# Multi-Criteria Fuzzy-Stochastic Diffusion Model of Groundwater Control System Selection

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

## 3. Test Site and Description of Alternatives and Criteria

#### 3.1. Alternative A_{1}

#### 3.2. Alternative A_{2}

#### 3.3. Alternative A_{3}

#### 3.4. Technical Criterion (C_{1})

#### 3.5. Energy Consumption (C_{2})

#### 3.6. Capital Expenditure (C_{3})

#### 3.7. Operating Costs (C_{4})

_{t}is the Brownian motion, and ${x}_{t}>0$; is the stochastic process. The following linear Itô-Doob type stochastic differential equation is used to describe the flow of operating costs:

_{t}is normalized Brownian motion. If the separation technique is applied, then Equation (16) becomes:

_{t}. Let $CO=\left\{C{O}_{t},t=0,1,\dots ,T\right\}$ denote a cost scenario with spot costs CO

_{t}, where CO

_{t}is determined by Equation (20). Figure 2 shows sample paths (s = 1, 2, ..., S) of the operating costs simulated using the above equation S times. This criterion needs to be minimized.

## 4. Decision-Making Model Using Simulation

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

_{i}, which has the greatest aggregated overall relative closeness value, is the most suitable alternative and rank z = 1 is assigned to it (for example:${A}_{2}=sup\left\{{Q}_{i=1,2,\dots ,m}^{ag}\right\}\to z=1$). The remaining alternatives are ranked accordingly, in descending order of aggregated overall relative closeness values, and values 2, 3, …, m are assigned to them, respectively. If we take into consideration the number of simulations, then there are S rank orders of the given alternatives. The space of rank order simulation is defined as follows:

**Definition**

**4.**

## 5. Results and Discussion

_{4}criterion (operating cost) over time is presented in Figure 3, while evaluations of information in different time episodes are shown in Table 2, Table 3, Table 4 and Table 5. The operating costs of Alternatives 1 and 3 vary over time considerably, given that the groundwater control system is comprised of 33 wells (Alternative 1) or 13 wells and a small impervious screen (Alternative 3). This means that systems made up of drainage wells are the most expensive option, because of the need for periodic pump replacement and well rehabilitation. This is not the case with the cut-off wall (Alternative 2). The operating costs of a groundwater control system comprised of a cut-off wall reflect solely maintenance labor.

_{1}(Ω

_{3}= 1093), A

_{2}(Ω

_{1}= 870) and A

_{3}(Ω

_{2}= 1037). The lowest total score (sum) represented the optimal solution, meaning that Alternative 2 most often came first in the 500 simulations over time.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Study area (

**A**) and dewatering systems: Alternative 1 (

**B**). Alternative 2 (

**C**), and Alternative 3 (

**D**) (Bajić et al. [9]; modified).

Alternative | Criterion | Value |
---|---|---|

A_{1} | C_{1} → max (FAHP Scale) | (t = 1) = (1, 1, 1); (t = 2) = (1.5, 2, 2.5); (t = 3) = (0.666,1,1.5); (t = 4) = (0.666, 1, 1.5) |

C_{2} → min (×10^{6} kWh) | (t = 1) = 1.445; (t = 2) = 5.913; (t = 3) = 6.044; (t = 4) = 5.913 | |

C_{3} → min (×10^{6} Euro) | (t = 1) = 0.1193; (t = 2) = 1.8351; (t = 3) = 0.2410; (t = 4) = 0.0000 | |

C_{4} → min (×10^{6} Euro) | spot value 0.30; drift 0.0291; cost volatility rate 0.0799 | |

A_{2} | C_{1} → max (FAHP Scale) | (t = 1) = (1, 1, 1); (t = 2) = (0.666, 1, 1.5); (t = 3) = (3.5, 4, 4.5); (t = 4) = (3.5, 4, 4.5) |

C_{2} → min (×10^{6} kWh) | (t = 1) = 1.445; (t = 2) = 1.576; (t = 3) = 1.078; (t = 4) = 1.578 | |

C_{3} → min (×10^{6} Euro) | (t = 1) = 0.1193; (t = 2) = 0.7094; (t = 3) = 9.4490; (t = 4) = 0.0000 | |

C_{4} → min (×10^{6} Euro) | spot value 0.30; drift 0.0014; cost volatility rate 0.0258 | |

A_{3} | C_{1} → max (FAHP Scale) | (t = 1) = (1, 1, 1); (t = 2) = (0.666, 1, 1.5); (t = 3) = (2, 3, 3.5); (t = 4) = (2.5, 3, 3.5) |

C_{2} → min (×10^{6} kWh) | (t = 1) = 1.445; (t = 2) = 1.576; (t = 3) = 3.416; (t = 4) = 3.285 | |

C_{3} → min (×10^{6} Euro) | (t = 1) = 0.1193; (t = 2) = 0.7094; (t = 3) = 7.8848; (t = 4) = 0.0000 | |

C_{4} → min (×10^{6} Euro) | spot value 0.30; drift 0,0142; cost volatility rate 0.0512 | |

Time slice | 4 years | |

Interval | 1 year | |

Sample | 500 simulations |

Criterion | C_{1} | C_{2} | C_{3} | C_{4} | ||
---|---|---|---|---|---|---|

Alternative | a_{1} | b_{1} | c_{1} | |||

A_{1} | 1 | 1 | 1 | 1.445 | 0.1194 | 0.3352 |

A_{2} | 1 | 1 | 1 | 1.445 | 0.1194 | 0.3031 |

A_{3} | 1 | 1 | 1 | 1.445 | 0.1194 | 0.296 |

Criterion | C_{1} | C_{2} | C_{3} | C_{4} | ||
---|---|---|---|---|---|---|

Alternative | a_{1} | b_{1} | c_{1} | |||

A_{1} | 1.5 | 2 | 2.5 | 5.913 | 1.8351 | 0.3213 |

A_{2} | 0.666 | 1 | 1.5 | 1.577 | 0.7095 | 0.3063 |

A_{3} | 0.666 | 1 | 1.5 | 1.577 | 0.7095 | 0.3012 |

Criterion | C_{1} | C_{2} | C_{3} | C_{4} | ||
---|---|---|---|---|---|---|

Alternative | a_{1} | b_{1} | c_{1} | |||

A_{1} | 0.666 | 1 | 1.5 | 6.044 | 0.2410 | 0.2975 |

A_{2} | 3.5 | 4 | 4.5 | 1.078 | 9.4490 | 0.3102 |

A_{3} | 2 | 3 | 3.5 | 3.416 | 7.8849 | 0.3132 |

Criterion | C_{1} | C_{2} | C_{3} | C_{4} | ||
---|---|---|---|---|---|---|

Alternative | a_{1} | b_{1} | c_{1} | |||

A_{1} | 0.666 | 1 | 1.5 | 5.913 | 0 | 0.307 |

A_{2} | 3.5 | 4 | 4.5 | 1.5786 | 0 | 0.32 |

A_{3} | 2.5 | 3 | 3.5 | 3.285 | 0 | 0.3366 |

Time | t_{1} | t_{2} | t_{3} | t_{4} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

a_{1} | b_{1} | c_{1} | a_{2} | b_{2} | c_{2} | a_{3} | b_{3} | c_{3} | a_{4} | b_{4} | c_{4} | |

C_{1} | 0 | 0 | 0 | 0.56693 | 0.15412 | 0.36484 | 0.27322 | 0.19051 | 0.29578 | 0.49894 | 0.49565 | 0.67533 |

C_{2} | 0 | 0 | 0 | 0.38014 | 0.55332 | 0.33111 | 0.24358 | 0.27796 | 0.2479 | 0.39354 | 0.50151 | 0.43985 |

C_{3} | 0 | 0 | 0 | 0.20031 | 0.29157 | 0.17448 | 0.46546 | 0.53115 | 0.4737 | 0 | 0 | 0 |

C_{4} | 1 | 1 | 1 | 0.00068 | 0.00099 | 0.0059 | 0.00033 | 0.00038 | 0.00034 | 0.00022 | 0.00285 | 0.0025 |

Time | t_{1} | t_{2} | t_{3} | t_{4} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Alternative | a_{1} | b_{1} | c_{1} | a_{2} | b_{2} | c_{2} | a_{3} | b_{3} | c_{3} | a_{4} | b_{4} | c_{4} |

A_{1} | 0 | 0 | 0 | 0.56693 | 0.15412 | 0.36484 | 0.27322 | 0.19051 | 0.29578 | 0.49894 | 0.49565 | 0.67533 |

A_{2} | 0.81921 | 0.81921 | 0.81921 | 0.38014 | 0.55332 | 0.33111 | 0.24358 | 0.27796 | 0.2479 | 0.39354 | 0.50151 | 0.43985 |

A_{3} | 1 | 1 | 1 | 0.20031 | 0.29157 | 0.17448 | 0.46546 | 0.53115 | 0.4737 | 0.12313 | 0.1741 | 0.32331 |

Time | t_{1} | t_{2} | t_{3} | t_{4} | ||||
---|---|---|---|---|---|---|---|---|

Alternative | DC | Rank | DC | Rank | DC | Rank | DC | Rank |

A_{1} | 0 | 3 | 0.33761 | 3 | 0.60837 | 1 | 0.18191 | 3 |

A_{2} | 0.81921 | 2 | 0.62429 | 2 | 0.38298 | 2 | 0.72725 | 1 |

A_{3} | 1 | 1 | 0.62429 | 1 | 0.31661 | 3 | 0.51349 | 2 |

Time | t_{1} | t_{2} | t_{3} | t_{4} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

a_{1} | b_{1} | c_{1} | a_{2} | b_{2} | c_{2} | a_{3} | b_{3} | c_{3} | a_{4} | b_{4} | c_{4} | |

λ(t) | 0.49541 | 0.48782 | 0.50520 | 0.13216 | 0.14230 | 0.12515 | 0.11764 | 0.13231 | 0.11793 | 0.23542 | 0.23774 | 0.27149 |

Alternative | AORC | DAORC | Rank | ||
---|---|---|---|---|---|

a_{1} | b_{1} | c_{1} | |||

A_{1} | 0.11657 | 0.10308 | 0.19474 | 0.13813 | 3 |

A_{2} | 0.53704 | 0.63622 | 0.66978 | 0.61435 | 2 |

A_{3} | 0.60760 | 0.69233 | 0.71725 | 0.67239 | 1 |

Simulation | Rank | ||
---|---|---|---|

A_{1} | A_{2} | A_{3} | |

1 | 1 | 3 | 2 |

2 | 1 | 2 | 3 |

3 | 3 | 1 | 2 |

4 | 1 | 3 | 2 |

5 | 3 | 2 | 1 |

6 | 2 | 3 | 1 |

7 | 1 | 2 | 3 |

8 | 3 | 1 | 2 |

9 | 1 | 3 | 2 |

10 | 3 | 1 | 2 |

⁝ | ⁝ | ⁝ | ⁝ |

495 | 1 | 3 | 2 |

496 | 1 | 2 | 3 |

497 | 3 | 2 | 1 |

498 | 1 | 2 | 3 |

499 | 3 | 1 | 2 |

500 | 3 | 1 | 2 |

Sum | 1093 | 870 | 1037 |

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## Share and Cite

**MDPI and ACS Style**

Polomčić, D.; Gligorić, Z.; Bajić, D.; Gligorić, M.; Negovanović, M.
Multi-Criteria Fuzzy-Stochastic Diffusion Model of Groundwater Control System Selection. *Symmetry* **2019**, *11*, 705.
https://doi.org/10.3390/sym11050705

**AMA Style**

Polomčić D, Gligorić Z, Bajić D, Gligorić M, Negovanović M.
Multi-Criteria Fuzzy-Stochastic Diffusion Model of Groundwater Control System Selection. *Symmetry*. 2019; 11(5):705.
https://doi.org/10.3390/sym11050705

**Chicago/Turabian Style**

Polomčić, Dušan, Zoran Gligorić, Dragoljub Bajić, Miloš Gligorić, and Milanka Negovanović.
2019. "Multi-Criteria Fuzzy-Stochastic Diffusion Model of Groundwater Control System Selection" *Symmetry* 11, no. 5: 705.
https://doi.org/10.3390/sym11050705