# Multi-Attribute Technological Modeling of Coal Deposits Based on the Fuzzy TOPSIS and C-Mean Clustering Algorithms

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{a}constraints where n

_{a}is the total number of block attributes. The technological model significantly reduces the number of constraints i.e., only one constraint is employed. This means that if the relative closeness of the mining cut is greater than a predefined value, all blocks belonging to the cut will be processed, otherwise they will be treated as waste. Most models are based on the maximization of the net present value and they are static models with respect to asset (metal) price and production costs. Including volatility of the asset price makes them impossible to solve. The technological model provides an opportunity to overcome this situation and makes the planning process more realistic. Forecasted price and costs are used to calculate the unit value of the each cluster for every year of the project time (${V}_{ij}^{cluster}$, where i denotes the i-th cluster in the j-th year of the project time). When we calculate the cash flow, it is only need to explore which blocks of the clusters can be mined for a given year. In this way we add a dynamic dimension to the problem of production planning and make it more realistic. The model also gives information about how much the quality of the block deviates from the desired value and helps planners deal with the process of blending.

## 2. Coal Deposit Partitioning

#### 2.1. The Concept of the Model

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- calculation of the relative closeness of every mineable block to the target values based on the Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) method,
- clustering the obtained values based on the fuzzy C-mean clustering method.

#### 2.2. The Relative Closeness

_{1}) is divided into two criteria with respect to the sign of the distance function (f

_{1}> 0; f

_{1}< 0). Suppose there is a sequence of mineable blocks with ascending order of the heating value and required (target) value, see Figure 1.

_{1}:

_{3}, f

_{4}) is also evaluated by Equation (11). Both criteria should be maximized. Finally we obtain the following decision matrix:

#### 2.3. Coal Deposit Partitioning Model

_{n}(n = 1, 2, …, N) and fuzzy partition matrix U by minimizing the following function:

- Step 1:
- select an integer number of technological mining cuts i.e., clusters (N) and threshold value $\epsilon $; let ω = 2;
- Step 2:
- input a set of initial cluster centers $\left[{c}_{1},{c}_{2},\dots ,{c}_{n}\right]$, composed of the increasing order values randomly chosen from the interval $\left[min\left\{{s}_{i}\right\},max\left\{{s}_{i}\right\}\right],i=1,\mathrm{2..},k$;
- Step 3:
- compute all $\sqrt{{\left({c}_{n}-{s}_{i}\right)}^{2}}$ and then all u
_{ni}according to Equation (26); - Step 4:
- update the set of initial cluster centers according to Equation (25);
- Step 5:
- compute the value of the objective function J according to Equation (21) and compare J
^{(t+1)}with J^{(t)}, where t is the iteration number. If $|{J}^{\left(t+1\right)}-{J}^{\left(t\right)}|<\epsilon $ then stop otherwise return to Step 2.

## 3. Numerical Example

- (1)
- Heating value:$${\tilde{r}}_{11}=\frac{\left(8440.659378.5010316.35\right)}{\left(620025688917757808\right)+\left(749488329715\right)}=\left(0.011000.013440.01644\right)$$$${\tilde{r}}_{1}=\frac{\left(749488329715\right)}{\left(620025688917757808\right)+\left(749488329715\right)}=\left(0.009760.012660.01548\right)$$
- (2)
- Sulfur content:$${\tilde{r}}_{12}=\frac{\left(1.421.581.74\right)}{\left(117.42130.43143.46\right)+\left(1.501.671.84\right)}=\left(0,09770.011960.01463\right)$$$${\tilde{r}}_{2}=\frac{\left(1.501.671.84\right)}{\left(117.42130.43143.46\right)+\left(1.501.671.84\right)}=\left(0.010320.012640.01547\right)$$
- (3)
- ash content:$${\tilde{r}}_{13}=\frac{\left(21.6324.0326.43\right)}{\left(1784.761983.122181.41\right)+\left(22.8825.4227.97\right)}=\left(0.009790.011960.01462\right)$$$${\tilde{r}}_{3}=\frac{\left(22.8825.4227.97\right)}{\left(1784.761983.122181.41\right)+\left(22.8825.4227.97\right)}=\left(0.010360.012660.01547\right)$$

- (1)
- Heating value:$${\tilde{w}}_{11}=\frac{\left(0.011000.013440.01644\right)}{\left(0.011000.013440.01644\right)+\left(0,09770.011960.01463\right)+\left(0.009790.011960.01462\right)}=\left(0.24060.35970.5379\right)$$$${\tilde{w}}_{1}=\frac{1}{3}=0.3333$$
- (2)
- Sulfur content:$${\tilde{w}}_{12}=\frac{\left(0,09770.011960.01463\right)}{\left(0.011000.013440.01644\right)+\left(0,09770.011960.01463\right)+\left(0.009790.011960.01462\right)}=\left(0.21380.32000.4787\right)$$$${\tilde{w}}_{2}=\frac{1}{3}=0.3333$$
- (3)
- Ash content:$${\tilde{w}}_{13}=\frac{\left(0.009790.011960.01462\right)}{\left(0.011000.013440.01644\right)+\left(0,09770.011960.01463\right)+\left(0.009790.011960.01462\right)}=\left(0.21420.32010.4784\right)$$$${\tilde{w}}_{3}=\frac{1}{3}=0.3333$$

_{1}, c

_{2}, c

_{3}, c

_{4}, c

_{5}] = [0.38, 0.42, 0.46, 0.50, 0.54]. Calculation of the first block membership degree indicating with what degree the relative closeness s

_{1}belongs to the initial cluster center vector [0.38, 0.42, 0.46, 0.50, 0.54] is represented in Table 4.

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

TOPSIS | Technique for order preference by similarity to ideal solution |

ARI | adjusted Rand index |

FS | Fukuyama-Sugeno validity functional |

AV | alternative matrix |

H(P), H(E) | entropy of cluster P and E |

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**Figure 5.**Technological model of coal deposit for matrix (78 × 4); 1—very small relative closeness; 2—small relative closeness; 3—medium relative closeness; 4—high relative closeness; 5—very high relative closeness.

**Figure 10.**Distribution of the technological mining cuts; (

**a**) for the (78 × 4) input data matrix—four criteria; (

**b**) for the (78 × 6) input data matrix—six criteria.

Block | Heating Value (kJ/kg) | Sulfur (%) | Ash (%) | Block | Heating Value (kJ/kg) | Sulfur (%) | Ash (%) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 8440.65 | 9378.50 | 10316.35 | 1.42 | 1.58 | 1.74 | 21.63 | 24.03 | 26.43 | 40 | 7338.38 | 8153.75 | 8969.13 | 1.61 | 1.79 | 1.96 | 24.45 | 27.17 | 29.88 |

2 | 7339.50 | 8155.00 | 8970.50 | 1.61 | 1.79 | 1.96 | 24.45 | 27.16 | 29.88 | 41 | 7928.10 | 8809.00 | 9689.90 | 1.51 | 1.68 | 1.84 | 22.94 | 25.49 | 28.04 |

3 | 8302.73 | 9225.25 | 10147.78 | 1.45 | 1.61 | 1.77 | 21.98 | 24.42 | 26.86 | 42 | 8298.00 | 9220.00 | 10142.00 | 1.45 | 1.61 | 1.77 | 21.99 | 24.44 | 26.88 |

4 | 8059.28 | 8954.75 | 9850.23 | 1.49 | 1.65 | 1.82 | 22.60 | 25.12 | 27.63 | 43 | 8059.05 | 8954.50 | 9849.95 | 1.49 | 1.65 | 1.82 | 22.60 | 25.12 | 27.63 |

5 | 8446.05 | 9384.50 | 10322.95 | 1.42 | 1.58 | 1.74 | 21.61 | 24.01 | 26.42 | 44 | 7814.48 | 8682.75 | 9551.03 | 1.53 | 1.70 | 1.87 | 23.23 | 25.81 | 28.39 |

6 | 8496.23 | 9440.25 | 10384.28 | 1.41 | 1.57 | 1.73 | 21.48 | 23.87 | 26.26 | 45 | 7881.75 | 8757.50 | 9633.25 | 1.52 | 1.68 | 1.85 | 23.06 | 25.62 | 28.18 |

7 | 7328.03 | 8142.25 | 8956.48 | 1.61 | 1.79 | 1.97 | 24.48 | 27.20 | 29.92 | 46 | 8073.45 | 8970.50 | 9867.55 | 1.48 | 1.65 | 1.81 | 22.57 | 25.08 | 27.58 |

8 | 7752.83 | 8614.25 | 9475.68 | 1.54 | 1.71 | 1.88 | 23.39 | 25.99 | 28.59 | 47 | 8237.48 | 9152.75 | 10068.03 | 1.46 | 1.62 | 1.78 | 22.15 | 24.61 | 27.07 |

9 | 8260.65 | 9178.50 | 10096.35 | 1.45 | 1.61 | 1.78 | 22.09 | 24.54 | 27.00 | 48 | 7917.53 | 8797.25 | 9676.98 | 1.51 | 1.68 | 1.85 | 22.97 | 25.52 | 28.07 |

10 | 7827.30 | 8697.00 | 9566.70 | 1.53 | 1.69 | 1.86 | 23.20 | 25.78 | 28.35 | 49 | 8177.63 | 9086.25 | 9994.88 | 1.47 | 1.63 | 1.79 | 22.30 | 24.78 | 27.26 |

11 | 8504.10 | 9449.00 | 10393.90 | 1.41 | 1.57 | 1.72 | 21.46 | 23.85 | 26.23 | 50 | 7709.18 | 8565.75 | 9422.33 | 1.55 | 1.72 | 1.89 | 23.50 | 26.11 | 28.72 |

12 | 7828.43 | 8698.25 | 9568.08 | 1.53 | 1.69 | 1.86 | 23.20 | 25.77 | 28.35 | 51 | 8378.33 | 9309.25 | 10240.18 | 1.43 | 1.59 | 1.75 | 21.79 | 24.21 | 26.63 |

13 | 7390.80 | 8212.00 | 9033.20 | 1.60 | 1.78 | 1.95 | 24.32 | 27.02 | 29.72 | 52 | 8415.00 | 9350.00 | 10285.00 | 1.43 | 1.58 | 1.74 | 21.69 | 24.10 | 26.51 |

14 | 8364.60 | 9294.00 | 10223.40 | 1.43 | 1.59 | 1.75 | 21.82 | 24.25 | 26.67 | 53 | 7890.08 | 8766.75 | 9643.43 | 1.51 | 1.68 | 1.85 | 23.04 | 25.60 | 28.16 |

15 | 8177.40 | 9086.00 | 9994.60 | 1.47 | 1.63 | 1.79 | 22.30 | 24.78 | 27.26 | 54 | 8494.43 | 9438.25 | 10382.08 | 1.41 | 1.57 | 1.73 | 21.49 | 23.88 | 26.26 |

16 | 7943.18 | 8825.75 | 9708.33 | 1.51 | 1.67 | 1.84 | 22.90 | 25.45 | 27.99 | 55 | 7746.08 | 8606.75 | 9467.43 | 1.54 | 1.71 | 1.88 | 23.41 | 26.01 | 28.61 |

17 | 7538.40 | 8376.00 | 9213.60 | 1.57 | 1.75 | 1.92 | 23.94 | 26.60 | 29.26 | 56 | 8512.43 | 9458.25 | 10404.08 | 1.41 | 1.57 | 1.72 | 21.44 | 23.83 | 26.21 |

18 | 8457.53 | 9397.25 | 10336.98 | 1.42 | 1.58 | 1.73 | 21.58 | 23.98 | 26.38 | 57 | 7334.55 | 8149.50 | 8964.45 | 1.61 | 1.79 | 1.97 | 24.46 | 27.18 | 29.90 |

19 | 7539.98 | 8377.75 | 9215.53 | 1.57 | 1.75 | 1.92 | 23.93 | 26.59 | 29.25 | 58 | 7255.58 | 8061.75 | 8867.93 | 1.63 | 1.81 | 1.99 | 24.30 | 27.00 | 29.70 |

20 | 7871.63 | 8746.25 | 9620.88 | 1.52 | 1.69 | 1.86 | 23.08 | 25.65 | 28.21 | 59 | 8372.93 | 9303.25 | 10233.58 | 1.43 | 1.59 | 1.75 | 21.80 | 24.22 | 26.65 |

21 | 8060.63 | 8956.25 | 9851.88 | 1.49 | 1.65 | 1.82 | 22.60 | 25.11 | 27.62 | 60 | 7354.58 | 8171.75 | 8988.93 | 1.60 | 1.78 | 1.96 | 24.41 | 27.12 | 29.83 |

22 | 8374.05 | 9304.50 | 10234.95 | 1.43 | 1.59 | 1.75 | 21.80 | 24.22 | 26.64 | 61 | 7630.20 | 8478.00 | 9325.80 | 1.56 | 1.73 | 1.90 | 23.70 | 26.34 | 28.97 |

23 | 7938.90 | 8821.00 | 9703.10 | 1.51 | 1.67 | 1.84 | 22.91 | 25.46 | 28.00 | 62 | 7576.65 | 8418.50 | 9260.35 | 1.57 | 1.74 | 1.92 | 23.84 | 26.49 | 29.14 |

24 | 8329.73 | 9255.25 | 10180.78 | 1.44 | 1.60 | 1.76 | 21.91 | 24.35 | 26.78 | 63 | 7299.00 | 8110.00 | 8921.00 | 1.61 | 1.79 | 1.97 | 24.55 | 27.28 | 30.01 |

25 | 8303.18 | 9225.75 | 10148.33 | 1.45 | 1.61 | 1.77 | 21.98 | 24.42 | 26.86 | 64 | 7261.65 | 8068.50 | 8875.35 | 1.62 | 1.80 | 1.98 | 24.65 | 27.39 | 30.13 |

26 | 8128.13 | 9031.25 | 9934.38 | 1.47 | 1.64 | 1.80 | 22.43 | 24.92 | 27.41 | 65 | 8215.20 | 9128.00 | 10040.80 | 1.46 | 1.62 | 1.78 | 22.20 | 24.67 | 27.14 |

27 | 7720.43 | 8578.25 | 9436.08 | 1.54 | 1.71 | 1.89 | 23.47 | 26.08 | 28.69 | 66 | 8488.58 | 9431.75 | 10374.93 | 1.41 | 1.57 | 1.73 | 21.50 | 23.89 | 26.28 |

28 | 7473.38 | 8303.75 | 9134.13 | 1.58 | 1.76 | 1.94 | 24.11 | 26.78 | 29.46 | 67 | 8082.23 | 8980.25 | 9878.28 | 1.48 | 1.65 | 1.81 | 22.55 | 25.05 | 27.56 |

29 | 8106.53 | 9007.25 | 9907.98 | 1.48 | 1.64 | 1.81 | 22.48 | 24.98 | 27.48 | 68 | 8502.53 | 9447.25 | 10391.98 | 1.41 | 1.57 | 1.73 | 21.47 | 23.85 | 26.24 |

30 | 7396.43 | 8218.25 | 9040.08 | 1.60 | 1.78 | 1.95 | 24.30 | 27.00 | 29.70 | 69 | 8256.38 | 9173.75 | 10091.13 | 1.45 | 1.61 | 1.78 | 22.10 | 24.55 | 27.01 |

31 | 8329.50 | 9255.00 | 10180.50 | 1.44 | 1.60 | 1.76 | 21.91 | 24.35 | 26.78 | 70 | 7704.00 | 8560.00 | 9416.00 | 1.55 | 1.72 | 1.89 | 23.51 | 26.13 | 28.74 |

32 | 8174.03 | 9082.25 | 9990.48 | 1.47 | 1.63 | 1.79 | 22.31 | 24.79 | 27.27 | 71 | 7906.73 | 8785.25 | 9663.78 | 1.51 | 1.68 | 1.85 | 23.00 | 25.55 | 28.11 |

33 | 7323.98 | 8137.75 | 8951.53 | 1.61 | 1.79 | 1.97 | 24.49 | 27.21 | 29.93 | 72 | 7907.85 | 8786.50 | 9665.15 | 1.51 | 1.68 | 1.85 | 22.99 | 25.55 | 28.10 |

34 | 8104.73 | 9005.25 | 9905.78 | 1.48 | 1.64 | 1.81 | 22.49 | 24.99 | 27.48 | 73 | 8406.90 | 9341.00 | 10275.10 | 1.43 | 1.59 | 1.74 | 21.71 | 24.13 | 26.54 |

35 | 7580.03 | 8422.25 | 9264.48 | 1.57 | 1.74 | 1.92 | 23.83 | 26.48 | 29.13 | 74 | 7394.18 | 8215.75 | 9037.33 | 1.60 | 1.78 | 1.95 | 24.31 | 27.01 | 29.71 |

36 | 7505.10 | 8339.00 | 9172.90 | 1.58 | 1.76 | 1.93 | 24.02 | 26.69 | 29.36 | 75 | 7868.03 | 8742.25 | 9616.48 | 1.52 | 1.69 | 1.86 | 23.09 | 25.66 | 28.23 |

37 | 7998.75 | 8887.50 | 9776.25 | 1.50 | 1.66 | 1.83 | 22.76 | 25.29 | 27.82 | 76 | 7858.80 | 8732.00 | 9605.20 | 1.52 | 1.69 | 1.86 | 23.12 | 25.69 | 28.26 |

38 | 8326.58 | 9251.75 | 10176.93 | 1.44 | 1.60 | 1.76 | 21.92 | 24.35 | 26.79 | 77 | 8165.03 | 9072.25 | 9979.48 | 1.47 | 1.63 | 1.79 | 22.33 | 24.81 | 27.30 |

39 | 7858.58 | 8731.75 | 9604.93 | 1.52 | 1.69 | 1.86 | 23.12 | 25.69 | 28.26 | 78 | 7410.15 | 8233.50 | 9056.85 | 1.60 | 1.77 | 1.95 | 24.27 | 26.96 | 29.66 |

Parameter | Value |
---|---|

Number of blocks | 78 |

Block dimension | 40 × 40 × 10 (m) |

Target heating value | 7494 8832 9715 (kJ/kg) |

Target sulfur content | 1.50 1.67 1.84 (%) |

Target ash content | 22.88 25.42 27.97 (%) |

Number of technological mining-cuts | Fukuyama-Sugeno |

The exponent $\omega $ | 2 |

Threshold value $\epsilon $ | 0.0001 |

Block | $\tilde{\mathit{S}}={\left\{{\tilde{\mathit{s}}}_{\mathit{i}}\right\}}_{\mathit{i}=1,2,\dots ,78}$ | Defuzzified | Block | $\tilde{\mathit{S}}={\left\{{\tilde{\mathit{s}}}_{\mathit{i}}\right\}}_{\mathit{i}=1,2,\dots ,78}$ | Defuzzified | ||||
---|---|---|---|---|---|---|---|---|---|

1 | 0.6014 | 0.6808 | 0.4539 | 0.5787 | 40 | 0.5243 | 0.1009 | 0.5204 | 0.3819 |

2 | 0.5244 | 0.1019 | 0.5203 | 0.3822 | 41 | 0.5689 | 0.6281 | 0.4794 | 0.5588 |

3 | 0.5935 | 0.7072 | 0.4598 | 0.5868 | 42 | 0.5932 | 0.7078 | 0.4600 | 0.5870 |

4 | 0.5780 | 0.6797 | 0.4720 | 0.5766 | 43 | 0.5780 | 0.6797 | 0.4720 | 0.5765 |

5 | 0.6017 | 0.6796 | 0.4537 | 0.5783 | 44 | 0.5607 | 0.5397 | 0.4864 | 0.5289 |

6 | 0.6043 | 0.6680 | 0.4517 | 0.5747 | 45 | 0.5656 | 0.5947 | 0.4822 | 0.5475 |

7 | 0.5235 | 0.0915 | 0.5212 | 0.3788 | 46 | 0.5789 | 0.6845 | 0.4712 | 0.5782 |

8 | 0.5562 | 0.4863 | 0.4904 | 0.5110 | 47 | 0.5895 | 0.7118 | 0.4628 | 0.5881 |

9 | 0.5910 | 0.7111 | 0.4617 | 0.5879 | 48 | 0.5682 | 0.6237 | 0.4801 | 0.5573 |

10 | 0.5617 | 0.5505 | 0.4856 | 0.5326 | 49 | 0.5858 | 0.7086 | 0.4657 | 0.5867 |

11 | 0.6047 | 0.6662 | 0.4514 | 0.5741 | 50 | 0.5529 | 0.4473 | 0.4933 | 0.4979 |

12 | 0.5618 | 0.5514 | 0.4855 | 0.5329 | 51 | 0.5979 | 0.6944 | 0.4564 | 0.5829 |

13 | 0.5284 | 0.1495 | 0.5163 | 0.3981 | 52 | 0.6000 | 0.6866 | 0.4549 | 0.5805 |

14 | 0.5971 | 0.6971 | 0.4570 | 0.5838 | 53 | 0.5662 | 0.6013 | 0.4817 | 0.5497 |

15 | 0.5857 | 0.7085 | 0.4658 | 0.5867 | 54 | 0.6042 | 0.6684 | 0.4518 | 0.5748 |

16 | 0.5700 | 0.6343 | 0.4786 | 0.5609 | 55 | 0.5557 | 0.4803 | 0.4909 | 0.5090 |

17 | 0.5399 | 0.2891 | 0.5052 | 0.4448 | 56 | 0.6052 | 0.6643 | 0.4511 | 0.5735 |

18 | 0.6023 | 0.6770 | 0.4532 | 0.5775 | 57 | 0.5240 | 0.0974 | 0.5207 | 0.3807 |

19 | 0.5401 | 0.2906 | 0.5051 | 0.4453 | 58 | 0.5213 | 0.0700 | 0.5234 | 0.3715 |

20 | 0.5649 | 0.5867 | 0.4829 | 0.5448 | 59 | 0.5976 | 0.6955 | 0.4567 | 0.5832 |

21 | 0.5781 | 0.6802 | 0.4719 | 0.5767 | 60 | 0.5256 | 0.1158 | 0.5192 | 0.3868 |

22 | 0.5977 | 0.6953 | 0.4566 | 0.5832 | 61 | 0.5470 | 0.3750 | 0.4987 | 0.4736 |

23 | 0.5697 | 0.6325 | 0.4788 | 0.5603 | 62 | 0.5429 | 0.3251 | 0.5025 | 0.4568 |

24 | 0.5951 | 0.7033 | 0.4585 | 0.5856 | 63 | 0.5212 | 0.0662 | 0.5236 | 0.3703 |

25 | 0.5935 | 0.7071 | 0.4597 | 0.5868 | 64 | 0.5183 | 0.0395 | 0.5266 | 0.3614 |

26 | 0.5826 | 0.6998 | 0.4683 | 0.5835 | 65 | 0.5881 | 0.7115 | 0.4639 | 0.5879 |

27 | 0.5538 | 0.4575 | 0.4926 | 0.5013 | 66 | 0.6039 | 0.6698 | 0.4520 | 0.5752 |

28 | 0.5349 | 0.2276 | 0.5100 | 0.4242 | 67 | 0.5795 | 0.6873 | 0.4707 | 0.5792 |

29 | 0.5811 | 0.6944 | 0.4694 | 0.5816 | 68 | 0.6046 | 0.6666 | 0.4515 | 0.5742 |

30 | 0.5289 | 0.1548 | 0.5159 | 0.3999 | 69 | 0.5907 | 0.7113 | 0.4619 | 0.5880 |

31 | 0.5951 | 0.7033 | 0.4586 | 0.5857 | 70 | 0.5526 | 0.4427 | 0.4937 | 0.4963 |

32 | 0.5855 | 0.7081 | 0.4659 | 0.5865 | 71 | 0.5674 | 0.6141 | 0.4807 | 0.5541 |

33 | 0.5232 | 0.0879 | 0.5216 | 0.3775 | 72 | 0.5675 | 0.6150 | 0.4807 | 0.5544 |

34 | 0.5810 | 0.6939 | 0.4695 | 0.5815 | 73 | 0.5995 | 0.6884 | 0.4552 | 0.5810 |

35 | 0.5431 | 0.3282 | 0.5022 | 0.4579 | 74 | 0.5287 | 0.1527 | 0.5161 | 0.3992 |

36 | 0.5374 | 0.2576 | 0.5077 | 0.4342 | 75 | 0.5646 | 0.5838 | 0.4831 | 0.5438 |

37 | 0.5738 | 0.6570 | 0.4753 | 0.5687 | 76 | 0.5640 | 0.5764 | 0.4836 | 0.5413 |

38 | 0.5949 | 0.7038 | 0.4587 | 0.5858 | 77 | 0.5849 | 0.7068 | 0.4664 | 0.5861 |

39 | 0.5639 | 0.5762 | 0.4837 | 0.5413 | 78 | 0.5300 | 0.1678 | 0.5148 | 0.4042 |

Euclidean Distance | |||||

s_{1} = 0.5787 | 0.38 | 0.42 | 0.46 | 0.50 | 0.54 |

0.1987 | 0.1587 | 0.1187 | 0.0787 | 0.0387 | |

Ω = 2 | Membership degree | ||||

c_{1} = 1 | c_{2} = 2 | c_{3} = 3 | c_{4} = 4 | c_{5} = 5 | |

u_{n}_{1} | 0.086304 | 0.108058 | 0.144476 | 0.21792 | 0.443242 |

b_{1}$\in $ | No | No | No | No | Yes |

C_{n}/Iteration | 1 | 2 | 3 | 4 | … | 9 |
---|---|---|---|---|---|---|

c_{1} | 0.38 | 0.38985 | 0.38975 | 0.38792 | … | 0.38439 |

c_{2} | 0.42 | 0.45535 | 0.46002 | 0.45586 | … | 0.44558 |

c_{3} | 0.46 | 0.49075 | 0.51084 | 0.51148 | … | 0.50674 |

c_{4} | 0.50 | 0.53099 | 0.54596 | 0.54964 | … | 0.55016 |

c_{5} | 0.54 | 0.56212 | 0.57287 | 0.57816 | … | 0.58086 |

J^{(t)} | 0.068953 | 0.029840 | 0.016992 | 0.012897 | … | 0.010989 |

$\epsilon $ | 0.039112 | 0.012848 | 0.004094 | … | 0.000054 |

Cluster | Number of Blocks | Heating Value (kJ/kg) | Sulfur Content (%) | Ash Content (%) | |||
---|---|---|---|---|---|---|---|

min | max | min | max | min | max | ||

c_{1} | 13 | 7255.5 8061.7 8867.9 | 7410.1 8233.5 9056.8 | 1.59 1.77 1.95 | 1.62 1.81 1.99 | 24.26 26.96 29.66 | 24.64 27.38 30.12 |

c_{2} | 7 | 7473.3 8303.7 9134.1 | 7630.2 8478.0 9325.8 | 1.55 1.73 1.90 | 1.58 1.76 1.93 | 23.70 26.33 28.97 | 24.10 26.78 29.46 |

c_{3} | 5 | 7704.0 8560.0 9416.0 | 7752.8 8614.3 9475.7 | 1.54 1.71 1.88 | 1.55 1.72 1.89 | 23.39 25.99 28.59 | 23.51 26.12 28.73 |

c_{4} | 15 | 7814.5 8682.8 9551.0 | 7943.1 8825.7 9708.3 | 1.50 1.67 1.84 | 1.53 1.70 1.87 | 22.90 25.44 27.99 | 23.23 25.81 28.39 |

c_{5} | 38 | 7998.7 8887.5 9776.2 | 8512.4 9458.2 10404.0 | 1.41 1.56 1.72 | 1.49 1.66 1.82 | 21.44 23.82 26.20 | 22.75 25.28 27.81 |

Block | $\tilde{\mathit{S}}={\left\{{\tilde{\mathit{s}}}_{\mathit{i}}\right\}}_{\mathit{i}=1,2,\dots ,78}$ | Defuzzified | Block | $\tilde{\mathit{S}}={\left\{{\tilde{\mathit{s}}}_{\mathit{i}}\right\}}_{\mathit{i}=1,2,\dots ,78}$ | Defuzzified | ||||
---|---|---|---|---|---|---|---|---|---|

1 | 0.5779 | 0.6534 | 0.4826 | 0.5713 | 40 | 0.5504 | 0.4781 | 0.5087 | 0.5124 |

2 | 0.5505 | 0.4787 | 0.5086 | 0.5125 | 41 | 0.5734 | 0.9456 | 0.4872 | 0.6687 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | |||||

39 | 0.5716 | 0.8878 | 0.4889 | 0.6494 | 78 | 0.5540 | 0.5156 | 0.5052 | 0.5249 |

Cluster | Parameter | Number of Blocks | Heating Value (kJ/kg) | Sulfur Content (%) | Ash Content (%) | Number of Blocks | Heating Value (kJ/kg) | Sulfur Content (%) | Ash Content (%) |
---|---|---|---|---|---|---|---|---|---|

c_{1} | min | 13 | 8061.7 | 1.77 | 26.96 | 14 | 8061.7 | 1.76 | 26.78 |

max | 8233.5 | 1.81 | 27.38 | 8303.7 | 1.81 | 27.39 | |||

expected | 8156.1 | 1.79 | 27.13 | 8166.7 | 1.78 | 27.10 | |||

standard deviation | 54.8 | 0.01 | 0.12 | 65.8 | 0.01 | 0.15 | |||

coefficient of variation (%) | 0.67 | 0.59 | 0.46 | 0.81 | 0.68 | 0.56 | |||

c_{2} | min | 9 | 8303.7 | 1.71 | 26.11 | 21 | 8339.0 | 1.56 | 23.83 |

max | 8565.7 | 1.76 | 26.78 | 9458.3 | 1.75 | 26.69 | |||

expected | 8426.8 | 1.74 | 26.46 | 9101.8 | 1.63 | 24.73 | |||

standard deviation | 91.92 | 0.02 | 0.23 | 457.0 | 0.08 | 1.17 | |||

coefficient of variation (%) | 1.09 | 0.89 | 0.89 | 5.1 | 4.73 | 4.73 | |||

c_{3} | min | 17 | 8578.3 | 1.67 | 25.45 | 27 | 8560.0 | 1.60 | 24.34 |

max | 8821.0 | 1.72 | 26.08 | 9255.2 | 1.71 | 26.12 | |||

expected | 8726.6 | 1.68 | 25.70 | 9006.5 | 1.64 | 24.98 | |||

standard deviation | 72.28 | 0.01 | 0.18 | 227.3 | 0.04 | 0.58 | |||

coefficient of variation (%) | 0.82 | 0.72 | 0.72 | 2.5 | 2.33 | 2.33 | |||

c_{4} | min | 39 | 8825.7 | 1.56 | 23.82 | 16 | 8682.8 | 1.66 | 25.28 |

max | 9458.2 | 1.67 | 25.44 | 8887.5 | 1.69 | 25.81 | |||

expected | 9197.3 | 1.61 | 24.49 | 8766.6 | 1.68 | 25.59 | |||

standard deviation | 180.9 | 0.03 | 0.46 | 54.5 | 0.01 | 0.14 | |||

coefficient of variation (%) | 1.96 | 1.89 | 1.89 | 0.62 | 0.54 | 0.55 |

© 2016 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**

Gligorić, M.; Gligorić, Z.; Beljić, Č.; Torbica, S.; Štrbac Savić, S.; Nedeljković Ostojić, J.
Multi-Attribute Technological Modeling of Coal Deposits Based on the Fuzzy TOPSIS and C-Mean Clustering Algorithms. *Energies* **2016**, *9*, 1059.
https://doi.org/10.3390/en9121059

**AMA Style**

Gligorić M, Gligorić Z, Beljić Č, Torbica S, Štrbac Savić S, Nedeljković Ostojić J.
Multi-Attribute Technological Modeling of Coal Deposits Based on the Fuzzy TOPSIS and C-Mean Clustering Algorithms. *Energies*. 2016; 9(12):1059.
https://doi.org/10.3390/en9121059

**Chicago/Turabian Style**

Gligorić, Miloš, Zoran Gligorić, Čedomir Beljić, Slavko Torbica, Svetlana Štrbac Savić, and Jasmina Nedeljković Ostojić.
2016. "Multi-Attribute Technological Modeling of Coal Deposits Based on the Fuzzy TOPSIS and C-Mean Clustering Algorithms" *Energies* 9, no. 12: 1059.
https://doi.org/10.3390/en9121059