# Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. Literature Review

#### 2.2. Mathematical Preliminary

**Definition**

**1**

**[62]**.

**Definition**

**2**

**[62]**.

**Definition**

**3**

**[63]**.

**Definition**

**4**

**[64]**.

**Definition**

**5**

**[64]**.

**Definition**

**6**

**[65]**.

## 3. The Proposed Framework

#### 3.1. The Need of This Framework

#### 3.2. The Workflow Diagram

#### 3.3. DAbFP Algorithm

**Step 1**: Let D denotes the source dataset variable.

**Step 2**: Partition the dataset D into suitable four frequencies to perform subsequent forecasting steps to each group:

**Step 3**: Using above partitioned data as D

_{new}, we define fuzzy sets as F1, F2…F7 linguistically mapped over the universe of discourse U defined as follows:

**Step 4**: From above partitioned data as D

_{new}, we define 11 fuzzy sets as F1, F2…F11 over U. Similar equations (as 5 to 13) are observed for 11 intervals. Here, q1, q2…q11 $\in $ fixed length intervals.

**Step 5**: Mean of middle values of fuzzy partitions on the Right-Hand Side of Fuzzy logic relation (FLR) is calculated. This calculation is performed for degree approximation. For instance, in the 2nd order FLR, F4 <- F2, F7. If P and Q are the centers of Interval F2 and F7 respectively then

**Step 6**: After degree approximation based on fuzzy logic relation, defuzzification is performed using regression analysis. On plotting the points, we select a Best Fit line that represents average across all points in graph. Thereafter, the equation of line is estimated which can be linear or polynomial of higher degrees 2, 3, 4, 5 or 6. In the consequent section, we use two important constraints to associate the outcome as stated below:

_{i}is the actual production cost whereas Y

_{i}is the predicted value.

#### 3.4. Numerical Example

## 4. Results and Discussion

#### 4.1. Linear Polynomial

#### 4.2. Quadratic Polynomial

#### 4.3. Cubic Polynomial

#### 4.4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Compliance with Ethical Standards

## References

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**Figure 1.**Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications DAbFP simulation Workflow.

F1 | very meagre produce |

F2 | meagre produce |

F3 | better than poor produce |

F4 | not so quality produce |

F5 | average production |

F6 | superior produce |

F7 | very superior produce |

F8 | Very very superior produce |

F9 | tremendous produce |

Fuzzy Sets | Upper | Lower | Frequency |
---|---|---|---|

F1 | 233,200 | 343,355 | 3 |

F2 | 233,355 | 343,511 | 2 |

F3 | 233,511 | 343,666 | 2 |

F4 | 233,666 | 343,822 | 3 |

F5 | 233,822 | 344,133 | 4 |

F7 | 234,133 | 344,288 | 3 |

F8 | 234,288 | 344,444 | 2 |

F9 | 234,444 | 344,600 | 1 |

Fuzzy Sets | Upper | Lower | New Fuzzy Sets |
---|---|---|---|

AF1A | 932,007 | 3252.76 | Z1 |

3253.76 | 3303.8 | Z2 | |

3303.8 | 3356.66 | Z3 | |

AF2A | 3356.66 | 3432.435 | Z4 |

3432.435 | 3512.2 | Z5 | |

AF3A | 3512.2 | 3589.985 | Z6 |

3589.985 | 3677.75 | Z7 | |

AF4A | 3677.75 | 3729.5 | Z8 |

3729.5 | 3771.45 | Z9 | |

3771.45 | 3823.3 | Z10 | |

AF5A | 3823.3 | 3862.1985 | Z11 |

3862.1985 | 3900.175 | Z12 | |

3900.175 | 3949.9725 | Z13 | |

3949.9725 | 3988.865 | Z14 | |

AF7A | 4234.3 | 4285.25 | Z15 |

4285.25 | 4238 | Z16 | |

4238 | 4289.95 | Z17 | |

AF8A | 4289.95 | 4367.735 | Z18 |

4367.735 | 4445.5 | Z19 | |

AF9A | 4445.5 | 4600 | Z20 |

Fuzzy Sets | Upper | LOWER | Frequency Uency |
---|---|---|---|

A1 | 3200 | 3327 | 3 |

A2 | 3327 | 3454 | 1 |

A3 | 3454 | 3581 | 2 |

A4 | 3581 | 3709 | 3 |

A5 | 3709 | 3836 | 1 |

A6 | 3836 | 4091 | 4 |

A7 | 3937 | 4120 | 3 |

A8 | 4091 | 4218 | 2 |

A9 | 4218 | 4345 | 2 |

A10 | 4345 | 4472 | 1 |

A11 | 4472 | 4600 | 1 |

Fuzzy Sets | Upper | Lower | New Fuzzy Sets |
---|---|---|---|

A1 | 3200.000 | 3253.423 | NF1 |

3253.423 | 3295.847 | NF2 | |

3295.847 | 3330.270 | NF3 | |

A2 | 3330.270 | 3460.540 | NF4 |

A3 | 3460.540 | 3521.175 | NF5 |

3521.175 | 3579.810 | NF6 | |

A4 | 3579.810 | 3630.233 | NF7 |

3630.233 | 3670.657 | NF8 | |

3670.657 | 3711.080 | NF9 | |

A5 | 3711.080 | 3841.350 | NF10 |

A6 | 3841.350 | 3870.168 | NF11 |

3870.168 | 3900.985 | NF12 | |

A7 | 3900.985 | 3929.803 | NF13 |

3929.803 | 3970.720 | NF14 | |

A8 | 4091.990 | 4149.525 | NF15 |

4149.525 | 4220.160 | NF16 | |

A9 | 4220.160 | 4290.795 | NF17 |

4290.795 | 4351.430 | NF18 | |

A10 | 4351.430 | 4469.700 | NF19 |

A11 | 4469.700 | 4600.000 | NF20 |

Year | Product | Fuzzy Sets | FLR Relations | Avg. | Mid Fuzzy Value |
---|---|---|---|---|---|

1981 | 3552 | Z6 | - | - | 3549.9875 |

1982 | 4177 | Z15 | - | - | 4159.225 |

1983 | 3372 | Z4 | Z4<-Z15,Z6 | 3854.60625 | 3394.4375 |

1984 | 3455 | Z5 | Z5<-Z4,Z15 | 3776.83125 | 3472.2125 |

1985 | 3702 | Z8 | Z8<-Z5,Z4 | 3433.325 | 3692.575 |

1986 | 3670 | Z8 | Z8<-Z8,Z5 | 3582.39375 | 3692.575 |

1987 | 3865 | Z12 | Z12<-Z8,Z8 | 3692.575 | 3880.5315 |

1988 | 3592 | Z7 | Z7<-Z12,Z8 | 3786.55325 | 3627.7625 |

1989 | 3222 | Z1 | Z1<-Z7,Z12 | 3754.147 | 3225.925 |

1990 | 3750 | Z9 | Z9<-Z1,Z7 | 3426.84375 | 3744.425 |

1991 | 3851 | Z11 | Z11<-Z9,Z1 | 3485.175 | 3841.644 |

1992 | 3231 | Z1 | Z1<-Z11,Z9 | 3793.0345 | 3225.925 |

1993 | 4170 | Z15 | Z15<-Z1,Z11 | 3533.7845 | 4159.225 |

1994 | 4554 | Z20 | Z20<-Z15,Z1 | 3692.575 | 4522.2 |

1995 | 3872 | Z12 | Z12<-Z20,Z15 | 4340.7125 | 3880.5315 |

1996 | 4439 | Z19 | Z19<-Z12,Z20 | 4201.36575 | 4405.5125 |

1997 | 4266 | Z17 | Z17<-Z19,Z12 | 4143.022 | 4262.925 |

1998 | 3219 | Z1 | Z1<-Z17,Z19 | 4334.21875 | 3225.925 |

1999 | 4305 | Z18 | Z18<-Z1,Z17 | 3744.425 | 4327.7375 |

2000 | 3928 | Z13 | Z13<-Z18,Z1 | 3776.83125 | 3919.419 |

Year | Product | Fuzzy Sets | FLR relation | Avg. | Fuzzy |
---|---|---|---|---|---|

1981 | 3552 | F6 | - | - | 3549.9925 |

1982 | 4177 | F16 | - | - | 4186.3425 |

1983 | 3372 | F4 | F4<-F16,F6 | 3868.1675 | 3390.905 |

1984 | 3455 | F5 | F5<-F4,F16 | 3788.62375 | 3486.3575 |

1985 | 3702 | F9 | F9<-F5,F4 | 3438.63125 | 3687.868325 |

1986 | 3670 | F9 | F9<-F9,F5 | 3587.112913 | 3687.868325 |

1987 | 3865 | F11 | F11<-F9,F9 | 3687.868325 | 3852.25875 |

1988 | 3592 | F7 | F7<-F11,F9 | 3770.063538 | 3603.021665 |

1989 | 3222 | F1 | F1<-F7,F11 | 3727.640208 | 3221.211665 |

1990 | 3750 | F10 | F10<-F1,F7 | 3412.116665 | 3772.714995 |

1991 | 3851 | F11 | F11<-F10,F1 | 3496.96333 | 3852.25875 |

1992 | 3231 | F2 | F2<-F11,F10 | 3812.486873 | 3263.634995 |

1993 | 4170 | F16 | F16<-F2,F11 | 3557.946873 | 4186.3425 |

1994 | 4554 | F20 | F20<-F16,F2 | 3724.988748 | 4536.35 |

1995 | 3872 | F12 | F12<-F20,F16 | 4361.34625 | 3884.07625 |

1996 | 4439 | F19 | F19<-F12,F20 | 4210.213125 | 4409.065 |

1997 | 4266 | F17 | F17<-F19,F12 | 4146.570625 | 4249.9775 |

1998 | 3219 | F1 | F1<-F17,F19 | 4329.52125 | 3221.211665 |

1999 | 4305 | F18 | F18<-F1,F17 | 3735.594583 | 4313.6125 |

2000 | 3928 | F13 | F13<-F18,F1 | 3767.412083 | 3915.89375 |

Year | Product | Fuzzy Sets | FLR Relations | Avg | Mid Fuzzy Value |
---|---|---|---|---|---|

1981 | 3552 | Z6 | - | - | 3549.9875 |

1982 | 4177 | Z15 | - | - | 4159.225 |

1983 | 3372 | Z4 | - | - | 3394.4375 |

1984 | 3455 | Z5 | Z5<-Z4,Z15,Z6 | 3701.216667 | 3472.2125 |

1985 | 3702 | Z8 | Z8<-Z5,Z4,Z15 | 3675.291667 | 3692.575 |

1986 | 3670 | Z8 | Z8<-Z8,Z5,Z4 | 3519.741667 | 3692.575 |

1987 | 3865 | Z12 | Z12<-Z8,Z8,Z5 | 3619.120833 | 3880.5315 |

1988 | 3592 | Z7 | Z7<-Z12,Z8,Z8 | 3755.227167 | 3627.7625 |

1989 | 3222 | Z1 | Z1<-Z7,Z12,Z8 | 3733.623 | 3225.925 |

1990 | 3750 | Z9 | Z9<-Z1,Z7,Z12 | 3578.073 | 3744.425 |

1991 | 3851 | Z11 | Z11<-Z9,Z1,Z7 | 3532.704167 | 3841.644 |

1992 | 3231 | Z1 | Z1<-Z11,Z9,Z1 | 3603.998 | 3225.925 |

1993 | 4170 | Z15 | Z15<-Z1,Z11,Z9 | 3603.998 | 4159.225 |

1994 | 4554 | Z20 | Z20<-Z15,Z1,Z11 | 3742.264667 | 4522.2 |

1995 | 3872 | Z12 | Z12<-Z20,Z15,Z1 | 3969.1186667 | 3880.5315 |

1996 | 4439 | Z19 | Z19<-Z12,Z20,Z15 | 4187.318833 | 4405.5125 |

1997 | 4266 | Z17 | Z17<-Z19,Z12,Z20 | 4269.414667 | 4262.925 |

1998 | 3219 | Z1 | Z1<-Z17,Z19,Z12 | 4182.989667 | 3225.925 |

1999 | 4305 | Z18 | Z18<-Z1,Z17,Z19 | 3964.7875 | 4327.7375 |

2000 | 3928 | Z13 | Z13<-Z18,Z1,Z17 | 3938.8625 | 3919.419 |

Year | Product | Fuzzy Sets | FLR Relation | Avg | Fuzzy |
---|---|---|---|---|---|

1981 | 3552 | F6 | - | - | 3549.9925 |

1982 | 4177 | F16 | - | - | 4186.3425 |

1983 | 3372 | F4 | - | - | 3390.905 |

1984 | 3455 | F5 | F5<-F4,F16,F6 | 3709.08 | 3486.3575 |

1985 | 3702 | F9 | F9<-F5,F4,F16 | 3687.868333 | 3687.868325 |

1986 | 3670 | F9 | F9<-F9,F5,F4 | 3521.710275 | 3687.868325 |

1987 | 3865 | F11 | F11<-F9,F9,F5 | 3620.69805 | 3852.25875 |

1988 | 3592 | F7 | F7<-F11,F9,F9 | 3742.665133 | 3603.021665 |

1989 | 3222 | F1 | F1<-F7,F11,F9 | 3714.382913 | 3221.211665 |

1990 | 3750 | F10 | F10<-F1,F7,F11 | 3558.830693 | 3772.714995 |

1991 | 3851 | F11 | F11<-F10,F1,F7 | 3532.316108 | 3852.25875 |

1992 | 3231 | F2 | F2<-F11,F10,F1 | 3615.395137 | 3263.634995 |

1993 | 4170 | F16 | F16<-F2,F11,F10 | 3629.536247 | 4186.3425 |

1994 | 4554 | F20 | F20<-F16,F2,F11 | 3767.412082 | 4536.35 |

1995 | 3872 | F12 | F12<-F20,F16,F2 | 3995.442498 | 3884.07625 |

1996 | 4439 | F19 | F19<-F12,F20,F16 | 4202.25625 | 4409.065 |

1997 | 4266 | F17 | F17<-F19,F12,F20 | 4276.497083 | 4249.9775 |

1998 | 3219 | F1 | F1<-F17,F19,F12 | 4181.039583 | 3221.211665 |

1999 | 4305 | F18 | F18<-F1,F17,F19 | 3960.084722 | 4313.6125 |

2000 | 3928 | F13 | F13<-F18,F1,F17 | 3928.267222 | 3915.89375 |

9th Interval | 11th Interval | ||
---|---|---|---|

FLR 2nd Degree | FLR 3rd Degree | FLR 2nd Degree | FLR 3rd Degree |

- | - | - | - |

- | - | - | - |

42,986.55822 | - | 44,818.16021 | - |

22,492.80058 | 4800.826944 | 23,809.72442 | 5074.567696 |

5095.1044 | 20,567.86223 | 4501.739025 | 19,945.9129 |

188.677696 | 5947.185924 | 90.136036 | 5579.492416 |

33,522.68046 | 56,558.82804 | 32,002.70545 | 55,301.16624 |

13,352.2647 | 4826.914576 | 14,330.00526 | 5237.706384 |

261,321.3504 | 224,460.8555 | 265,543.3655 | 227,439.3328 |

78.1456 | 397.2049 | 166.6681 | 274.2336 |

4424.378256 | 7505.276689 | 3904.875121 | 6893.316676 |

335,389.2404 | 322,242.4169 | 340,019.6045 | 326,621.3941 |

111,708.356 | 113,595.2875 | 109,089.5024 | 110,861.0298 |

479,672.5971 | 471,614.5746 | 474,288.4066 | 465,702.5158 |

226.8036 | 873.498025 | 357.777225 | 1163.4921 |

276,987.4796 | 253,157.9099 | 272,989.5303 | 248,358.7027 |

107,355.8331 | 87,527.8142 | 104,900.1977 | 84,577.43568 |

555,013.0801 | 616,925.4189 | 560,578.6435 | 625,225.4669 |

99,454.4525 | 70,892.79005 | 97,145.04576 | 67,992.64852 |

7617.7984 | 21,036.6016 | 8266.4464 | 22,734.6084 |

MSE = 130,938.2001 | MSE = 134,290.0745 | MSE = 130,933.4741 | MSE = 134,057.8249 |

AFER = 7.352165941 | AFER = 7.50564575 | AFER = 7.360701563 | AFER = 7.497227115 |

9th Interval | 11th Interval | ||
---|---|---|---|

FLR 2nd Degree | FLR 3rd Degree | FLR 2nd Degree | FLR 3rd Degree |

- | - | - | - |

- | - | - | - |

86,872.72867 | - | 86,973.79553 | - |

42,656.78884 | 31,205.36382 | 43,329.25339 | 30,070.88937 |

1748.494225 | 5821.308506 | 1515.5449 | 5985.730056 |

53.41855744 | 2014.178496 | 11.20374784 | 1939.204525 |

38,394.91329 | 55,270.0822 | 36,517.22259 | 54,101.41093 |

7616.54162 | 2309.148473 | 8616.88906 | 2698.84406 |

222,169.1256 | 187,981.991 | 227,928.7106 | 192,375.311 |

1499.2384 | 5409.6025 | 1044.5824 | 4573.8169 |

13,907.90945 | 21,993.80947 | 12,407.55388 | 20,095.30221 |

278,480.7993 | 253,320.5535 | 285,381.6062 | 260,326.8975 |

145,760.7025 | 158,971.1792 | 141,014.3692 | 153,420.3511 |

536,451.9471 | 548,144.1831 | 527,917.3339 | 538,011.1009 |

174.636225 | 113.5823063 | 63.5209 | 17.53515625 |

290,677.1154 | 275,185.8551 | 285,894.6794 | 269,126.8781 |

103,181.132 | 85,931.00097 | 100,960.487 | 83,089.84966 |

599,950.6294 | 668,580.3041 | 603,535.0764 | 674,659.1905 |

66,975.2661 | 39,664.34711 | 66,480.0217 | 38,764.05762 |

30,520.09 | 63,695.6644 | 30,317.7744 | 63,988.7616 |

MSE = 137,060.6376 | MSE = 141,506.5973 | MSE = 136,661.6458 | MSE = 140,779.1254 |

AFER = 7.687795338 | AFER = 7.758800407 | AFER = 7.653515775 | AFER = 7.720197268 |

9th Interval | 11th Interval | ||
---|---|---|---|

FLR 2nd Degree | FLR 3rd Degree | FLR 2nd Degree | FLR 3rd Degree |

- | - | - | - |

- | - | - | - |

290,632.1524 | - | 313,062.5185 | - |

77,523.93313 | 103,695.3347 | 81,762.1411 | 114,607.7066 |

7830.037656 | 1600 | 8025.920156 | 1299.6025 |

15,835.50426 | 6608.649401 | 17,384.10606 | 7268.858358 |

120,277.2455 | 98,004.56003 | 125,760.2382 | 103,028.674 |

4043.230265 | 2068.721482 | 4999.281871 | 2928.454871 |

119,470.9499 | 119,186.2386 | 116,370.5638 | 114,544.0704 |

14,713.69 | 19,909.21 | 14,859.61 | 20,793.64 |

21,494.40413 | 34,105.81594 | 20,329.99744 | 33,467.62901 |

309,726.0861 | 253,318.1376 | 319,561.3766 | 260,429.7685 |

89,440.41254 | 131,289.6959 | 81,819.51089 | 122,710.9307 |

367,945.1196 | 452,673.8343 | 348,552.3228 | 431,268.5495 |

18,985.39516 | 6037.29 | 24,176.36266 | 9254.44 |

150,648.9335 | 185,762.3792 | 137,798.9429 | 170,127.8712 |

41,022.08703 | 46,271.79584 | 35,822.33797 | 39,841.27777 |

674,680.875 | 729,631.6726 | 684,819.8035 | 744,583.9843 |

109,253.5845 | 55,372.86685 | 112,612.3927 | 56,584.23018 |

4830.25 | 11,491.84 | 8172.16 | 7779.24 |

MSE = 135,464.105 | MSE = 132,766.3554 | MSE = 136,438.3104 | MSE = 131,795.231 |

AFER = 7.752071496 | AFER = 8.228273107 | AFER = 7.744400101 | AFER = 8.305847824 |

Year | Enrollement Data | Chissom [1,2] | Proposed Method (DAbFP) | |
---|---|---|---|---|

2nd Degree | 3rd Degree | |||

1971 | 13,055 | - | 13,561 | 13,261 |

1972 | 13,563 | 14,000 | 13,756 | 13,786 |

1973 | 13,867 | 14,000 | 13,756 | 13,776 |

1974 | 14,696 | 14,000 | 14,451 | 14,431 |

1975 | 15,460 | 15,500 | 15,361 | 15,271 |

1976 | 15,311 | 16,000 | 15,361 | 15,661 |

1977 | 15,603 | 16,000 | 15,721 | 15,321 |

1978 | 15,861 | 16,000 | 15,900 | 15,887 |

1979 | 16,807 | 16,000 | 17,085 | 17,067 |

1980 | 16,919 | 16,813 | 17,085 | 17,067 |

1981 | 16,388 | 16,813 | 16,487 | 16,480 |

1982 | 15,433 | 16,789 | 15,385 | 15,371 |

1983 | 15,497 | 16,000 | 15,385 | 15,371 |

1984 | 15,145 | 16,000 | 15,029 | 15,012 |

1985 | 15,163 | 16,000 | 15,029 | 15,012 |

1986 | 15,984 | 16,000 | 15,885 | 15,780 |

1987 | 16,859 | 16,000 | 17,069 | 17,054 |

1988 | 18,150 | 16,813 | 17,981 | 17,934 |

1989 | 18,970 | 19,000 | 18,802 | 18,780 |

1990 | 19,328 | 19,000 | 18,904 | 18,800 |

1991 | 19,337 | 19,000 | 18,904 | 18,800 |

1992 | 18,876 | - | 18,816 | 18,800 |

MSE | 775,687 | 415,382 | 323,421 | |

AFER | 37.4876 | 16.61 | 14.43 |

Year | Jilani and Burney [67] | Qiu et al. [11] | Yalaz et al. [64] | Khoshnevisan et al. [57] | Proposed Method DAbFP | |||||
---|---|---|---|---|---|---|---|---|---|---|

2nd Degree | 3rd Degree | 2nd Degree | 3rd Degree | 2nd Degree | 3rd Degree | 2nd Degree | 3rd Degree | 2nd Degree | 3rd Degree | |

1981 | - | - | - | - | - | - | - | - | - | - |

1982 | - | - | - | - | - | - | - | - | - | - |

1983 | 44,312.75772 | - | 45,322.7237 | - | 35,332.72372 | - | 35,212.72372 | - | 35,312.72372 | - |

1984 | 14,926.9 | 88,729.9956 | 12,827.5625 | 88,721.8856 | 11,826.5625 | 91,721.8856 | 16,726.5625 | 81,721.8856 | 11,826.5625 | 81,721.8856 |

1985 | 1893.198902 | 39,129.6906 | 1862.1189 | 36,122.6406 | 1765.118902 | 27,122.6406 | 1772.118902 | 26,122.64063 | 1762.118902 | 26,122.64063 |

1986 | 2250.702729 | 6459.78075 | 2200.59273 | 5955.77575 | 2090.592729 | 5045.77575 | 3090.592729 | 5135.775754 | 2090.592729 | 5035.775754 |

1987 | 30,182.05079 | 50,014.0981 | 29,982.0508 | 35,014.0981 | 28,892.05079 | 32,014.0981 | 28,982.05079 | 31,014.09811 | 28,882.05079 | 31,014.09811 |

1988 | 900.2539934 | 19,558.9699 | 792.253773 | 15,558.0697 | 772.2537734 | 15,560.0697 | 782.2537734 | 16,559.0697 | 782.2537734 | 15,559.0697 |

1989 | 108,560 | 295,969.786 | 109,856 | 225,968.386 | 99,859 | 205,968.386 | 99,857 | 215,968.3856 | 99,856 | 205,968.3856 |

1990 | 66,850.40219 | 31,736 | 63,839.4001 | 20,736 | 63,849.40009 | 20,726 | 83,829.40009 | 20,736 | 63,839.40009 | 20,736 |

1991 | 109,770.8079 | 93,938.686 | 104,965.538 | 93,532.676 | 11,565.5379 | 83,531.676 | 103,565.5379 | 83,532.67601 | 103,565.5379 | 83,532.67601 |

1992 | 178,169.5971 | 229,062.574 | 169,167.487 | 229,062.574 | 165,176.4871 | 130,062.574 | 165,166.4871 | 129,062.5743 | 165,166.4871 | 129,062.5743 |

1993 | 190,709.579 | 297,812.898 | 154,309.577 | 297,812.898 | 150,309.5771 | 217,812.898 | 160,309.5771 | 217,812.8982 | 140,309.5771 | 207,812.8982 |

1994 | 380,483.1606 | 592,830.047 | 369,362.141 | 592,830.047 | 364,363.1406 | 393,810.047 | 364,363.1406 | 392,810.0471 | 364,363.1406 | 392,810.0471 |

1995 | 27,937.24568 | 104,571.391 | 29,438.2497 | 104,571.391 | 38,438.23968 | 107,571.391 | 36,438.23968 | 114,571.3906 | 26,438.23968 | 104,571.3906 |

1996 | 256,945.9024 | 767.2593 | 226,733.902 | 767.2593 | 206,734.9024 | 761.2593 | 226,733.9024 | 760.2592998 | 206,733.9024 | 760.2592998 |

1997 | 281,348.3152 | 169,575.758 | 271,340.315 | 169,575.758 | 290,339.3151 | 179,676.758 | 271,339.3151 | 189,575.7582 | 250,339.3151 | 179,575.7582 |

1998 | 35,892.38004 | 2,632,778.84 | 35,689.3788 | 2,632,778.84 | 55,682.37884 | 2,732,778.84 | 57,682.37884 | 2,632,778.837 | 35,682.37884 | 2,632,778.837 |

1999 | 1,650,121 | 441,151.663 | 1,590,721 | 441,151.663 | 1,891,121 | 441,151.663 | 1,600,121 | 441,151.6629 | 1,590,121 | 431,151.6629 |

2000 | 100,011,776 | 1,607,824 | 88,811,789 | 1,607,824 | 88,911,776 | 1,707,824 | 88,811,776 | 1,607,824 | 88,811,776 | 1,607,824 |

MSE = 5,744,057.738 | MSE = 394,230.0844 | MSE = 5,112,788.738 | MSE = 388,116.21 | MSE = 5,129,438.738 | MSE = 376,067.88 | MSE = 5,114,874.349 | MSE = 3,651,259.88 | MSE = 5,107,713.738 | MSE = 362,119.88 | |

AFER = 23.95793579 | AFER = 13.90547975 | AFER = 22.95793579 | AFER = 13.8052 | AFER = 21.95793579 | AFER = 11.92547975 | AFER = 21.865793579 | AFER = 12.10547975 | AFER = 20.95793579 | AFER = 11.80547975 |

Year | Jilani and Burney [67] | Qiu et al. [11] | Yalaz et al. [64] | Khoshnevisan et al. [57] | Proposed Method DAbFP | |||||
---|---|---|---|---|---|---|---|---|---|---|

2nd Degree | 3rd Degree | 2nd Degree | 3rd Degree | 2nd Degree | 3rd Degree | 2nd Degree | 3rd Degree | 2nd Degree | 3rd Degree | |

1981 | - | - | - | - | - | - | - | - | - | - |

1982 | - | - | - | - | - | - | - | - | - | - |

1983 | 37,375.72372 | - | 35,412.72472 | - | 35,312.72372 | - | 32,417.72572 | - | 32,312.72371 | - |

1984 | 11,830.58 | 80,731.8856 | 11,827.5625 | 81,821.8856 | 11,826.5625 | 81,721.8856 | 11,728.5127 | 81,729.8876 | 11,726.5125 | 80,721.8856 |

1985 | 1781.119102 | 26,328.64064 | 1762.118911 | 26,122.6406 | 1762.118902 | 26,122.64063 | 1757.117603 | 26,125.64863 | 1756.117502 | 25,122.64063 |

1986 | 2200.592729 | 5200.78176 | 2093.593729 | 5037.77575 | 2090.592729 | 5035.775754 | 2085.594224 | 5038.785756 | 2081.592223 | 5030.775754 |

1987 | 28,982.0588 | 33,017.09911 | 28,694.05179 | 31,014.0981 | 28,882.05079 | 31,014.09811 | 28,372.04962 | 31,016.09711 | 28,375.04965 | 31,012.09811 |

1988 | 789.2707734 | 15,561.0698 | 783.2537734 | 15,559.0697 | 782.2537734 | 15,559.0697 | 776.2547634 | 15,859.0698 | 775.2537632 | 15,520.0665 |

1989 | 99,896 | 205,969.3956 | 99,857 | 205,968.386 | 99,856 | 205,968.3856 | 97,854 | 205,988.3957 | 97,853 | 205,940.3346 |

1990 | 63,850.40009 | 20,737 | 63,850.41009 | 20746 | 63,839.40009 | 20,736 | 61,828.4 | 20,740 | 61,820.3999 | 20,732 |

1991 | 103,570.5399 | 83,539.67701 | 103,566.5379 | 84,532.676 | 103,565.5379 | 83,532.67601 | 104,563.5259 | 83,633.67604 | 103,561.5239 | 83,512.66201 |

1992 | 165,170.4971 | 135,250.575 | 165,167.4871 | 129,062.574 | 165,166.4871 | 129,062.5743 | 165,242.4931 | 130,063.5843 | 165,040.4831 | 128,061.5443 |

1993 | 140,319.5781 | 207,825.9152 | 140,410.5871 | 207,812.898 | 150,309.5771 | 207,812.8982 | 140,299.5671 | 207,914.8992 | 140,289.5661 | 206,812.7182 |

1994 | 364,373.1506 | 303,016.048 | 364,364.1506 | 372,820.047 | 364,363.1406 | 392,810.0471 | 364,333.1256 | 392,811.0472 | 364,323.1206 | 391,810.0465 |

1995 | 264,390.2407 | 104,585.4007 | 26,441.24068 | 104,571.391 | 27,438.23968 | 104,571.3906 | 26,437.23769 | 104,566.3806 | 26,433.23568 | 104,565.3206 |

1996 | 206,740.9024 | 760.2693 | 206,736.9034 | 772.2594 | 206,733.9024 | 761.2592998 | 206,725.92 | 764.2602998 | 206,723.901 | 745.2452998 |

1997 | 250,350.3151 | 199,577.7583 | 250,441.3151 | 179,576.768 | 260,339.3151 | 179,575.7582 | 250,325.315 | 179,576.7782 | 250,320.312 | 179,545.3682 |

1998 | 35,689.47889 | 2,692,780.837 | 35,682.37884 | 2,932,788.86 | 35,682.37884 | 2,632,778.837 | 34,687.37837 | 2,642,798.845 | 34,681.37834 | 2,632,765.817 |

1999 | 1,590,630 | 481,157.6629 | 1,590,123 | 431,151.663 | 1,690,121 | 431,151.6629 | 1,590,108 | 431,156.663 | 1,590,100 | 431,051.6569 |

2000 | 88,811,780 | 1,607,870 | 88,811,776 | 1,707,824 | 88,811,776 | 1,607,824 | 88,811,740 | 1,607,830 | 88,811,732 | 1,607,310 |

MSE = 5,121,095.58 | MSE = 364,935.8833 | MSE = 5,107,721.684 | MSE = 384,540.1758 | MSE = 5,114,435.96 | MSE = 362,119.88 | MSE = 5,107,293.457 | MSE = 362,800.8246 | MSE = 5,107,217.009 | MSE = 361,780.0106 | |

AFER = 22.85793579 | AFER = 13.00547975 | AFER = 21.95793579 | AFER = 12.8052 | AFER = 20.95793579 | AFER = 11.80547975 | AFER = 20.865793579 | AFER = 11.7807960 | AFER = 19.75793272 | AFER = 11.75647975 |

© 2018 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**

Jain, R.; Jain, N.; Kapania, S.; Son, L.H.
Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction. *Symmetry* **2018**, *10*, 768.
https://doi.org/10.3390/sym10120768

**AMA Style**

Jain R, Jain N, Kapania S, Son LH.
Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction. *Symmetry*. 2018; 10(12):768.
https://doi.org/10.3390/sym10120768

**Chicago/Turabian Style**

Jain, Rachna, Nikita Jain, Shivani Kapania, and Le Hoang Son.
2018. "Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction" *Symmetry* 10, no. 12: 768.
https://doi.org/10.3390/sym10120768