Volume 14, Issue 04 (April 2025)

Fuzzy Mathematical Fatigue Life Model for the Effect of North West India Rainfall

DOI : 10.17577/IJERTV14IS040069

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Prakasam. S, Venkatesh. A, 2025, Fuzzy Mathematical Fatigue Life Model for the Effect of North West India Rainfall, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 14, Issue 04 (April 2025),

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Fuzzy Mathematical Fatigue Life Model for the Effect of North West India Rainfall

Prakasam. S

Assistant Professor of Mathematics,

A. V. V. M. Sri Pushpam College (Autonomous) (Affiliated to Bharathidasan University, Tiruchirappalli),

Thanjavur (Dt), Tamilnadu, India.

Venkatesh. A

Assistant Professor of Mathematics,

A. V. V. M. Sri Pushpam College (Autonomous) (Affiliated to Bharathidasan University, Tiruchirappalli),

Thanjavur (Dt), Tamilnadu, India.

ABSTRACT:

In this paper, we use the mathematical Fatigue Life distribution model to calculate the rainfall in the northwestern states of India, namely Uttar Pradesh, Uttarakhand, Delhi, Punjab, Himachal Pradesh, Jammu and Kashmir, and Rajasthan, and see which year received the most rainfall.

  1. INDRODUCTION:

    The North West Indian subdivisions by the Meteorological Department include 7 states in India.

    • Uttar Pradesh

    • Uttarakhand

    • Delhi

    • Punjab

    • Himachal Pradesh

    • Jammu & Kashmir

    • Rajasthan

      1. History of North West India:

        Rainfall and temperature are the most important environmental factors influencing crop growth, development, and yield. The northwestern (NW) part of India is one of the main regions of food grain production of the country. North-East India displays, to a large extent, the character of tropical climate, especially in the valleys. The region has a monsoon climate with heavy to very heavy rains, confined within four summer months from June to September. There are three distinct seasons in the region: winter, summer, and the rainy season.

      2. Uttar Pradesh:

        Uttar Pradesh is a state in India, stretches back several millennia. The region shows the presence of human habitation dating back to between 85,000 and 73,000 years ago. Additionally, the region seems to have been domesticated as early as 6,000 BC. After independence in 1947, the United Provinces were renamed Uttar Pradesh in 1950.

        Uttar Pradesh is a state in northern India. With over 241 million inhabitants, it is the most populated state in India as well as the most populous country subdivision in the world more populous than all but four other countries outside of India and accounting for 16.5 percent of the total population of India. The state is divided into 18 divisions and 75 districts. On 9 November 2000, a new state, Uttaranchal (now Uttarakhand), was created from Uttar Pradesh's western Himalayan hill region. The two major rivers of the state, the Ganges and its tributary Yamuna, meet at the Triveni Sangam in Prayagraj (formerly Allahabad), a Hindu pilgrimage site. Other notable rivers are Gomti and Saryu. The forest cover in the state is 6.1 per cent of the state's geographical area.

        The climate of Uttar Pradesh (U.P) is primarily defined as humid subtropical with dry winter (Cwa) type with parts of Western U.P. as hot semi- arid (BSh) type. Alternatively, some authors refer to it as tropical monsoon. Variations do exist in different parts of the large state. The average annual rainfall is approximately 650 millimetres in the southwestern part of UP. The precipitation is around 1000 millimetres in the southeast and east of UP.

      3. Uttarakhand:

        Uttarakhand is a Himalayan state in North India, nestled between the Tibetan Plateau and the Indo- Gangetic Plains. The name, which means "northern land" or "section" or "northern part" in Sanskrit was made popular in the 80s as part of the wider statehood struggle within the region. Initially, it was believed that due to harsh climate and mountainous terrain, this was a barren and uninhabited land. But after various excavations and the study of ancient literature, it is now established that the history of Uttarakhand goes back to Stone Age. Uttarakhand is popularly referred to as "the land of the gods" (Devbhumi) owing to its huge number of Hindu pilgrimage sites and proximity to Mount Kailash.

        Uttarakhand state is located between 28°43 – 31°27 N latitudes and 77°34 – 81°02 E longitudes in the northern part of India. The state is mostly hilly and has international boundary with China (Tibet) in the north and Nepal in the east and state boundary with Himachal Pradesh in the northwest. The state has foothills areas in the south and southwest which are bounded by Uttar Pradesh.

        Uttarakhand state has total geographical area of 53,485 sq. km, of which 86% is mountainous and 65% is covered by forest.The state is rich in natural resources especially water and forests with many glaciers, perennial rivers, dense forests and snow- capped mountain peaks. Annual rainfall ranges from 92 to 237 cm.

      4. Delhi:

        Delhi has been an important political centre of India as the capital of several empires. The recorded history of Delhi begins with the 8th century Tomar Rajputs kingdom. Delhi, city and national capital, and union territory, north-central India. The city of Delhi actually consists of two components: Old Delhi, in the north, the historic city; and New Delhi, in the south, since 1947 the capital of India, built in the first part of the 20th century as the capital of British India.

        One of the countrys largest urban agglomerations, Delhi sits astride (but primarily on the west bank of) the Yamuna River, a tributary of the Ganges (Ganga) River, about 100 miles (160 km) south of the Himalayas. The national capital territory embraces Old and New Delhi and the surrounding metropolitan region, as well as adjacent rural areas. Delhi receives an average annual precipitation of 774.4 mm.

        The climate of Delhi is an overlap between monsoon-influenced and semi-arid, with high variation between summer and winter temperatures and precipitation. Summer starts in early April and peaks in late May or early June, with average

        temperatures near 38 °C (100 °F) although occasional heat waves can result in highs close to 45 °C (113 °F) on some days and therefore higher apparent temperature. The monsoon starts in late June and lasts until mid-September, with about

        797.3 mm (31.39 inches) of rain. The average temperatures are around 29 °C (84 °F), although they can vary from around 25 °C (77 °F) on rainy days to 3540 °C (95104 °F) during dry spells.

      5. Punjab:

        The History of Punjab refers to the past history of Punjab region which is a geopolitical, cultural, and historical region in the northwest of South Asia, comprising western Punjab province in Pakistan and eastern Punjab state in India. It is believed that the earliest evidence of human habitation in Punjab traces to the Soan valley of the Pothohar,between the Indus and the Jhelum rivers, where Soanian culture developed between 774,000 BC and 11,700 BC.

        Some two-fifths of Punjabs population is engaged in the agricultural sector, which accounts for a significant segment of the states gross product. Punjab produces an important portion of Indias food grain and contributes a major share of the wheat and rice stock held by the Central Pool . Much of the states agricultural progress and productivity is attributable to the so-called Green Revolution, an international movement launched in the 1960s that introduced not only new agricultural technologies but also high-yielding varieties ofwheat and rice.

        Punjab has an inland subtropical location, and its climate is continental, being semiarid to subhumid. In June, the warmest month, daily temperatures in Ludhiana usually reach about 100 °F (upper 30s C) from a low in the upper 70s F (mid-20s C). In January, the coolest month, daily temperatures normally rise from the mid-40s (about 7 °C) into the mid-60s F (upper 10s C). The average annual rainfall is 273 to 676 mm .

      6. Himachala Pradesh:

        Himachal Pradesh was established in 1948 as a Chief Commissioner's Province within the Union of India. The province comprised the hill districts around Shimla and southern hill areas of the former Punjab region. Thus Himachal emerged as the eighteenth state of the Indian Union. After independence the Chief Commissioner's province of H.P. came into being on 15 April 1948. Himachal became a part C state on 26 January 1950 with the implementation of the Constitution of India. Himachal Pradesh became Union Territory on 1 November 1956. On 18 December 1970 the State of Himachal Pradesh Act was passed by Parliament and the new state came into being on 25 January 1971.

        There is a huge variation in the climatic conditions of Himachal Pradesh due to variation in altitude (3606500 metres). The climate varies from hot and sub-humid tropical (450900 metres) in the southern low tracts, warm and temperate (900 1800 metres), cool and temperate (18002400 metres) and cold glacial and alpine (24004800 metres) in the northern and eastern high elevated mountain. Pollution is affecting the climate of almost all the states of India. The average annual rainfall is 1,251(mm). The rainy season start at the end of the month of June.

        The spring season starts from mid February to mid April. The weather is pleasant and comfortable in the season. The rainy season start at the end of the month of June. The landscape lushes green and fresh. During the season streams and natural springs are replenished. The average annual rainfall is 1,251(mm).

      7. Jammu & Kashmir:

        Jammu and Kashmir, also known as Kashmir and Jammu, was a princely state in a subsidiary alliance under British East India Company rule from 1846 to 1858 and under the paramountcy of the British Crown, from 1858 until the Partition of India in 1947, when it became a disputed territory, now administered by three countries: China, India, and Pakistan. Jammu and Kashmir is an State of India with population of Approximate 1.25 Crores. The population of Jammu and Kashmir state is 12,541,302. The density of Jammu and Kashmir state is 56 per sq km. Jammu and Kashmir State is spread over 222,236 Sq Km.

        The inter mountainous valley of Kashmir has unique geographical setting and it is located 130 between the Greater Himalayas in the north and Pir Panjal ranges in the south, roughly within the latitude and longitude ranges of 330 55´ to 34o 50´ and 74o 30´ to 75o 131 35´ respectively. The heights of these 132 mountains range from about 3,000 to 5,000 m and the mountains strongly influence the weather and 133 climate of the region.

        State of Jammu and Kashmir is home to several Himalayan glaciers. Jhelum, Indus, Tawi, Ravi and Chenab are major rivers flowing through the state. the state of Jammu comprises of three distinct climatic regions: cold arid desert areas of Ladakh, temperate Kashmir Valley, and the humid sub-tropical region of Jammu. The mean annual rainfall in cold arid zone varies from 100 mm to

        300 mm received by western disturbance. The average rainfall annually is usually between 700 and 750 mm.

      8. Rajasthan:

    The history of human settlement in the western Indian state of Rajasthan dates back to

    about 100,000 years ago. Around 5000 to 2000 BCE many regions of Rajasthan belonged as the site of the Indus Valley Civilization. Kalibangan is the main Indus site of Rajasthan, here fire altars have been discovered, similar to those found at Lothal. Around 2000 BCE, Sarasvati River flowed through the Aravalli mountain range in the state.

    Rajasthan state of northwestern India, located in the northwestern part of the Indian subcontinent. It is bounded to the north and northeast by the states of Punjab and Haryana, to the east and southeast by the states of Uttar Pradesh and Madhya Pradesh, to the southwest by the state of Gujarat, and to the west and northwest by the provinces of Sindh and Punjab in Pakistan.

    The capital city is Jaipur, in the east-central part of the state. The Aravalli (Aravali) Range forms a line across the state running roughly from Guru Peak on Mount Abu (5,650 feet [1,722 metres]), near the town of Abu in the southwest, to the town of Khetri in the northeast. About three-fifths of the state lies northwest of that line, leaving the remaining two- fifths in the southeast. Those are the two natural divisions of Rajasthan. The northwestern tract is generally arid and unproductive, although its character shifts gradually from desert in the far west and northwest to comparatively fertile and habitable land toward the east. The region includes the Thar (Great Indian) Desert. The annual rainfall in the East and West Rajasthan is about 64.9 cm and 32.7 cm respectively.

  2. FATIGUE LIFE DISTRIBUTION

    The BirnbaumSaunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. There are several alternative formulations of this distribution in the literature. It is named after Z. W. Birnbaum and S. C. Saunders. The Fatigue life distribution is closed under scalar multiplication and under reciprocation; its median coincides with the BS scale parameter; it has different shapes for its probability density function (PDF), which cover high, medium, and low asymmetry levels .

    This distribution was developed to model failures due to cracks. A material is placed under repeated cycles of stress. The jth cycle leads to an increase in the crack by Xj amount. The sum of the Xj is assumed to be normally distributed with mean n and variance n2. The probability that the crack does not exceed a critical length is

    where is the cdf of normal distribution. The more usual form of this distribution is:

    Here is the shape parameter and is the scale parameter.

    1. Proprerties

      The BirnbaumSaunders distribution is unimodal with a median of .

      Since the general form of probability functions can be expressed in terms of the standard distribution, all of the subsequent formulas are given for the standard form of the function.

      2.4 Cumulative Distribution Function

      The formula for the cumulative distribution function is

      Mean =

      ²

      (1 + )

      Variance = ( )² (1 + 5² )

      4

      2

      Given a data set that is thought to be Birnbaum Saunders distributed the parameters' values are best estimated by maximum likelihood.

      If T is BirnbaumSaunders distributed with parameters and then T1 is also Birnbaum- Saunders distributed with parameters and 1.

    2. Probability density function

      The general formula for the probability density function (pdf) is

      where is the shape parameter, is the location parameter, is the scale parameter, and s the probability density function of the standard normal distribution.

    3. Standard Fatigue Life Distrinution

      The case where = 0 and = 1 is called the standard fatigue life distribution. The pdf for the standard fatigue life distribution reduces to

      where is the cumulative distribution function of the standard normal distribution.

      2.5 Quantile Function

      The formula for the quantile function is

      where 1 is the quantile function of the standard normal distribution.

  3. APPLICATION

    1. Application – I

      In this work, We observe the normal monthly rainfall for the state Uttar Pradesh which is the state of North West India from the month January to December during the year 2012. We present a model of Rainfall for the state Uttar Pradesh in India using the Fatigue Life distribution with two parameters.

      We have constructed a model of Rainfall that produces result that qualitatively and quantitatively agree with experimental observation.

      Month

      Jan

      Feb

      Mar

      Apr

      May

      Jun

      Jul

      Aug

      Sep

      Oct

      Nov

      Dec

      Rainfall (mm)

      26

      5.75

      3.95

      5.55

      0.45

      16.35

      273.45

      233.9

      163.7

      1.0

      0.2

      1.9

      Table 3.1.1 Rainfall of Uttar Pradesh in 2012

      The scale and shape parameters of Fatigue life distribution in Table 3.1.1, = 3.6399 = 8.46115. The triangular fuzzy number is = [ 3.1021, 3.6399, 4.1777 ] and = [ 7.92335, 8.46115, 8.99895 ] are corresponding cuts are = [3.1021+0.5378 , 4.1777-0.5378 ] = [ 7.92335+0.5378 , 8.99895-0.5378 ]

      low

      up

      low

      up

      low E(X)

      up E(X)

      0

      3.1021

      4.1777

      7.92335

      8.99895

      46.04665

      87.52908

      0.1

      3.15588

      4.12392

      7.97713

      8.94517

      47.70156

      85.00915

      0.2

      3.20966

      4.07014

      8.03091

      8.89139

      49.3978

      82.53895

      0.3

      3.26344

      4.01636

      8.08469

      8.83761

      51.13583

      80.11801

      0.4

      3.31722

      3.96258

      8.13847

      8.78383

      52.91612

      77.74586

      0.5

      3.371

      3.9088

      8.19225

      8.73005

      54.73914

      75.42203

      0.6

      3.42478

      3.85502

      8.24603

      8.67627

      56.60536

      73.14607

      0.7

      3.47856

      3.80124

      8.29981

      8.62249

      58.51524

      70.9175

      0.8

      3.53234

      3.74746

      8.35359

      8.56871

      60.46924

      68.73586

      0.9

      3.58612

      3.69368

      8.40737

      8.51493

      62.46784

      66.60068

      1

      3.6399

      3.6399

      8.46115

      8.46115

      64.5115

      64.5115

      Table 3.1.2 Lower and Upper alpha cut for mean

    2. Application – II

      In this work, We observe the normal monthly rainfall for the state Uttarakhand which is the state of North West India from the month January to December during the year 2012.

      We present a model of Rainfall for the state Uttarakhand in India using the Fatigue Life distribution with two parameters.

      Month

      Jan

      Feb

      Mar

      Apr

      May

      Jun

      Jul

      Aug

      Sep

      Oct

      Nov

      Dec

      Rainfall (mm)

      53.4

      13.4

      30.8

      46.6

      10.7

      48.0

      389.8

      461.2

      223.3

      8.8

      4.6

      19.2

      Table 3.2.1 Rainfall of Uttarakhand in 2012

      The scale and shape parameter of Fatigue life distribution in Table 3.2.1, = 1.6981 = 47.012. The triangular fuzzy number is = [ 1.1603, 1.6981, 2.2359 ] & = [ 46.4742, 47.012, 47,5498 ] are corresponding cuts are = [1.1603+0.5378 , 2.2359-0.5378 ] & = [ 46.4742+0.5378 , 47.5498-0.5378 ]

      low

      up

      low

      up

      low E(X)

      up E(X)

      0

      1.1603

      2.2359

      46.4742

      47.5498

      77.75822

      166.4064

      0.1

      1.21408

      2.18212

      46.52798

      47.49602

      80.81887

      160.5757

      0.2

      1.26786

      2.12834

      46.58176

      47.44224

      84.02113

      154.8949

      0.3

      1.32164

      2.07456

      46.63554

      47.38846

      87.36544

      149.3637

      0.4

      1.37542

      2.02078

      46.68932

      47.33468

      90.85229

      143.9815

      0.5

      1.4292

      1.967

      46.7431

      47.2809

      94.48212

      138.7479

      0.6

      1.48298

      1.91322

      46.79688

      47.22712

      98.25542

      133.6624

      0.7

      1.53676

      1.85944

      46.85066

      47.17334

      102.1727

      128.7247

      0.8

      1.59054

      1.80566

      46.90444

      47.11956

      106.2343

      123.9341

      0.9

      1.64432

      1.75188

      46.95822

      47.06578

      110.4408

      119.2902

      1

      1.6981

      1.6981

      47.012

      47.012

      114.7926

      114.7926

      Table 3.2.2 Lower and Upper alpha cut for mean

    3. Application – III

      In this work, We observe the normal monthly rainfall for the state Delhi which is the state of North West India from the month January to December during the year 2012. We present a model of Rainfall for the state Delhi in India using the Fatigue Life distribution with two parameters. We have constructed a model of Rainfall that produces result that qualitatively and quantitatively agree with experimental observation.

      Month

      Jan

      Feb

      Mar

      Apr

      May

      Jun

      Jul

      Aug

      Sep

      Oct

      Nov

      Dec

      Rainfall (mm)

      4.6

      0.1

      0.3

      13.6

      3.6

      4.4

      57.9

      163.5

      57.1

      4.0

      0.6

      3.9

      Table 3.3.1 Rainfall of Delhi in 2012

      The scale and shape parameter of Fatigue life distribution in table 3.3.1 = 3.1529 =4.2156, The triangular fuzzy number is = [ 2.6151, 3.1529, 3.6907 ] & = [ 3.6778, 4.2156, 4.753 ] are corresponding cuts are = [ 2.6151+0.5378 , 3.6907-0.5378 ] = [ 3.6778+0.5378 , 4.7534-0.5378 ]

      low

      up

      low

      up

      low E(X)

      up E(X)

      0

      2.6151

      3.6907

      3.6778

      47.5498

      16.25357

      371.394

      0.1

      2.66888

      3.63692

      3.73158

      47.49602

      17.02145

      361.6154

      0.2

      2.72266

      3.58314

      3.78536

      47.44224

      17.81556

      351.9951

      0.3

      2.77644

      3.52936

      3.83914

      47.38846

      18.63637

      342.5328

      0.4

      2.83022

      3.47558

      3.89292

      47.33468

      19.48435

      333.228

      0.5

      2.884

      3.4218

      3.9467

      47.2809

      20.35995

      324.0802

      0.6

      2.93778

      3.36802

      4.00048

      47.22712

      21.26365

      315.0889

      0.7

      2.99156

      3.31424

      4.05426

      47.17334

      22.19592

      306.2537

      0.8

      3.04534

      3.26046

      4.10804

      47.11956

      23.15722

      297.5741

      0.9

      3.09912

      3.20668

      4.16182

      47.06578

      24.14801

      289.0497

      1

      3.1529

      3.1529

      4.2156

      47.012

      25.16877

      280.6799

      Table 3.3.2 Lower and Upper alpha cut for mean

    4. Application – IV

      In this work, We observe the normal monthly rainfall for the state Punjab which is the state of North West India from the month January to December during the year 2012. We present a model of Rainfall for the state Punjab in India using the Fatigue Life distribution with two parameters. We have constructed a model of Rainfall that produces result that qualitatively and quantitatively agree with experimental observation.

      Month

      Jan

      Feb

      Mar

      Apr

      May

      Jun

      Jul

      Aug

      Sep

      Oct

      Nov

      Dec

      Rainfall (mm)

      35.8

      2.9

      1.9

      20.2

      0.8

      9.6

      67.7

      105.1

      83.5

      2.8

      0.4

      8.2

      Table 3.4.1 Rainfall of Punjab in 2012

      The scale and shape parameter of Fatigue life distribution in Table 3.4.1 =2.2535 =7.7669. The triangular fuzzy number is = [ 1.7157, 2.2535, 2.7913 ] & = [ 7.2291, 7.7669, 8.3047 ] are corresponding cuts are

      = [1.7157+0.5378 , 2.7913-0.5378 ] & = [ 7.2291+0.5378 , 8.3047-0.5378 ]

      low

      up

      low

      up

      low E(X)

      up E(X)

      0

      1.7157

      2.7913

      7.2291

      8.3047

      17.86899

      40.65714

      0.1

      1.76948

      2.73752

      7.28288

      8.25092

      18.68445

      39.16718

      0.2

      1.82326

      2.68374

      7.33666

      8.19714

      19.53121

      37.71693

      0.3

      1.87704

      2.62996

      7.39044

      8.14336

      20.40973

      36.30591

      0.4

      1.93082

      2.57618

      7.44422

      8.08958

      21.32049

      34.93365

      0.5

      1.9846

      2.5224

      7.498

      8.0358

      22.26395

      33.5997

      0.6

      2.03838

      2.46862

      7.55178

      7.98202

      23.24058

      32.30357

      0.7

      2.09216

      2.41484

      7.60556

      7.92824

      24.25084

      31.04482

      0.8

      2.14594

      2.36106

      7.65934

      7.87446

      25.29519

      29.82296

      0.9

      2.19972

      2.30728

      7.71312

      7.82068

      26.37412

      28.63754

      1

      2.2535

      2.2535

      7.7669

      7.7669

      27.48808

      27.48808

      Table 3.4.2 Lower and Upper alpha cut for mean

    5. Application – V

      In this work, We observe the normal monthly rainfall for the state Himachal Pradesh which is the state of North West India from the month January to December during the year 2012. We present a model of Rainfall for the state Himachal Pradesh in India using the Fatigue Life distribution with two parameters. We have constructed a model of Rainfall that produces result that qualitatively and quantitatively agree with experimental observation.

      Month

      Jan

      Feb

      Mar

      Apr

      May

      Jun

      Jul

      Aug

      Sep

      Oct

      Nov

      Dec

      Rainfall (mm)

      112.1

      66.4

      43.2

      64.1

      11.5

      25.5

      207.1

      311.6

      152.1

      2.9

      6.7

      31.6

      Table 3.5.1 Rainfall of Himachal Pradesh in 2012

      The scale and shape parameter of Fatigue life distribution in Table 3.5.1, =1.6073 =35.913. The triangular fuzzy number is = [ 1.0695, 1.6073, 2.1451 ] & = [ 35,3752, 35.913, 36.4508 ] are corresponding cuts are = [1.0695+0.5378 , 2.1451-0.5378 ] = [ 35.3752+0.5378 , 36.4508-0.5378 ]

      low

      up

      low

      up

      low E(X)

      up E(X)

      0

      1.0695

      2.1451

      35.3752

      36.4508

      55.60681

      120.3141

      0.1

      1.12328

      2.09132

      35.42898

      36.39702

      57.78038

      115.9904

      0.2

      1.17706

      2.03754

      35.48276

      36.34324

      60.06291

      111.784

      0.3

      1.23084

      1.98376

      35.53654

      36.28946

      62.45488

      107.6945

      0.4

      1.28462

      1.92998

      35.59032

      36.23568

      64.95676

      103.7214

      0.5

      1.3384

      1.8762

      35.6441

      36.1819

      67.569

      99.86433

      0.6

      1.39218

      1.82242

      35.69788

      36.12812

      70.29207

      96.12274

      0.7

      1.44596

      1.76864

      35.75166

      36.07434

      73.12645

      92.49619

      0.8

      1.49974

      1.71486

      35.80544

      36.02056

      76.0726

      88.9842

      0.9

      1.55352

      1.66108

      35.85922

      35.96678

      79.13098

      85.58631

      1

      1.6073

      1.6073

      35.913

      35.913

      82.30206

      82.30206

      Table 3.5.2 Lower and Upper alpha cut for mean

    6. Application VI

      In this work, we observe the normal monthly rainfall for the state Jammu & Kashmir which is the state of North West India from the month January to December during the year 2012. We present a model of Rainfall for the state Jammu & Kashmir in India using the Fatigue Life distribution with two parameters.

      Month

      Jan

      Feb

      Mar

      Apr

      May

      Jun

      Jul

      Aug

      Sep

      Oct

      Nov

      Dec

      Rainfall (mm)

      120.2

      116.7

      53.3

      112.5

      60.1

      42.9

      121.0

      256.4

      138.5

      14.9

      12.7

      67.5

      Table 3.6.1 Rainfall of Jammu & Kashmir in 2012

      The scale and shape parameter of Fatigue life distribution in Table 3.6.1, =0.92999 =64.065 The triangular fuzzy number is = [ 0.39219, 0.92999, 1.46779 ] & = [ 63.5272, 64.065, 64.6028 ] are corresponding cuts are

      = [0.39219+0.5378 , 1.46779-0.5378 ] = [ 63.5272+0.5378 , 64.6028-0.5378 ]

      low

      up

      low

      up

      low E(X)

      up E(X)

      0

      0.39219

      1.46779

      63.5272

      64.6028

      68.41285

      134.1932

      0.1

      0.44597

      1.41401

      63.58098

      64.54902

      69.90377

      129.0795

      0.2

      0.49975

      1.36023

      63.63476

      64.49524

      71.58115

      124.1606

      0.3

      0.55353

      1.30645

      63.68854

      64.44146

      73.44548

      119.4362

      0.4

      0.60731

      1.25267

      63.74232

      64.38768

      75.49721

      114.9057

      0.5

      0.66109

      1.19889

      63.7961

      64.3339

      77.73682

      110.5687

      0.6

      0.71487

      1.14511

      63.84988

      64.28012

      80.16477

      106.4246

      0.7

      0.76865

      1.09133

      63.90366

      64.22634

      82.78153

      102.4732

      0.8

      0.82243

      1.03755

      63.95744

      64.17256

      85.58756

      98.71376

      0.9

      0.87621

      0.98377

      64.01122

      64.11878

      88.58333

      95.14597

      1

      0.92999

      0.92999

      64.065

      64.065

      91.76931

      91.76931

      Table 3.6.2 Lower and Upper alpha cut for mean

    7. Application – VII

      In this work, We observe the normal monthly rainfall for the state Rajasthan which is the state of North West India from the month January to December during the year 2012. We present a model of Rainfall for the state Rajasthan in India using the Fatigue Life distribution with two parameters.

      Month

      Jan

      Feb

      Mar

      Apr

      May

      Jun

      Jul

      Aug

      Sep

      Oct

      Nov

      Dec

      Rainfall (mm)

      1.1

      0.0

      0.0

      6.35

      12.05

      10.5

      104.4

      261.8

      109.1

      0.95

      0.0

      0.75

      Table 3.7.1 Rainfall of Rajasthan in 2012

      The scale and shape parameter of Fatigue life distribution =2.53515 =11.39415. The triangular fuzzy number is = [ 1.99735, 2.53515, 3.07295 ] & = [ 10.85635, 11.39415, 11.93195 ] are

      corresponding cuts are = [1.99735+0.5378 , 3.07295-0.5378 ] & = [ 10.85635+0.5378 , 11.93195- 0.5378 ]

      low

      up

      low

      up

      low E(X)

      up E(X)

      0

      1.99735

      3.07295

      10.85635

      11.93195

      32.51155

      68.26878

      0.1

      2.05113

      3.01917

      10.91013

      11.87817

      33.86032

      66.01523

      0.2

      2.10491

      2.96539

      10.96391

      11.82439

      35.25251

      63.8135

      0.3

      2.15869

      2.91161

      11.01769

      11.77061

      36.68859

      61.66312

      0.4

      2.21247

      2.85783

      11.07147

      11.71683

      38.16902

      59.56363

      0.5

      2.26625

      2.80405

      11.12525

      11.66305

      39.69427

      57.51456

      0.6

      2.32003

      2.75027

      11.17903

      11.60927

      41.26481

      55.51544

      0.7

      2.37381

      2.69649

      11.23281

      11.55549

      42.88111

      53.56581

      0.8

      2.42759

      2.64271

      11.28659

      11.50171

      44.54362

      51.6652

      0.9

      2.48137

      2.58893

      11.34037

      11.44793

      46.25282

      49.81314

      1

      2.53515

      2.53515

      11.39415

      11.39415

      48.00917

      48.00917

      Table 3.7.2 Lower and Upper alpha cut for mean

    8. Discussion

      UP

      Uttarakhand

      Delhi

      Punjab

      HP

      J & K

      Rajasthan

      0

      46.04665

      77.75822

      16.25357

      17.86899

      55.60681

      68.41285

      32.51155

      0.1

      47.70156

      80.81887

      17.02145

      18.68445

      57.78038

      69.90377

      33.86032

      0.2

      49.3978

      84.02113

      17.81556

      19.53121

      60.06291

      71.58115

      35.25251

      0.3

      51.13583

      87.36544

      18.63637

      20.40973

      62.45488

      73.44548

      36.68859

      0.4

      52.91612

      90.85229

      19.48435

      21.32049

      64.95676

      75.49721

      38.16902

      0.5

      54.73914

      94.48212

      20.35995

      22.26395

      67.569

      77.73682

      39.69427

      0.6

      56.60536

      98.25542

      21.26365

      23.24058

      70.29207

      80.16477

      41.26481

      0.7

      58.51524

      102.1727

      22.19592

      24.25084

      73.12645

      82.78153

      42.88111

      0.8

      60.46924

      106.2343

      23.15722

      25.29519

      76.0726

      85.58756

      44.54362

      0.9

      62.46784

      110.4408

      24.14801

      26.37412

      79.13098

      88.58333

      46.25282

      1

      64.5115

      114.7926

      25.16877

      27.48808

      82.30206

      91.76931

      48.00917

      lower E(X)

      500

      450

      400

      350

      300

      250

      200

      150

      100

      50

      0

      Rajasthan

      J & K HP

      Punjab Delhi

      Uttarakhand

      0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

      alpha value

      1

      UP

      Rainfall

      Table 3.8.1 Lower fuzzy mean value of North West India

      Fig: 3.8.1 Fuzzy lower mean value of North West India

      UP

      Uttarakhand

      Delhi

      Punjab

      HP

      J & K

      Rajasthan

      0

      87.52908

      166.4064

      371.394

      40.65714

      120.3141

      134.1932

      68.26878

      0.1

      85.00915

      160.5757

      361.6154

      39.16718

      115.9904

      129.0795

      66.01523

      0.2

      82.53895

      154.8949

      351.9951

      37.71693

      111.784

      124.1606

      63.8135

      0.3

      80.11801

      149.3637

      342.5328

      36.30591

      107.6945

      119.4362

      61.66312

      0.4

      77.74586

      143.9815

      333.228

      34.93365

      103.7214

      114.9057

      59.56363

      0.5

      75.42203

      138.7479

      324.0802

      33.5997

      99.86433

      110.5687

      57.51456

      0.6

      73.14607

      133.6624

      315.0889

      32.30357

      96.12274

      106.4246

      55.51544

      0.7

      70.9175

      128.7247

      306.2537

      31.04482

      92.49619

      102.4732

      53.56581

      0.8

      68.73586

      123.9341

      297.5741

      29.82296

      88.9842

      98.71376

      51.6652

      0.9

      66.60068

      119.2902

      289.0497

      28.63754

      85.58631

      95.14597

      49.81314

      1

      64.5115

      114.7926

      280.6799

      27.48808

      82.30206

      91.76931

      48.00917

      Table 3.8.2 Upper fuzzy mean value of North West India

      Upper E(X)

      1200

      1000

      800

      600

      400

      200

      0

      0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

      Alpha value

      1

      rajasthan

      J&K HP

      punjab delhi uttarakhand

      up

      Rainfall

      Fig: 3.8.2 Fuzzy Upper mean value of North West India

  4. CONCLUSION

This chapter has provided a brief introduction to the fatigue Life distribution and their application in fuzzy Mathematical modeling. Finally in this paper, if the fuzzy mean is increasing in lower alpha cut values and it decreases in upper alpha cut values. In this work, we observe the normal monthly rainfall of the North West India from the month January to December during the year 2012. Using the Fatigue life distribution we present the lower and upper fuzzy mean value of North West India. In this observation the state Rajasthan has the highest lower and upper mean value in North east India . Finally we conclude that Rajasthan gets highest level of rainfall during the year 2012.

REFERENCE:

[1]. Andras Kornai, Mathematical Linguistics (Advanced Information and Knowledge Processing), Springer, ISBN 978-1849966948.

[2]. Birnbaum, Z. W.; Saunders, S. C. "A new family of life distributions", Journal of Applied Probability, 6 (2): 319

327, doi:10.2307/3212003, JSTOR 3212003, archived

from the original on September 23, 2017.

[3]. Cordeiro, G. M.; Lemonte, A. J, "The -Birnbaum Saunders distribution: An improved distribution for fatigue life. modeling", Computational Statistics and Data Analysis, 55 (3): 1445

1461, doi:10.1016/j.csda.2010.10.007(2011).

[4]. D. Dubois and H. Prade , Fuzzy Sets and Systems, Academic Press, New York, 1988.

[5]. D. Dubois and H. Prade, Operations on Fuzzy Numbers, International Journal of SystemsScience, Vol. 9, No. 6, pp. 613- 626,1978.

[6]. Edwards, Dilwyn; Hamson, Mike, Guide to Mathematical Modelling, 2 edition, New York: Industrial Press Inc. ISBN 978-0-8311 3337-5, 2007.

J. J. Buckley and E. Eslami, Fuzzy Trigonometry, An Introduction to Fuzzy Logic andFuzzySets,Vol.13,Physica-Verlag,Heidelberg,pp.185-194,2002.

[7].

[8]. Johnson, N.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions, vol. 2 (2nd ed.), New York: Wiley, 1995.

[9]. L. A. Zadeh, "Fuzzy sets" Archived 2015-08-13 at the Wayback Machine. Information and Control 8 (3) 338

353, 1965.

[10]. Novák, V.; Perfilieva, I.; Moko, J., Mathematical principles of fuzzy logic Dordrecht: Kluwer Academic. ISBN 978-0-7923-8595-0, 1999.

[11]. A. Venkatesh, S. Prakasam, A Mathematical Model for the Effect of Parathyroid Hormone using Fuzzy Log- Normal Distribution, International Journal of Recent Technology and Engineering, Volume-7, Issue-6S (2019), pp.971-973. DOI:

http://www.doi.org/10.35940/ijrte.F0398.0376S19.

[12]. Prakasam. S, Venkatesh.A, Ramesh.R, Seenivasan. M , Sakthivel. K, A Fuzzy Log- Normal Distribution Model for the Rainfall level of Nmakkal District, Communications on Applied Nonlinear Analysis, Vol 31, No.6s (2024), pp.172-178.

DOI: https://doi.org/10.52783/cana.v31.1176

[13]. S. Prakasam, Venkatesh. A, A Fuzzy Mathematical Model for the Parathyroid Hormone Secretion in Health Using LOG Gamma Distribution, Advances in Nonlinear Variational Inequalities, Vol 28, No.1s (2025), pp.470-

478. DOI: https://doi.org/10.52783/anvi.v28.2447

[14]. Prakasam. S, Syamala. P, Venkatesh. A, Exploring the Mathematical Landscape of p-adic Numbers: Unveiling Uniqueness and Ostrowskis Theorem, Advances in Nonlinear Variational Inequalities, Vol 28, No.3s (2025), pp.93-108. DOI: https://doi.org/10.52783/anvi.v28.2852

[15]. R. Praveena, P.Gajendran, L. Jeyanthi, S. Prakasam, Khalil Al-Basyouni, K. Sudarmozhi, Fluid Flow Numerical Study Considering the Impact of Deborah Number and Slip Condition, Advances in Nonlinear Variational Inequalities, Vol 28, No. 5s (2025), pp.315-335. DOI: https://doi.org/10.52783/anvi.v28.3907

Author: Dr. S. Prakasam

Dr. S. Prakasam is working as an Assistant Professor in A.V.V.M Sri Pushpam College (Autonomous). Thanjavur, Tamilnadu. He awarded Ph.D degree in Mathematics from the Bharathidasan University, Trichy, India. He has more than 15 years of teaching and research experience. He has published two International Books in Mathematics area. He has also published many research papers in National and International various UGC, Scopus and peer- reviewed journals. His areas of research are Mathematical modelling and fuzzy sets.