Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 172 https://internationalpubls.com A Fuzzy Log-Normal Distribution Model for the Rainfall level of Namakkal District Prakasam. S1, Venkatesh.A2, Ramesh.R3*, Seenivasan. M4, Sakthivel. K5 1Department of Mathematics, A. V. V. M. Sri Pushpam College (Affiliated to BharathidasanUniversity, Tiruchirappalli), Poondi, Thanjavur (Dt), Tamilnadu, India. Email: prakasamspcmaths@gmail.com 2 Department of Mathematics, A. V. V. M. Sri Pushpam College, Poondi, Thanjavur (Dt), Tamilnadu, India. Email: a_venkatesh03@yahoo.co.in 3 Department of Mathematics, Arignar Anna Govt. Arts College, Musiri, Tamilnadu, India. *Corresponding author: rameshsanju123@gmail.com 4 Mathematics Wing -CDOE, Annamalai University, Annamalai Nagar, Tamil Nadu, India. Email: emseeni@yahoo.com 5 Department of Mathematics, Govt. Arts and Science College, Veppanthattai, Perambalur, Tamilnadu, India. E mail :ksvgac@gmail.com Article History: Received: 28-05-2024 Revised: 19-07-2024 Accepted: 01-08-2024 Abstract: Models of mathematics can accurately and precisely obtain solutions to many complex unsolvable problems in all fields. In this paper using a few mathematical models we can take the rainfall levels in Namakkal district from 2014 to 2018 and find the highest rainfall year by the fuzzy mean and the fuzzy variance in the Log-Normal distribution. Keywords: Log - Normal distribution, Mean, Variance, Mathematical Modelling, Fuzzy Set 2010 AMS subject classification: 01-06, 62E10, 62E86, 97M10. 1. Introduction A model may help to explain a system and to study the effect of different components, and to make predictions about behavior [1].The Namakkal area is referred to as 'Thiruvaraikkal' in the inscriptions on the north-western and southern walls of the Paladaina temple located on the hill. Namakkal is also known as the 'City of Chickens' and the 'Egg City' as the eggs sent to most parts of the country is produced in Namakkal. Namakkaldistrict was separated from Salem district and emerged as a separate district from 01.01.1997. In Namakkal district, there are eight circles namely Namakkal, Rasipuram, Kollimalai, Chenthamangalam, Paramathi Vellore, Tiruchengode, Kumarapalayam and Mohanur. Namakkal district is bounded on the north by Salem district, on the south by Karur district, on the east by Trichy and Salem districts and on the west by Erode district. According to the 2011 census, the population of Namakkal district is 17, 26,601. This includes 8, 69,280 males and 8, 57,321 females [6]. Some authors [2, 3, 4] instructed with some distributions in their books. The geographical area of Namakkal district is 3368.21 sq. Km. Is. The district is located between 11.00 ′ and 11.360 ′ north latitude and 77.28′ and 78.300′ east longitude.The main waterfalls of the district are Kaviriyaru, Ayyaru, Karippottan River and Thirumanimuttaru. Of these, the Cauvery River flows in a south-southwesterly direction, bordering the district. The famous Cauvery River Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 173 https://internationalpubls.com flows through the Paramathi Vellore Circle. Thus irrigated lands are located in Paramathi and Mohanur Unions. Agriculture continues to be an important part of the economy of the district. 70% of the people are engaged in agriculture and its related occupations as their livelihood. The total area of the district is 3363.35 sq km. The total crop area is 3.367 lakh hectares. Sustainable agricultural production, increasing productivity in sustainable agriculture, meeting the demand for food in line with the growing population, meeting the demand for raw materials for agro-based industries and providing employment to the rural population have always been key principles and principles. The total geographical area of Namakkal district is 3, 36,719 hectares. Of this, the area under net crop cultivation is 1, 41,537 hectares. Of these, 60,939 hectares are irrigated. The remaining 80,598 hectares are irrigated area. The Pallipalayam area is irrigated on an area of 4585 hectares through the Mato East Coast Canal. The average annual rainfall is 716.54 mm [5]. Due to the cultivation of various crops in Namakkal district, agriculture is the main occupation of the people of this district. The district receives rainfall in all seasons. Most rainfall, however, is available through the northeast monsoon. Syamala et. al [7, 8, 9, 10] discussed about some fuzzy concepts. In this paper, we can find out about calculating the maximum rainfall year in Namakkal district by using fuzzy mean and the fuzzy variance in the Log-Normal distribution using mathematical models with rainfall patterns for 2014-2018. 2. Fuzzy Log-Normal Distribution One way to quality a random variable follows a log-normal distribution is to say that its logarithm is normally distributed [4]. The p.d.f of log-normal distribution is given by f( n ; μ , σ ) = 1 𝑛𝜎√2𝜋 𝑒− 1 2 ( 𝐼𝑛 𝑛−𝜇 𝜎 ) 2 where the variable n> 0 and the parameter μ and σ > 0 all are real numbers. It is sometimes denoted Λ( μ, σ2 ) in the same spirit as we often a normally distributed variable by N( μ, σ2 ) [2]. The c.d.f of the log-normal distribution is given by F(n) = 1 𝜎√2𝜋 ∫ 1 𝑡 𝑛 0 𝑒− 1 2 ( 𝐼𝑛 𝑛−𝜇 𝜎 ) 2 dt The mean of the log-normal distribution is given by 𝜇𝑘 1 = E( nk ) = 𝑒𝑘𝜇+ 𝑘2𝜎2 2 The 𝛼 cut of fuzzy mean is �̅�(n) = {�̅�𝑙(𝑛), �̅�𝑢(𝑛)} where�̅�𝑙(𝑛) = min {𝑒�̅�+ �̅�2 2 } and �̅�𝑢(𝑛) = max {𝑒�̅�+ �̅�2 2 }. The 𝛼 cut of fuzzy variance is �̅�(n) = {�̅�𝑙(𝑛), �̅�𝑢(𝑛)} where�̅�𝑙(𝑛) = min {𝑒2�̅�+ �̅�2 ( 𝑒�̅�2 − 1)} and �̅�𝑢(𝑛) = max {𝑒2�̅�+ �̅�2 ( 𝑒�̅�2 − 1)}. 3. Application Let us take the Namakkal district rainfall levels from 2014 to 2018 [3]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 174 https://internationalpubls.com Table.3.ARainfall levels for 2014 Months Jan Feb Mar April May June July Aug Sep Oct Nov Dec Rainfall levels (mm) 0 0.6 0 21.4 96.2 35.7 34.8 67.3 81.9 195.5 44.8 11.3 Table.3.B Rainfall levels for 2015 Months Jan Feb Mar April May June July Aug Sep Oct Nov Dec Rainfall levels (mm) 24.5 1.1 2.7 107.8 75.4 45.2 23.6 65.7 164.3 115 162.3 31.9 Table.3.C Rainfall levels for 2016 Months Jan Feb Mar April May June July Aug Sep Oct Nov Dec Rainfall levels (mm) 0 0 0 5.3 85.6 35.4 125.2 49.4 31.3 24.2 7.4 25.9 Table.3.D Rainfall levels for 2017 Months Jan Feb Mar April May June July Aug Sep Oct Nov Dec Rainfall levels (mm) 6.3 0 16.8 26.9 98 24 45.1 115.2 194.8 149.8 43.6 63.1 Table.3.E Rainfall levels for 2018 Months Jan Feb Mar April May June July Aug Sep Oct Nov Dec Rainfall levels (mm) 0 26.8 14 13.6 155.8 33.2 41.5 61.1 120.9 106.5 63.7 12.4 4. Results The double parameter of the log-normal distribution for Table.3.A is σ = 0.57667, μ = 4.1952. Let the matching fuzzy triangular numbers are 𝜎 = [0.34547, 0.57667, 0.80787] and �̅� = [3.7415, 4.1952, 4.6489] and the matching α- cut are given by 𝜎= [0.34547 + 0.2312α, 0.80787 – 0.2312α] and �̅� = [3.7415 + 0.4537α, 4.6489 – 0.4537α] Table 4.A Fuzzy Mean and variance value for lower and upper alpha values α low σ low μ up σ up μ El(x) Eu(x) Vl(x) Vu(x) 0 0.34547 3.7415 0.80787 4.6489 44.75373 144.7817 253.8945 19298.03 0.1 0.36859 3.78687 0.78475 4.60353 47.21914 135.8357 324.458 15705.73 0.2 0.39171 3.83224 0.76163 4.55816 49.847 127.5107 412.0517 12782.51 0.3 0.41483 3.87761 0.73851 4.51279 52.64925 119.7599 520.5086 10402.46 0.4 0.43795 3.92298 0.71539 4.46742 55.63876 112.5403 654.5131 8463.763 0.5 0.46107 3.96835 0.69227 4.42205 58.82946 105.8125 819.7913 6883.961 0.6 0.48419 4.01372 0.66915 4.37668 62.2364 99.54013 1023.346 5596.23 0.7 0.50731 4.05909 0.64603 4.33131 65.87584 93.68962 1273.746 4546.353 0.8 0.53043 4.10446 0.62291 4.28594 69.76539 88.23012 1581.479 3690.304 0.9 0.55355 4.14983 0.59979 4.24057 73.92409 83.13319 1959.391 2992.299 1 0.57667 4.1952 0.57667 4.1952 78.37258 78.37258 2423.228 2423.228 The double parameter of the log-normal distribution for Table.3.B is σ = 0.65456, μ = 4.3452. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 175 https://internationalpubls.com Let the matching fuzzy triangular numbers are 𝜎 = [0.42336, 0.65456, 0.88576] and �̅� = [3.8915, 4.3452, 4.7989] and the matching α- cut are given by 𝜎= [0.42336 + 0.2312α, 0.88576 – 0.2312α] and �̅� = [3.8915 + 0.4537α, 4.7989 – 0.4537α] Table 4.B Fuzzy Mean and variance value for lower and upper alpha values α low σ low μ up σ up μ El(x) Eu(x) Vl(x) Vu(x) 0 0.42336 3.8915 0.88576 4.7989 53.57684 179.6814 563.4755 38466.97 0.1 0.44648 3.93687 0.86264 4.75353 56.63019 168.2757 707.4666 31279.98 0.2 0.4696 3.98224 0.83952 4.70816 59.88955 157.6783 884.9611 25444.96 0.3 0.49272 4.02761 0.8164 4.66279 63.37038 147.8273 1103.46 20704.03 0.4 0.51584 4.07298 0.79328 4.61742 67.08936 138.6658 1372.141 16849.36 0.5 0.53896 4.11835 0.77016 4.57205 71.06457 130.1416 1702.243 13713.31 0.6 0.56208 4.16372 0.74704 4.52668 75.31558 122.2068 2107.535 11160.47 0.7 0.5852 4.20909 0.72392 4.48131 79.86355 114.8171 2604.894 9081.355 0.8 0.60832 4.25446 0.7008 4.43594 84.73143 107.9319 3215.026 7387.357 0.9 0.63144 4.29983 0.67768 4.39057 89.94409 101.5138 3963.351 6006.681 1 0.65456 4.3452 0.65456 4.3452 95.52848 95.52848 4881.101 4881.101 The double parameter of the log-normal distribution for Table.3.C is σ = 1.0416, μ = 3.9543. Let the matching fuzzy triangular numbers are 𝜎 = [0.8104, 1.0416, 1.2728] and �̅� = [3.5006, 3.9543, 4.4070] and the matching α- cut are given by 𝜎= [0.8104 + 0.2312α, 1.2728 – 0.2312α] and �̅� = [3.5006 + 0.4537α, 4.4070 – 0.4537α] Table 4.C Fuzzy Mean and variance value for lower and upper alpha values α low σ low μ up σ up μ El(x) Eu(x) Vl(x) Vu(x) 0 0.8104 3.5006 1.2728 4.408 46.01531 184.5665 1966.037 138071.1 0.1 0.83352 3.54597 1.24968 4.36263 49.0749 171.3109 2416.1 110549.4 0.2 0.85664 3.59134 1.22656 4.31726 52.36592 159.0923 2969.923 88628.67 0.3 0.87976 3.63671 1.20344 4.27189 55.90752 147.8242 3651.929 71144.36 0.4 0.90288 3.68208 1.18032 4.22652 59.72055 137.4277 4492.455 57179.18 0.5 0.926 3.72745 1.1572 4.18115 63.82775 127.8306 5529.222 46009.63 0.6 0.94912 3.77282 1.13408 4.13578 68.25389 118.9673 6809.197 37064.2 0.7 0.97224 3.81819 1.11096 4.09041 73.02598 110.7778 8390.929 29890.69 0.8 0.99536 3.86356 1.08784 4.04504 78.1735 103.2071 10347.5 24130.86 0.9 1.01848 3.90893 1.06472 3.99967 83.72861 96.20529 12770.24 19500.42 1 1.0416 3.9543 1.0416 3.9543 89.72642 89.72642 15773.46 15773.46 The double parameter of the log-normal distribution for Table.3.D is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 176 https://internationalpubls.com σ = 0.70643, μ = 4.3393. Let the matching fuzzy triangular numbers are 𝜎 = [0.47523, 0.70643, 0.93763] and �̅� = [3.8856, 4.3393, 4.7930] and the matching α- cut are given by 𝜎= [0.47523 + 0.2312α, 0.93763 – 0.2312α] and �̅� = [3.8856 + 0.4537α, 4.7930 – 0.4537α] Table 4.D Fuzzy Mean and variance value for lower and upper alpha values α low σ low μ up σ up μ El(x) Eu(x) Vl(x) Vu(x) 0 0.47523 3.8856 0.93763 4.793 54.5175 187.2744 753.0841 49410.82 0.1 0.49835 3.93097 0.91451 4.74763 57.69361 175.1765 938.3589 40134.62 0.2 0.52147 3.97634 0.89139 4.70226 61.08739 163.9478 1166.133 32617.59 0.3 0.54459 4.02171 0.86827 4.65689 64.7154 153.5209 1445.926 26520.79 0.4 0.56771 4.06708 0.84515 4.61152 68.59553 143.8339 1789.403 21571.84 0.5 0.59083 4.11245 0.82203 4.56615 72.74717 134.8303 2210.864 17551.54 0.6 0.61395 4.15782 0.79891 4.52078 77.19134 126.4578 2727.856 14283.33 0.7 0.63707 4.20319 0.77579 4.47541 81.9508 118.6686 3361.928 11624.8 0.8 0.66019 4.24856 0.75267 4.43004 87.05024 111.4188 4139.556 9460.951 0.9 0.68331 4.29393 0.72955 4.38467 92.51643 104.6678 5093.308 7698.846 1 0.70643 4.3393 0.70643 4.3393 98.37844 98.37844 6263.272 6263.272 The double parameter of the log-normal distribution for Table.3.E is σ = 0.52939, μ = 4.2867. Let the matching fuzzy triangular numbers are 𝜎 = [0.29819, 0.52939, 0.76059] and �̅� = [3.8330, 4.2867, 4.7404] and the matching α- cut are given by 𝜎= [0.29819 + 0.2312α, 0.76059 – 0.2312α] and �̅� = [3.8330 + 0.4537α, 4.7404 – 0.4537α] Table 4.E Fuzzy Mean and variance value for lower and upper alpha values α low σ low μ up σ up μ El(x) Eu(x) Vl(x) Vu(x) 0 0.29819 3.833 0.76059 4.7404 48.30131 152.8793 216.9477 18308.66 0.1 0.32131 3.87837 0.73747 4.69503 50.90647 143.5899 281.842 14899.57 0.2 0.34443 3.92374 0.71435 4.64966 53.68083 134.937 362.9574 12122.59 0.3 0.36755 3.96911 0.69123 4.60429 56.63666 126.8734 463.9738 9859.662 0.4 0.39067 4.01448 0.66811 4.55892 59.7872 119.3554 589.3857 8015.078 0.5 0.41379 4.05985 0.64499 4.51355 63.14673 112.3429 744.6849 6511.193 0.6 0.43691 4.10522 0.62187 4.46818 66.7307 105.799 936.5822 5284.951 0.7 0.46003 4.15059 0.59875 4.42281 70.55579 99.68954 1173.281 4285.101 0.8 0.48315 4.19596 0.57563 4.37744 74.64003 93.9831 1464.815 3469.948 0.9 0.50627 4.24133 0.55251 4.33207 79.00291 88.65068 1823.457 2805.546 1 0.52939 4.2867 0.52939 4.2867 83.66552 83.66552 2264.233 2264.233 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 177 https://internationalpubls.com Table 4.F Fuzzy Mean value of the Rainfall levels for 2014 to 2018 α El(x) Eu(x) 2014 2015 2016 2017 2018 2014 2015 2016 2017 2018 0 44.7537 3 53.5768 4 46.0153 1 54.5175 48.3013 1 144.781 7 179.681 4 184.566 5 187.274 4 152.879 3 0. 1 47.2191 4 56.6301 9 49.0749 57.6936 1 50.9064 7 135.835 7 168.275 7 171.310 9 175.176 5 143.589 9 0. 2 49.847 59.8895 5 52.3659 2 61.0873 9 53.6808 3 127.510 7 157.678 3 159.092 3 163.947 8 134.937 0. 3 52.6492 5 63.3703 8 55.9075 2 64.7154 56.6366 6 119.759 9 147.827 3 147.824 2 153.520 9 126.873 4 0. 4 55.6387 6 67.0893 6 59.7205 5 68.5955 3 59.7872 112.540 3 138.665 8 137.427 7 143.833 9 119.355 4 0. 5 58.8294 6 71.0645 7 63.8277 5 72.7471 7 63.1467 3 105.812 5 130.141 6 127.830 6 134.830 3 112.342 9 0. 6 62.2364 75.3155 8 68.2538 9 77.1913 4 66.7307 99.5401 3 122.206 8 118.967 3 126.457 8 105.799 0. 7 65.8758 4 79.8635 5 73.0259 8 81.9508 70.5557 9 93.6896 2 114.817 1 110.777 8 118.668 6 99.6895 4 0. 8 69.7653 9 84.7314 3 78.1735 87.0502 4 74.6400 3 88.2301 2 107.931 9 103.207 1 111.418 8 93.9831 0. 9 73.9240 9 89.9440 9 83.7286 1 92.5164 3 79.0029 1 83.1331 9 101.513 8 96.2052 9 104.667 8 88.6506 8 1 78.3725 8 95.5284 8 89.7264 2 98.3784 4 83.6655 2 78.3725 8 95.5284 8 89.7264 2 98.3784 4 83.6655 2 Fig.4.A Lower mean value of rainfall levels in 2014 - 2018 Fig.4.B Upper mean value of rainfall levels in 2014 - 2018 5. Conclusion This paper gets the results through a mathematical model of rainfall levels in Namakkal district from 2014 to 2018. These results show that the alpha cut value continues to increase in the lower average and the alpha cut value decreases in the upper average. By this approach the conclusion holds that in 2017, Namakkal district received heavy rainfall. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 178 https://internationalpubls.com References [1]. Chandran D, Copeland W.B, Sleight S.C, Sauro H.M, “ Mathematical modeling and synthetic biology’’ , Elsevier, vol-5, No.4, 2008. [2]. Chirstian Walck, “ Hand book on statistical distribution for experimentalists’’, InternationalReport SUF-PFY, 86,87, 1996. [3]. Customized Rainfall Information System (CRIS) Hydromet Division India Meteorological Department Ministry Of Earth Sciences New Delhi-110 003, 2020 [4]. 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R, “3D Based CT Scan Retrial Queuing Models by Fuzzy Ordering Approach”, Second International Conference on Electrical, Electronics, Information and Communication Technologies (ICEEICT),IEEE Explore, pp. 01-05, doi: 10.1109/ICEEICT56924.2023.10157305, 2023. https://namakkal.nic.in/departments/agriculture/ https://www.census2011.co.in/census/district/29-namakkal.html