reservoir’s capacities at month, (×10^{3}m^{3}). The constraints (5e) and (5f) reflected the water connections among treatment plants, reservoirs and tributaries, respectively. The constraint (5g) is used to ensure that the water is transferred in prescribed paths. The constraint (5h) regulated that water amounts provided by the reservoirs should be lower than the maximum capacities of reservoirs. - (3)
- Treatment plant constraints:$$I{T}_{t1}=IT{O}_{t}+{\displaystyle \sum _{j=1}^{J}(1-LX{J}_{jt})}*XJ{T}_{jt1}-{\displaystyle \sum _{r=1}^{R}XT{R}_{tr1}},\text{\hspace{0.17em}\hspace{1em}}\forall t$$$$I{T}_{tk}=I{T}_{t,k-1}+{\displaystyle \sum _{j=1}^{J}(1-LX{J}_{jt})}*XJ{T}_{jtk}-{\displaystyle \sum _{r=1}^{R}XT{R}_{trk}},\text{\hspace{0.17em}\hspace{1em}}\forall t,\text{\hspace{0.17em}}k=2,\cdots ,K$$$$XJ{T}_{jtk}\le ZJ{T}_{jt}*{U}_{JT}\text{\hspace{0.17em}\hspace{1em}}\forall j,t,\text{\hspace{0.17em}}k$$$$I{T}_{tk}\le V{T}_{tk}\text{\hspace{1em}}\forall t,\text{\hspace{0.17em}}k$$
_{t}= inventory amounts of treatment plants at the first planning phase, (×10^{3}m^{3}); IT_{tk}= inventory amounts of treatment plants at the end of month, (×10^{3}m^{3}); ZJT_{jt}= binary variable (0 or 1) used to define paths from source to treatment plant; U_{jt}= the maximum capacity of prescribed paths from water source to treatment plant, (×10^{3}m^{3}); VT_{tk}= treatment capacities at month, (×10^{3}m^{3}). The constraints (5i) and (5j) are used to reflect the hydraulic relation among water sources, treatment plants and reservoirs, respectively. The constraint (5k) regulated the water must be allocated in the prescribed paths. The constraint (5l) regulated that treated water amounts by the plants should be lower than the maximum capacities of treatment plants. - (4)
- Water source constraints:$$I{J}_{j1}=IJ{O}_{j}-{\displaystyle \sum _{t=1}^{T}XJ{T}_{jt1}}+B{J}_{j1},\text{\hspace{0.17em}\hspace{1em}}\forall j$$$$I{J}_{jk}=I{J}_{j,k-1}-{\displaystyle \sum _{t=1}^{T}XJ{T}_{jtk}}+B{J}_{jk},\text{\hspace{0.17em}\hspace{1em}}\forall j,\text{\hspace{0.17em}}k=2,\cdots ,K$$$$\sum _{t=1}^{T}XJ{T}_{jtk}}\le M{J}_{jk}\text{\hspace{0.17em}\hspace{1em}}\forall j,\text{\hspace{0.17em}}k$$
_{j}= inventory amounts of each water source at the first phase (×10^{3}m^{3}); IJ_{jk}= inventory amounts at the end of month (×10^{3}m^{3}); BJ_{jk}= recovered water amount from water source, (×10^{3}m^{3}); MJ_{jk}= the maximum water amount extracted from water source at month, (×10^{3}m^{3}). The constraints (5m) and (5n) are used to reflect the relationship among water sources and treatment plants, respectively. The constraint (5o) is used to ensure that the water extracted from water sources should be lower than their maximum capacities. - (5)
- Technical constraints$$XJ{T}_{jtk}\ge 0,\text{\hspace{0.17em}}XT{R}_{trk}\ge 0,\text{\hspace{0.17em}}XR{Z}_{rzk}\ge 0,\text{\hspace{1em}}\forall j,\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}z,\text{\hspace{0.17em}}k$$
_{1,opt}, f_{2,opt}, XJT_{jtk}, XTR_{trk}and XRZ_{rzk}) are obtained under various weight combinations and constraint-violation levels.

## 4. Result Analysis and Discussion

#### 4.1. Result Analysis

_{1}= 0.9 and w

_{2}= 0.1) and the probabilistic level (i.e., q

_{i}) increases over twelve months, the total water amounts supplied to seven tributaries would decrease. For example, as shown in Table 4, at q

_{i}levels of 0.01, 0.05 and 0.1, the water amounts transferred to tributary 2 in the first period are 240.15, 197.44 and 177.88 × 10

^{3}m

^{3}, respectively; similarly, the water amounts allocated to tributary 7 in the second period are 156.46, 135.88 and 126.05 × 10

^{3}m

^{3}, respectively. The reason behind such a difference is that the water demand constraint is involved in the stochastic variables, where the required water amounts of the tributaries were expressed as random variables with log-normal distributions. Therefore, the increase in violation level of q

_{i}means that the satisfaction level of the constraint would decrease, leading to a decrease of the water demand. The intrinsic balanced relationship between water supply and demand determined that a decrease in water demand must be accompanied with a decrease in the water amounts extracted from water sources. As shown in Figure 4, as the probability level increases, the total water amounts extracted from the groundwater would decrease (i.e., 26,418.10, 16,381.95 and 12,724.73 × 10

^{3}m

^{3}, respectively); similarly, the water amounts provided by surface water would be 22,587.89, 17,600.33 and 15,454.30 × 10

^{3}m

^{3}, respectively. In fact, the decrease in required water amounts leads to the decrease in transferred water amounts in the entire water management system, such that the total leakage loss and total system costs (including supplied, treated and transferred costs) would decrease. At three q

_{i}values (i.e., 0.01, 0.05 and 0.1), the total leakage loss amounts are 19,055.62, 12,813.74 and 10,440.01 × 10

^{3}m

^{3}, respectively. Correspondingly, the total system costs are 51.56, 34.48 and 28.02 × 10

^{6}RMB, respectively.

_{1}and w

_{2}are equal to 0.1 and 0.9, respectively. As the q

_{i}value increases from 0.01 to 0.1, the water amounts allocated to tributary 4 in the third period would be 631.59, 451.01 and 376.90 × 10

^{3}m

^{3}, respectively (see Table 5). As shown in Figure 4, the water amounts drawn from the surface water are 14,658.54, 11,545.56 and 10,428.10 × 10

^{3}m

^{3}, respectively; the water amounts sourced from groundwater are 34,268.16, 22,376.18 and 17,700.67 × 10

^{3}m

^{3}, respectively. The total leakage loss rates are 18,976.33, 12,753.19 and 10,389.75 × 10

^{3}m

^{3}, respectively, and the total system costs are 54.01, 36.34 and 29.57 × 10

^{6}RMB, respectively. The above variations in the objective functions and decision variables reflect the trade-off between system objective realization and constraint satisfaction degree. A low water requirement is associated with a reduced amount of water supply, a low leakage loss and a low system cost, which means an improvement in system efficiency. Nevertheless, the system-failure risk would become high due to insufficient water provision. Conversely, a higher system cost could ensure that the water demand is better satisfied and the system remains more stable.

_{1}= 0.9 and w

_{2}= 0.1). Under q

_{i}values of 0.01, 0.05 and 0.1, the difference values between extracted surface water amounts and groundwater amounts are −3830.21, 1218.38 and 2729.57 × 10

^{3}m

^{3}, respectively. Conversely, when the leakage loss is considered as a critical factor, the groundwater becomes a preferred source, where the difference values are −19,609.62, −10,830.62 and −7272.57 × 10

^{3}m

^{3}, respectively. This is because the leakage loss situation occurs in the transportation path between surface water and treatment plants. Moreover, the selection of the transportation path is also dependent on the weight coefficients. For example, it is required that the tributary 7 is able to receive the water drawn from treatment plants 2 and 4, respectively. The path between treatment plant 2 and tributary 7 is adopted due to its low leakage loss. At three probabilistic levels, the transferred water amounts are 2281.57, 1890.30 and 1711.34 × 10

^{3}m

^{3}, respectively. A similar situation is also reflected in tributary 5, which receives the water sourced from treatment plant 1, rather than plant 2. The received water amounts are 1533.50, 1361.36 and 1277.75 × 10

^{3}m

^{3}, respectively. The variations in the weight coefficients not only affect the decision variables, but also the objective values. Under the economic-prior condition (w

_{1}= 0.9 and w

_{2}= 0.1), the low costs are expected (namely 51.56, 34.48 and 28.02 ×10

^{6}RMB, respectively). Meanwhile, the high leakage losses are unavoidable (i.e., 19,055.62, 12,813.74 and 10,440.01 × 10

^{3}m

^{3}, respectively). Conversely, when the resource protection obtains more attention (w

_{1}= 0.1 and w

_{2}= 0.9), the low leakage loss amounts and the high operational costs would be expected (i.e., the two groups of objective values are 18,976.33, 12,753.19 and 10,389.75 × 10

^{3}m

^{3}and 54.01, 36.34 and 29.57 × 10

^{6}RMB, respectively).

_{1}= 0.1 and w

_{2}= 0.9 is recommended as the decision basis for decision-making due to their robust characteristics, although the high system costs are inevitable. The successful application of the SMOCCP model in the Xiaoqing River watershed provides a good example for other watersheds in solving similar problems.

#### 4.2. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Formulation and solution framework of the stochastic multi-objective chance-constrained programming model (SMOCCP) model.

Type | Item | System Parameters (×10^{3} m^{3}) | |
---|---|---|---|

Beginning Inventory | Maximum Capacities | ||

Water source | Surface water | 19,000 | 2950 |

Groundwater | 4200 | 3600 | |

Treatment Plant | First Water Purification Plant | 7.5 | 1300 |

Second Water Purification Plant | 13 | 2100 | |

Dajin Sewage Treatment Plant | 0 | 10,000,000 | |

Tantou Sewage Treatment Plant | 0 | 10,000,000 | |

Reservoir | Dazhan Reservoir | 26 | 3150 |

Duzhuang Reservoir | 13.5 | 580 | |

Mengshan Reservoir | 3.5 | 185 | |

Duozhuang Reservoir | 13.5 | 345 | |

Xinglin Reservoir | 4.5 | 185 | |

Langmaoshan Reservoir | 38 | 560 | |

Taihe Reservoir | 6.5 | 345 |

Type | Parameters | Item | ||||||
---|---|---|---|---|---|---|---|---|

r = 1 | r = 2 | r = 3 | r = 4 | r = 5 | r = 6 | r = 7 | ||

t = 1 | transportation cost | 0 | 320 | 420 | 0 | 418 | 0 | 0 |

leakage rate | 0 | 0.05 | 0.05 | 0 | 0.01 | 0 | 0 | |

t = 2 | transportation cost | 0 | 220 | 0 | 222 | 315 | 350 | 110 |

leakage rate | 0 | 0.05 | 0 | 0.03 | 0.03 | 0.07 | 0.01 | |

t = 3 | transportation cost | 4400 | 1400 | 1350 | 530 | 0 | 0 | 0 |

leakage rate | 0.51 | 0.25 | 0.25 | 0.08 | 0 | 0 | 0 | |

t = 4 | transportation cost | 330 | 0 | 0 | 386 | 0 | 0 | 110 |

leakage rate | 0.01 | 0 | 0 | 0.03 | 0 | 0 | 0.06 | |

b = 1 | transportation cost | 580 | 0 | 0 | 0 | 0 | 0 | 0 |

leakage rate | 0.45 | 0 | 0 | 0 | 0 | 0 | 0 | |

b = 2 | transportation cost | 0 | 180 | 0 | 0 | 0 | 0 | 0 |

leakage rate | 0 | 0.15 | 0 | 0 | 0 | 0 | 0 | |

b = 3 | transportation cost | 0 | 0 | 180 | 0 | 0 | 0 | 0 |

leakage rate | 0 | 0 | 0.10 | 0 | 0 | 0 | 0 | |

b = 4 | transportation cost | 0 | 0 | 0 | 180 | 0 | 0 | 0 |

leakage rate | 0 | 0 | 0 | 0.38 | 0 | 0 | 0 | |

b = 5 | transportation cost | 0 | 0 | 0 | 0 | 180 | 0 | 0 |

leakage rate | 0 | 0 | 0 | 0 | 0.15 | 0 | 0 | |

b = 6 | transportation cost | 0 | 0 | 0 | 0 | 0 | 340 | 0 |

leakage rate | 0 | 0 | 0 | 0 | 0 | 0.42 | 0 | |

b = 7 | transportation cost | 0 | 0 | 0 | 0 | 0 | 0 | 180 |

leakage rate | 0 | 0 | 0 | 0 | 0 | 0 | 0.05 |

Planning Period | Required Water Amounts (×10^{3} m^{3}) | Recovered Water Amounts (×10^{3} m^{3}) | |||||||
---|---|---|---|---|---|---|---|---|---|

b = 1 | b = 2 | b = 3 | b = 4 | b = 5 | b = 6 | b = 7 | Surface Water | Groundwater | |

1 | 6.10 ^{m} | 4.65 | 4.55 | 4.85 | 4.28 | 4.61 | 4.58 | 3172 | 629 |

0.63 ^{s} | 0.29 | 0.23 | 0.52 | 0.17 | 0.28 | 0.25 | |||

2 | 5.75 | 4.62 | 4.52 | 4.82 | 4.28 | 4.60 | 4.52 | 10,343 | 2059 |

0.75 | 0.25 | 0.21 | 0.49 | 0.15 | 0.26 | 0.21 | |||

3 | 5.34 | 4.62 | 4.52 | 4.82 | 4.28 | 4.60 | 4.52 | 14,359 | 2914 |

0.61 | 0.25 | 0.21 | 0.49 | 0.15 | 0.26 | 0.21 | |||

4 | 5.27 | 4.62 | 4.52 | 4.82 | 4.28 | 4.60 | 4.52 | 8492 | 1648 |

0.75 | 0.25 | 0.21 | 0.49 | 0.15 | 0.26 | 0.21 | |||

5 | 5.36 | 4.65 | 4.55 | 4.85 | 4.28 | 4.61 | 4.58 | 13,267 | 2676 |

0.59 | 0.29 | 0.23 | 0.52 | 0.17 | 0.28 | 0.25 | |||

6 | 5.35 | 4.70 | 4.61 | 4.90 | 4.29 | 4.63 | 4.61 | 16,782 | 3432 |

0.86 | 0.35 | 0.28 | 0.54 | 0.21 | 0.37 | 0.29 | |||

7 | 5.10 | 4.45 | 4.28 | 4.70 | 4.10 | 4.52 | 4.29 | 15,455 | 3169 |

0.81 | 0.32 | 0.21 | 0.35 | 0.14 | 0.21 | 0.20 | |||

8 | 5.34 | 4.62 | 4.52 | 4.82 | 4.28 | 4.60 | 4.52 | 11,782 | 2387 |

0.61 | 0.25 | 0.21 | 0.49 | 0.15 | 0.26 | 0.21 | |||

9 | 5.35 | 4.70 | 4.58 | 4.90 | 4.29 | 4.61 | 4.61 | 2606 | 541 |

0.78 | 0.35 | 0.25 | 0.54 | 0.20 | 0.29 | 0.28 | |||

10 | 5.47 | 4.80 | 4.61 | 5.00 | 4.32 | 4.65 | 4.65 | 225 | 65 |

0.75 | 0.40 | 0.29 | 0.54 | 0.20 | 0.40 | 0.40 | |||

11 | 5.47 | 4.75 | 4.61 | 5.00 | 4.31 | 4.65 | 4.63 | 239 | 53 |

0.67 | 0.37 | 0.28 | 0.54 | 0.20 | 0.40 | 0.37 | |||

12 | 5.51 | 4.72 | 4.60 | 4.90 | 4.31 | 4.63 | 4.61 | 74 | 56 |

0.40 | 0.37 | 0.26 | 0.54 | 0.20 | 0.37 | 0.29 |

**Table 4.**Part of the solutions from the SMOCCP model under three constraint-violation levels at w

_{1}= 0.9 and w

_{2}= 0.1 (×10

^{3}m

^{3}).

p | Transferred Path | k = 1 | k = 2 | k = 3 | k = 4 | k = 5 | k = 6 | k = 7 | k = 8 | k = 9 | k = 10 | k = 11 | k = 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.01 | S1→T1 | 2026.92 | 400.41 | 148.43 | 521.78 | 566.54 | 438.24 | 359.36 | 419.15 | 558.86 | 666.17 | 978.77 | 20.84 |

S1→T2 | 4274.80 | 1514.01 | 48.12 | 48.13 | 48.23 | 2085.75 | 535.48 | 1622.93 | 1160.58 | 4042.63 | 101.58 | 0.19 | |

S2→T4 | 3539.77 | 22,874.71 | 0.76 | 0.55 | 0.43 | 0.36 | 0.32 | 0.29 | 0.26 | 0.24 | 0.21 | 0.19 | |

T1→R5 | 310.55 | 0.17 | 242.93 | 121.55 | 128.23 | 140.34 | 98.85 | 121.55 | 136.63 | 140.79 | 91.78 | 0.14 | |

T2→R4 | 1049.58 | 651.13 | 295.46 | 651.13 | 707.83 | 785.01 | 407.88 | 651.13 | 785.01 | 867.57 | 867.57 | 785.01 | |

T2→R6 | 915.28 | 341.22 | 341.22 | 93.06 | 0.00 | 722.87 | 0.00 | 943.37 | 363.59 | 494.48 | 339.48 | 0.17 | |

T2→R7 | 180.19 | 506.52 | 154.35 | 0.00 | 0.00 | 557.01 | 122.25 | 12.20 | 0.37 | 540.15 | 0.00 | 208.53 | |

T4→R1 | 3539.32 | 3283.20 | 1577.32 | 2021.45 | 1531.93 | 2864.25 | 1965.14 | 1577.32 | 2375.16 | 2475.19 | 2052.53 | 1155.28 | |

R2→B2 | 240.15 | 215.08 | 215.08 | 215.08 | 240.15 | 288.58 | 213.04 | 215.08 | 288.58 | 364.58 | 320.05 | 310.59 | |

R4→B4 | 686.59 | 631.59 | 631.59 | 631.59 | 686.59 | 761.46 | 395.64 | 631.59 | 761.46 | 841.54 | 841.54 | 761.46 | |

R7→B7 | 184.89 | 156.46 | 156.46 | 156.46 | 184.89 | 206.44 | 121.03 | 156.46 | 200.99 | 280.76 | 253.98 | 206.44 | |

0.05 | S1→T1 | 1940.36 | 341.72 | 66.21 | 451.33 | 288.19 | 319.54 | 340.47 | 392.49 | 452.19 | 603.60 | 488.84 | 255.12 |

S1→T2 | 3971.32 | 1248.16 | 0.00 | 0.01 | 0.11 | 1343.04 | 389.65 | 999.62 | 851.09 | 2856.74 | 0.35 | 0.19 | |

S2→T4 | 2291.94 | 14,086.40 | 0.76 | 0.55 | 0.43 | 0.36 | 0.32 | 0.29 | 0.26 | 0.24 | 0.21 | 0.19 | |

T1→R5 | 296.34 | 0.17 | 219.40 | 109.78 | 114.02 | 121.89 | 89.98 | 109.78 | 119.60 | 123.24 | 56.84 | 0.32 | |

T2→R4 | 839.35 | 464.96 | 109.29 | 464.96 | 497.59 | 543.28 | 322.47 | 464.96 | 543.28 | 600.41 | 600.41 | 543.28 | |

T2→R6 | 854.63 | 284.97 | 284.97 | 284.97 | 26.61 | 614.88 | 0.00 | 524.30 | 298.94 | 375.93 | 121.77 | 0.17 | |

T2→R7 | 150.64 | 485.74 | 137.26 | 137.26 | 157.20 | 171.45 | 63.28 | 0.37 | 0.37 | 215.61 | 199.68 | 171.45 | |

T4→R1 | 2291.48 | 1973.42 | 1041.50 | 1215.03 | 1027.50 | 1591.93 | 1135.82 | 1041.50 | 1394.48 | 1487.76 | 1303.23 | 878.31 | |

R2→B2 | 197.44 | 181.04 | 181.04 | 181.04 | 197.44 | 228.16 | 171.09 | 181.04 | 228.16 | 277.17 | 249.11 | 241.74 | |

R4→B4 | 482.67 | 451.01 | 451.01 | 451.01 | 482.67 | 526.98 | 312.80 | 451.01 | 526.98 | 582.40 | 582.40 | 526.98 | |

R7→B7 | 155.63 | 135.88 | 135.88 | 135.88 | 155.63 | 169.73 | 105.94 | 135.88 | 166.55 | 213.45 | 197.68 | 169.73 | |

0.1 | S1→T1 | 1899.90 | 314.07 | 7.55 | 322.75 | 315.04 | 339.78 | 255.06 | 303.34 | 481.05 | 536.80 | 405.23 | 223.30 |

S1→T2 | 3843.06 | 1123.63 | 0.00 | 0.01 | 0.11 | 839.73 | 317.66 | 878.60 | 723.41 | 2323.68 | 0.35 | 0.19 | |

S2→T4 | 1816.46 | 10,904.65 | 0.76 | 0.55 | 0.43 | 0.36 | 0.32 | 0.29 | 0.26 | 0.24 | 0.21 | 0.19 | |

T1→R5 | 289.42 | 0.17 | 207.80 | 103.98 | 107.09 | 113.06 | 85.58 | 103.98 | 111.41 | 114.80 | 40.12 | 0.32 | |

T2→R4 | 754.13 | 388.56 | 32.89 | 388.56 | 412.37 | 446.49 | 284.51 | 388.56 | 446.49 | 493.45 | 493.45 | 446.49 | |

T2→R6 | 826.66 | 256.45 | 261.32 | 258.89 | 265.37 | 304.62 | 0.00 | 480.88 | 269.31 | 324.83 | 27.13 | 0.17 | |

T2→R7 | 136.84 | 475.81 | 127.32 | 127.32 | 143.41 | 154.46 | 29.97 | 0.37 | 0.37 | 186.30 | 174.71 | 154.46 | |

T4→R1 | 1816.00 | 1504.45 | 834.78 | 926.29 | 830.46 | 1163.99 | 848.01 | 834.78 | 1049.86 | 1134.20 | 1022.99 | 758.92 | |

R2→B2 | 177.88 | 165.15 | 165.15 | 165.15 | 177.88 | 201.30 | 152.22 | 165.15 | 201.30 | 239.49 | 217.96 | 211.51 | |

R4→B4 | 400.00 | 376.90 | 376.90 | 376.90 | 400.00 | 433.09 | 275.98 | 376.90 | 433.09 | 478.64 | 478.64 | 433.09 | |

R7→B7 | 141.97 | 126.05 | 126.05 | 126.05 | 141.97 | 152.91 | 98.68 | 126.05 | 150.68 | 184.44 | 172.96 | 152.91 |

**Table 5.**Part of the solutions from the SMOCCP model under three constraint-violation levels at w

_{1}= 0.1 and w

_{2}= 0.9 (×10

^{3}m

^{3}).

p | Transferred Path | k = 1 | k = 2 | k = 3 | k = 4 | k = 5 | k = 6 | k = 7 | k = 8 | k = 9 | k = 10 | k = 11 | k = 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.01 | S1→T1 | 921.66 | 2366.99 | 0.00 | 0.00 | 0.02 | 713.33 | 598.08 | 471.55 | 282.45 | 1071.53 | 0.06 | 679.81 |

S1→T2 | 4227.46 | 504.30 | 145.88 | 0.00 | 0.00 | 0.00 | 0.00 | 263.33 | 205.07 | 785.94 | 758.62 | 662.47 | |

S2→T4 | 3579.00 | 30,689.08 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | |

T1→R5 | 310.55 | 56.22 | 0.00 | 308.42 | 128.23 | 0.00 | 239.18 | 71.31 | 0.00 | 142.84 | 139.14 | 137.60 | |

T2→R4 | 654.23 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

T2→R6 | 915.28 | 341.22 | 324.96 | 331.68 | 379.79 | 447.32 | 275.55 | 102.66 | 0.00 | 494.48 | 494.48 | 447.32 | |

T2→R7 | 528.68 | 158.04 | 154.45 | 155.93 | 174.74 | 0.00 | 0.00 | 158.04 | 203.02 | 283.60 | 256.55 | 208.53 | |

T4→R1 | 3539.32 | 3283.20 | 1577.32 | 2021.45 | 1531.93 | 2864.25 | 1965.14 | 1577.32 | 2375.16 | 2475.19 | 2052.53 | 1155.28 | |

R2→B2 | 240.15 | 215.08 | 215.08 | 215.08 | 240.15 | 288.58 | 213.04 | 215.08 | 288.58 | 364.58 | 320.05 | 310.59 | |

R4→B4 | 686.59 | 631.59 | 631.59 | 631.59 | 686.59 | 761.46 | 395.64 | 631.59 | 761.46 | 841.54 | 841.54 | 761.46 | |

R7→B7 | 184.89 | 156.46 | 156.46 | 156.46 | 184.89 | 206.44 | 121.03 | 156.46 | 200.99 | 280.76 | 253.98 | 206.44 | |

0.05 | S1→T1 | 835.10 | 2284.77 | 0.00 | 0.00 | 0.02 | 277.48 | 626.91 | 305.46 | 172.92 | 882.87 | 0.06 | 554.48 |

S1→T2 | 3475.50 | 136.05 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 426.50 | 243.39 | 217.79 | 581.42 | 524.86 | |

S2→T4 | 2775.17 | 19,600.92 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | |

T1→R5 | 296.34 | 32.69 | 0.00 | 296.65 | 114.02 | 121.89 | 89.98 | 42.51 | 0.00 | 125.29 | 121.77 | 120.22 | |

T2→R4 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

T2→R6 | 854.63 | 265.88 | 287.89 | 273.23 | 321.26 | 348.16 | 239.32 | 284.97 | 72.72 | 0.00 | 375.93 | 348.16 | |

T2→R7 | 499.12 | 133.04 | 137.90 | 134.67 | 93.33 | 0.00 | 0.00 | 137.26 | 168.24 | 215.61 | 199.68 | 171.45 | |

T4→R1 | 2291.48 | 1973.42 | 1041.50 | 1215.03 | 1027.50 | 1591.93 | 1135.82 | 1041.50 | 1394.48 | 1487.76 | 1303.23 | 878.31 | |

R2→B2 | 197.44 | 181.04 | 181.04 | 181.04 | 197.44 | 228.16 | 171.09 | 181.04 | 228.16 | 277.17 | 249.11 | 241.74 | |

R4→B4 | 482.67 | 451.01 | 451.01 | 451.01 | 482.67 | 526.98 | 312.80 | 451.01 | 526.98 | 582.40 | 582.40 | 526.98 | |

R7→B7 | 155.63 | 135.88 | 135.88 | 135.88 | 155.63 | 169.73 | 105.94 | 135.88 | 166.55 | 213.45 | 197.68 | 169.73 | |

0.1 | S1→T1 | 794.64 | 2245.52 | 0.00 | 0.00 | 0.02 | 149.45 | 482.89 | 167.37 | 269.16 | 796.94 | 0.06 | 497.84 |

S1→T2 | 3433.32 | 106.31 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 844.40 | 176.47 | 463.72 | |

S2→T4 | 2214.48 | 15486.11 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | |

T1→R5 | 289.42 | 21.10 | 0.00 | 290.85 | 107.09 | 113.06 | 2.70 | 0.00 | 111.41 | 116.84 | 113.41 | 111.86 | |

T2→R4 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

T2→R6 | 826.66 | 242.81 | 272.14 | 243.23 | 283.85 | 304.62 | 148.05 | 0.00 | 0.00 | 649.66 | 0.00 | 304.62 | |

T2→R7 | 485.33 | 123.78 | 130.25 | 123.87 | 0.00 | 0.00 | 53.13 | 127.32 | 152.20 | 186.30 | 174.71 | 154.46 | |

T4→R1 | 1816.00 | 1504.45 | 834.78 | 926.29 | 830.46 | 1163.99 | 848.01 | 834.78 | 1049.86 | 1134.20 | 1022.99 | 758.92 | |

R2→B2 | 177.88 | 165.15 | 165.15 | 165.15 | 177.88 | 201.30 | 152.22 | 165.15 | 201.30 | 239.49 | 217.96 | 211.51 | |

R4→B4 | 400.00 | 376.90 | 376.90 | 376.90 | 400.00 | 433.09 | 275.98 | 376.90 | 433.09 | 478.64 | 478.64 | 433.09 | |

R7→B7 | 141.97 | 126.05 | 126.05 | 126.05 | 141.97 | 152.91 | 98.68 | 126.05 | 150.68 | 184.44 | 172.96 | 152.91 |

P | Weighted Combination | Objective Function | Variation Levels of Targeted Parameters | ||||||
---|---|---|---|---|---|---|---|---|---|

−75% | −50% | −25% | 1 | +25% | +50% | +75% | |||

0.01 | w_{1} = 0.9 | TC (×10^{6} RMB) | 12.89 | 25.78 | 38.67 | 51.56 | 64.46 | 77.35 | 90.24 |

w_{2} = 0.1 | LL (×10^{3} m^{3}) | 19,055.62 | 19,055.62 | 19,055.62 | 19,055.62 | 19,055.62 | 19,055.62 | 19,055.62 | |

w_{1} = 0.1 | TC (×10^{6} RMB) | 13.50 | 27.01 | 40.51 | 54.01 | 67.51 | 81.01 | 94.51 | |

w_{2} = 0.9 | LL (×10^{3} m^{3}) | 18,976.33 | 18,976.33 | 18,976.33 | 18,976.33 | 18,976.33 | 18,976.33 | 18,976.33 | |

0.05 | w_{1} = 0.9 | TC (×10^{6} RMB) | 8.62 | 17.24 | 25.86 | 34.48 | 43.10 | 51.72 | 60.34 |

w_{2} = 0.1 | LL (×10^{3} m^{3}) | 12,813.74 | 12,813.74 | 12,813.74 | 12,813.74 | 12,813.74 | 12,813.74 | 12,813.74 | |

w_{1} = 0.1 | TC (×10^{6} RMB) | 9.09 | 18.17 | 27.26 | 36.34 | 45.43 | 54.51 | 63.60 | |

w_{2} = 0.9 | LL (×10^{3} m^{3}) | 12,753.19 | 12,753.19 | 12,753.19 | 12,753.19 | 12,753.19 | 12,753.19 | 12,753.19 | |

0.1 | w_{1} = 0.9 | TC (×10^{6} RMB) | 7.01 | 14.01 | 21.02 | 28.02 | 35.03 | 42.03 | 49.04 |

w_{2} = 0.1 | LL (×10^{3} m^{3}) | 10,440.01 | 10,440.01 | 10,440.01 | 10,440.01 | 10,440.01 | 10,440.01 | 10,440.01 | |

w_{1} = 0.1 | TC (×10^{6} RMB) | 7.39 | 14.79 | 22.18 | 29.57 | 36.96 | 44.36 | 51.75 | |

w_{2} = 0.9 | LL (×10^{3} m^{3}) | 10,389.75 | 10,389.75 | 10,389.75 | 10,389.75 | 10,389.75 | 10,389.75 | 10,389.75 |

**Notes:**TC is the abbreviation of the total costs; LL is the abbreviation of the leakage loss.

P | Weighted Combination | Objective Function | Variation Levels of Targeted Parameters | ||||||
---|---|---|---|---|---|---|---|---|---|

−0.99% | −0.66% | −0.33% | 1 | 0.33% | 0.66% | 0.99% | |||

0.01 | w_{1} = 0.9 | TC (×10^{6} RMB) | 51.19 | 51.32 | 51.44 | 51.56 | 51.69 | 51.81 | 51.94 |

w_{2} = 0.1 | LL (×10^{3} m^{3}) | 18732.59 | 18839.74 | 18947.42 | 19055.62 | 19164.35 | 19273.62 | 19383.43 | |

w_{1} = 0.1 | TC (×10^{6} RMB) | 53.63 | 53.75 | 53.88 | 54.01 | 54.13 | 54.26 | 54.39 | |

w_{2} = 0.9 | LL (×10^{3} m^{3}) | 18,654.29 | 18,761.11 | 18,868.46 | 18,976.33 | 19,084.73 | 19,193.66 | 19,303.14 | |

0.05 | w_{1} = 0.9 | TC (×10^{6} RMB) | 34.24 | 34.32 | 34.40 | 34.48 | 34.56 | 34.64 | 34.72 |

w_{2} = 0.1 | LL (×10^{3} m^{3}) | 12,598.37 | 12,669.82 | 12,741.60 | 12,813.74 | 12,886.22 | 12,959.05 | 13,032.23 | |

w_{1} = 0.1 | TC (×10^{6} RMB) | 36.09 | 36.17 | 36.26 | 36.34 | 36.43 | 36.51 | 36.60 | |

w_{2} = 0.9 | LL (×10^{3} m^{3}) | 12,538.81 | 12,609.93 | 12,681.39 | 12,753.19 | 12,825.34 | 12,897.84 | 12,970.69 | |

0.1 | w_{1} = 0.9 | TC (×10^{6} RMB) | 27.83 | 27.89 | 27.96 | 28.02 | 28.09 | 28.15 | 28.22 |

w_{2} = 0.1 | LL (×10^{3} m^{3}) | 10,265.40 | 10,323.33 | 10,381.53 | 10,440.01 | 10,498.77 | 10,557.81 | 10,617.13 | |

w_{1} = 0.1 | TC (×10^{6} RMB) | 29.37 | 29.44 | 29.50 | 29.57 | 29.64 | 29.71 | 29.78 | |

w_{2} = 0.9 | LL (×10^{3} m^{3}) | 10,215.96 | 10,273.62 | 10,331.54 | 10,389.75 | 10,448.23 | 10,507.00 | 10,566.04 |

**Notes:**TC is the abbreviation of the total costs; LL is the abbreviation of the leakage loss.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, Y.; Li, W.; Ding, X.
A Stochastic Multi-Objective Chance-Constrained Programming Model for Water Supply Management in Xiaoqing River Watershed. *Water* **2017**, *9*, 378.
https://doi.org/10.3390/w9060378

**AMA Style**

Xu Y, Li W, Ding X.
A Stochastic Multi-Objective Chance-Constrained Programming Model for Water Supply Management in Xiaoqing River Watershed. *Water*. 2017; 9(6):378.
https://doi.org/10.3390/w9060378

**Chicago/Turabian Style**

Xu, Ye, Wei Li, and Xiaowen Ding.
2017. "A Stochastic Multi-Objective Chance-Constrained Programming Model for Water Supply Management in Xiaoqing River Watershed" *Water* 9, no. 6: 378.
https://doi.org/10.3390/w9060378