Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

90 https://internationalpubls.com 

Application of the Inverse Weibull Distribution to Agricultural Data 

Based on Intuitionistic Fuzzy Sets 

 

S. Sujatha1*, A. Dinesh Kumar2, R. Sivaraman3 & M. Vasuki4 

1* Research Scholar, Department of Mathematics, Khadir Mohideen College (Affiliated to Bharathidasan University), 

Adhirampattinam, Tamil Nadu, India. Email: suja01rakshi@gmail.com  

2 Assistant Professor, Department of Mathematics, Khadir Mohideen College (Affiliated to Bharathidasan University), 

Adhirampattinam, Tamil Nadu, India. Email: dradineshkumar@gmail.com  

3 Associate Professor, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Chennai, Tamil 

Nadu, India. Email: rsivaraman1729@yahoo.co.in  

4 Assistant Professor, Department of Mathematics, Srinivasan College of Arts and Science (Affiliated to Bharathidasan 

University), Perambalur, Tamil Nadu, India. Email: vasuki.scas@gmail.com  

*Corresponding Author: suja01rakshi@gmail.com 

 

Article History: 

Received: 08-07-2024 

Revised: 21-08-2024 

Accepted: 04-09-2024 

Abstract:  

The inverse Weibull distribution (IWD) is a frequently used model in dependability 

analysis that finds widespread use in a variety of scientific domains. This work examines 

the intuitionistic fuzzy lifespan data-based dependability estimation of the IWD. Prior to 

deriving the ideas of intuitionistic fuzzy conditional expectation, intuitionistic fuzzy 

probability function, and intuitionistic fuzzy conditional density, the associated concepts of 

fuzzy set theory are examined. In conventional estimations, the maximum likelihood 

estimators for reliability and parameters are obtained. The maximum likelihood estimates 

are obtained using the EM algorithm because of the nonlinearity. The gamma prior is 

chosen in the Bayesian estimation process, and the symmetric entropy and scale square 

error loss functions, respectively, are used to estimate the parameters and reliability. The 

Lindley approximation is used to approximate the Bayesian estimates due to the complexity 

of the integrals. According to the simulation results, the maximum likelihood estimate is 

not as appropriate for reliability estimation as the Bayesian estimation. Ultimately, the 

efficacy of these suggested techniques is demonstrated using a collection of agriculture 

production data. These techniques yield an accurate evaluation of the intuitive fuzzy life 

data's reliability, serving as a crucial point of reference for reliability analysis in the 

scientific community. Using intuitionistic fuzzy values for real-time data, the present work 

investigated the Reliability Estimation, cumulative density function, and probability density 

function of the inverse Weibull distribution. We examined Andra agricultural output in 

2019 in this analysis. The comparison thus showed that the estimation of fuzziness values is 

better than the real-time data. 

Keywords: Agriculture data, Inverse Weibull distribution, intuitionistic fuzzy set, 

Reliability Estimation. 

Mathematical Subject Classification (2020): 03E72, 62A86.  

 

 

 

 

mailto:suja01rakshi@gmail.com
mailto:dradineshkumar@gmail.com
mailto:rsivaraman1729@yahoo.co.in
mailto:vasuki.scas@gmail.com
mailto:suja01rakshi@gmail.com


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1. Introduction  

Real-world models can be more accurate when fuzzy sets of numbers are used, especially when it 

comes to uncertainty models like statistical fluctuations in observed lifetime. Voids are another type 

of uncertainty that arises when an observation is not a precise number but rather a fuzzy one. Zadeh 

used the term "fuzzy variable" in 1965 [1] to refer to incorrect linguistic idiom and vernacular. Fuzzy 

set theory got its start with this. A fuzzy set is made up of elements with different membership levels.  

 The Rayleigh lifetime distribution to study the dependability properties of systems, where the 

lifespan parameter is taken to be a generalized intuitionistic fuzzy number, When the systems follow 

a generalised intuitionistic fuzzy Rayleigh lifetime distribution, generalised intuitionistic fuzzy 

dependability, generalised intuitionistic fuzzy hazard function, and generalised intuitionistic fuzzy 

mean time to failure are explored along with their cut sets. With this method, the danger and 

dependability curves for each unique cut set resemble a band with upper and lower bounds. A 

numerical example is provided to demonstrate the suggested methodology. Additional reliability 

study is conducted for both the parallel and series systems [2].  

An intuitionistic fuzzy Weibull lifetime distribution to examine the fuzzy dependability of a few 

systems, Because of data imperfection and uncertainty, fuzzy lifetime parameters are postulated. 

When systems follow intuitionistic fuzzy Weibull lifetime distribution, expressions for fuzzy 

dependability, fuzzy mean time to failure, fuzzy hazard function, and their 𝛼-cut have been 

addressed. To demonstrate the process for calculating the fuzzy reliability characteristics of systems, 

a numerical example is also provided by [3]. 

 

An intuitionistic fuzzy random variable with exact parameters was created and used to assess the 

reliability functions of a k-out-of-n system. Some reliability evaluation criteria were explored and 

interpreted. Numerical assessments were also presented to demonstrate the determination of system 

dependability criteria in the form of intuitionistic fuzzy numbers [4]. Finally, many potential 

engineering applications for the suggested technique were discussed. 

Generalized intuitionistic fuzzy numbers are used to assess the reliability of various systems. The 

reliability features of systems using the Pareto lifespan distribution are explored, under the 

assumption that the lifetime scale parameter is a generalized intuitionistic fuzzy number. In general, 

the cut sets for the generalized intuitionistic fuzzy reliability function, generalised intuitionistic fuzzy 

conditional reliability function, generalized intuitionistic fuzzy hazard function, and generalized 

intuitionistic fuzzy mean time to failure are examined. The reliability functions listed above are 

discussed for generalized intuitionistic fuzzy Pareto lifespan systems. Furthermore, reliability 

analysis of series and parallel systems is performed, and a numerical example is provided [5]. 

The intuitionistic fuzzy sets (IFS) idea, along with the triangle fuzzy number (TFN) and Weibull 

lifetime distribution, is used to determine the same system's fuzzy dependability. In addition, an 

averaging operator with equal weights is applied to a set of three triangular intuitionistic fuzzy 

numbers. A numerical example is solved as an illustration [6]. 



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The simulation results demonstrated [7] that the fuzzy reliability at the estimation stage of the 

Maximum Likelihood Method and the Mixed Thompson Method outperform the other methods in 

terms of Mean Squared Error (MSE), indicating that the use of this type of estimation is 

recommended. The fuzzy values also outperform the real values for all sample sizes. 

A new technique for analysing fuzzy system reliability based on intuitionistic trapezoidal fuzzy set 

theory is described, in which the reliabilities of system components are represented by intuitionistic 

trapezoidal fuzzy numbers to simulate uncertainty and vagueness in real-world circumstances [8]. 

The suggested method may describe and analyse fuzzy system reliability in a more adaptable and 

intelligent way. The suggested method also compares different complex systems using the score and 

accuracy functions. The score function can assist the decision maker in making his decision more 

efficiently in a decision-making dilemma. 

This method is then applied to a wireless communication system to evaluate its intuitionistic fuzzy 

reliability and availability, with each performance state and probability represented by a TIFN. The 

acquired results are also shown graphically for better understanding. This work contributes to a 

better understanding of FMSS, making them more dependable [9]. 

The foundation of reliability engineering is reliability theory. Reliability analysis is vital for 

developing several system improvement alternatives during the design, configuration, and tuning 

stages, all of which are required for a complex system to operate efficiently. This work focuses on 

estimating the reliability of a degradable system with imperfect coverage and unknown information 

about its components[10]. The Weibull intuitionistic fuzzy set (WIFS) idea was utilised to address 

data uncertainty. The trapezoidal intuitionistic fuzzy number (TrIFN) and its arithmetic operations 

are discussed. Trapezoidal intuitionistic fuzzy numbers (TrIFN) are used to represent the failure rate 

of the system. 

Simulation results indicate that Bayesian estimation outperforms maximum likelihood estimation for 

estimating dependability. The proposed approaches are tested against real data to demonstrate their 

usefulness. These approaches reliably evaluate the reliability of intuitive fuzzy life data, serving as a 

valuable reference for reliability analysis in science [11]. 

The current work used a fuzzy intuitionistic fuzzy set to analyze the inverse Weibull distribution's 

probability density function, cumulative density function, and Reliability Estimation with real-time 

data. This study focused on Andra's agriculture in 2019. 

2. Fuzzy Mathematical Approach. 

The probability density function (pdf), cumulative distribution function (cdf), and reliability function 

of IWD are defined as follows: 

𝑦(𝑡; 𝜆, 𝜂) = 𝜆𝜂𝑡−𝜂−1 exp(−𝜆𝑡−𝜂) , 𝑡 > 0

𝑌(𝑡; 𝜆, 𝜂) = exp(−𝜆𝑡−𝜂) , 𝑡 > 0

𝑅(𝑡) = 1 − exp(−𝜆𝑡−𝜂) , 𝑡 > 0

 

where 𝜆 > 0 is the scale parameter and 𝜂 > 0 is the shape parameter, 

 



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 A fuzzy set expresses the concept of fuzziness. Similarly to the 

definition of the characteristic function of a classical set, the definition of a fuzzy set can be achieved 

by expanding its domain. 

Definition 2.1 

        Let T represent a non-empty universal set. The fuzzy set �̃� is defined as �̃� = {< 𝑡, 𝜇�̃�(𝑡)⟩ ∣ 𝑡 ∈

𝕋}, where 𝜇�̃�: 𝕋 → [0,1]] represents the degree of membership of t in �̃�. . 

Definition 2.2  

Let T represent a non-empty universal set. IFS A ̃ is defined as �̃� = {⟨𝑡, 𝜇�̃�(𝑡), 𝑣�̃�(𝑡) >|𝑡 ∈ 𝕋}, 

where 𝜇�̃�: 𝕋 → [0,1] is the degree of membership of t in �̃�  and 𝑣�̃�: 𝕋 → [0,1] is the degree of non-

membership of t in �̃�. They satisfy 0 ≤ 𝜇�̃�(𝑡) + 𝑣�̃�(𝑡) ≤ 1 for each t. When T has only one element, 

�̃� =< 𝜇�̃�, 𝑣�̃� >  is frequently referred to as intuitionistic fuzzy number. 

The membership and non-membership functions are: 

𝜇Ã(𝑡) =

{
 
 

 
 𝛼

𝑡 − 𝑎

𝑏 − 𝑎
𝑡 ∈ [𝑎, 𝑏]

𝛼 𝑡 ∈ (𝑏, 𝑐)

𝛼
𝑑 − 𝑡

𝑑 − 𝑐
𝑡 ∈ [𝑐, 𝑑]

0  else 

𝑣Ã(𝑡) =

{
 
 

 
 
𝑏 − 𝑡

𝑏 − 𝑎
+ 𝛽

𝑡 − 𝑎

𝑏 − 𝑎
𝑡 ∈ [𝑎, 𝑏]

𝛽 𝑡this ∈ (𝑏, 𝑐)

𝑡 − 𝑐

𝑑 − 𝑐
+ 𝛽

𝑑 − 𝑡

𝑑 − 𝑐
𝑡 ∈ [𝑐, 𝑑]

1  else 

 

where α is the maximum membership degree and β is the minimum membership degree. 

In paper, we assume T be a set of real numbers, which is T=R. Additionally, we assumed that the 

IFSs discussed in this paper were TraIFNs. To better investigate the estimation problem on the basis 

of intuitionistic fuzzy data, some concepts in the probability theory were extended to intuitionistic 

fuzzy random variables. 

Definition 2.3 

Consider a probability space (ℝ𝑛, 𝔄, 𝒫), The probability of an intuitionistic fuzzy observation �̃� in 

ℝ𝑛 is defined as 

𝑃(�̃�) = ∫  
ℝ𝑛
 
1 − 𝑣�̃�(𝑡) + 𝜇�̃�(𝑡)

2
𝑑𝒫  

The continuous random variable 𝑇 = (𝑇1, 𝑇2, … , 𝑇𝑛) follows the IW (𝜆, 𝜂),  and its intuitionistic 

fuzzy observations are denoted by �̃� = (�̃�1, �̃�2, … , �̃�𝑛). The conditional density of random variables 

in probability theory is introduced, and the intuitionistic fuzzy conditional density is given as 

follows: 



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𝑦(𝑡 ∣ �̃�) =
𝑠(𝑡)𝑦(𝑡; 𝜆, 𝜂)

∫  
ℝ
 𝑠(𝑡)𝑦(𝑡; 𝜆, 𝜂)𝑑𝑡

 

where 𝑠(𝑡) =
1−𝑣�̃�(𝑡)+𝜇�̃�(𝑡)

2
. In this case, the intuitionistic fuzzy likelihood function of IW⁡(𝜆, 𝜂) is: 

ℎ(𝜆, 𝜂 ∣ �̃�) =∏ 

𝑛

𝑖=1

 𝑃(�̃�𝑖 ∣ 𝜆, 𝜂) =∏  

𝑛

𝑖=1

 ∫  
ℝ

  𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)𝑑𝑡  

where 𝑠𝑖(𝑡) =
1−𝑣�̃�𝑖(𝑡)+𝜇�̃�𝑖(𝑡)

2
. 

Finally, intuitionistic fuzzy conditional expectation is defined. Using the intuitionistic fuzzy 

conditional density and observation �̃� = (�̃�1, �̃�2, … , �̃�𝑛), the intuitionistic fuzzy conditional 

expectation of a random variable 𝑇 = (𝑇1, 𝑇2, … , 𝑇𝑛) is: 

 

𝐸(𝑇 ∣ �̃�)= ∫  
ℝ

 𝑡𝑦(𝑡 ∣ �̃�)𝑑𝑡

= ∫  
ℝ

 𝑡
𝑠(𝑡)𝑦(𝑡; 𝜆, 𝜂)

ℎ(𝜆, 𝜂 ∣ �̃�)
𝑑𝑡

 

2.4 Maximum Likelihood Estimation 

The above Equation shows the intuitionistic fuzzy likelihood function of IW (𝜆, 𝜂) Thus, the 

intuitionistic fuzzy log-likelihood function is given as follows: 

𝐻(𝜆, 𝜂 ∣ �̃�) = ln ℎ(𝜆, 𝜂 ∣ �̃�) =∑  

𝑛

𝑖=1

 ln [∫  
ℝ

  𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)𝑑𝑡]  

The MLEs �̂�𝑀𝐿 and �̂�𝑀𝐿 are obtained by the below equations: 

{
 

 
∂𝐻(𝜆, 𝜂 ∣ �̃�)

∂𝜆
= 0

∂𝐻(𝜆, 𝜂 ∣ �̃�)

∂𝜂
= 0

 

where 
∂𝐻(𝜆,𝜂∣�̃�)

∂𝜆
 and 

∂𝐻(𝜆,𝜂∣�̃�)

∂𝜂
 are shown in below 

∂𝐻(𝜆, 𝜂 ∣ �̃�)

∂𝜆
=∑  

𝑛

𝑖=1

 
1

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)
∫  
ℝ

 𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜆
𝑑𝑡

∂𝐻(𝜆, 𝜂 ∣ �̃�)

∂𝜂
=∑  

𝑛

𝑖=1

 
1

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)
∫  
ℝ

 𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜂
𝑑𝑡

 

Here, ℎ(𝜆, 𝜂 ∣ �̃�𝑖) = ∫0
+∞
 𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)𝑑𝑡,

∂𝑦(𝑡;𝜆,𝜂)

∂𝜆
 and 

∂𝑦(𝑡;𝜆,𝜂)

∂𝜂
 are shown in below: 



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∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜆
=
1

𝜆
𝑦(𝑡; 𝜆, 𝜂) − 𝑡−𝜂𝑦(𝑡; 𝜆, 𝜂)

∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜂
=
1

𝜂
𝑦(𝑡; 𝜆, 𝜂) − 𝑦(𝑡; 𝜆, 𝜂) ln 𝑡 + 𝜆𝑡−𝜂𝑦(𝑡; 𝜆, 𝜂) ln 𝑡

 

It is evident that the equations above are nonlinear and difficult to solve. Then we considered the EM 

algorithm. 

The EM technique is equally applicable to intuitionistic fuzzy data because the observed 

intuitionistic fuzzy data can be viewed as imperfect characterizations of the completed data. To 

better illustrate the iterative process of the EM method,  

From above equation we get, 

∂𝐻(𝜆, 𝜂 ∣ �̃�)

∂𝜆
=∑  

𝑛

𝑖=1

 
1

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅
∫  
+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜆
𝑑𝑡

=∑  

𝑛

𝑖=1

 
1

ℎ‾(𝜆, 𝜂 ∣ �̃�𝑖)
∫  
+∞

0

  𝑠𝑖(𝑡) [
1

𝜆
𝑦(𝑡; 𝜆, 𝜂) − 𝑡−𝜂𝑦(𝑡; 𝜆, 𝜂)] 𝑑𝑡

=∑  

𝑛

𝑖=1

 ∫  
+∞

0

 
1

𝜆

𝑠𝑖(𝑡)𝑦(𝑡, 𝜆, 𝜂)

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)
𝑑𝑡 −∑  

𝑛

𝑖=1

 ∫  
+∞

0

  𝑡−𝜂
𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)
𝑑𝑡

= 𝑛
1

𝜆
−∑  

𝑛

𝑖=1

 𝐸1𝑖.

 

 Let 
∂𝐻(𝜆, 𝜂 ∣ �̃�)

∂𝜆
= 0, 

where 

𝐸1𝑖 = 𝐸(𝑇
−𝜂 ∣ �̃�𝑖) = ∫  

+∞

0

𝑡−𝜂
𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)
𝑑𝑡 

∂𝐻(𝜆, 𝜂 ∣ �̃�)

∂𝜂
=∑  

𝑛

𝑖=1

 
1

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅
∫  
+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜂
𝑑𝑡

=𝑛
1

𝜂
−∑  

𝑛

𝑖=1

 𝐸2𝑖 + 𝜆∑  

𝑛

𝑖=1

 𝐸3𝑖

 

Let 
∂𝐻(𝜆, 𝜂 ∣ �̃�)

∂𝜂
= 0, 

where 

𝐸2𝑖 = 𝐸(ln 𝑇 ∣ �̃�𝑖) = ∫  
+∞

0

𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)
(ln 𝑡)𝑑𝑡 

and 



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𝐸3𝑖 = 𝐸(𝑇−𝜂ln 𝑇 ∣ �̃�𝑖) = ∫  
+∞

0

𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)

ℎ(𝜆, 𝜂 ∣ �̃�𝑖)
(𝑡−𝜂ln 𝑡)𝑑𝑡 

The iterative processes for obtaining MLEs with the EM method are as follows: 

Step 1: Let the initial value be 𝜃(0) = (𝜆(0), 𝜂(0)), and set 𝑗 = 0. Give the accuracy ε > 0.  

Step 2: At the (𝑗 + 1)th iteration, compute the intuitionistic fuzzy conditional expectations shown 

below.  

𝐸1𝑖 = ∫ 𝑡−𝜂
𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)

ℎ( 𝜆, 𝜂 ∣∣ �̃�𝑖 )

+∞

0

  |

𝜃(𝑗+1)=𝜃(𝑗)

𝑑𝑡,

𝐸2𝑖 = ∫
𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)

ℎ( 𝜆, 𝜂 ∣∣ �̃�𝑖 )(ln 𝑡)

+∞

0

  |

𝜃(𝑗+1)=𝜃(𝑗)

𝑑𝑡,

𝐸3𝑖 = ∫
𝑠𝑖(𝑡)𝑦(𝑡; 𝜆, 𝜂)

ℎ( 𝜆, 𝜂 ∣∣ �̃�𝑖 )
(𝑡−𝜂 ln 𝑡)

+∞

0

  |
𝜃(𝑗+1)&=𝜃(𝑗)

𝑑𝑡.

 

Step 3.  

𝜆(𝑗+1) = 𝑛(∑  

𝑛

𝑖=1

 𝐸1𝑖)

−1

 

From above equations we get,  

𝜂(𝑗+1) = 𝑛(∑  

𝑛

𝑖=1

 𝐸2𝑖 − 𝜆
(𝑗)∑ 

𝑛

𝑖=1

 𝐸3𝑖)

−1

 

Step 4: If |𝜃(𝑗+1) − 𝜃(𝑗)| < 𝜀, the MLEs are produced by �̂�𝑀𝐿 = 𝜆
(𝑗) and �̂�𝑀𝐿 = 𝜂

(𝑗). If not, set 𝑗 =

𝑗 + 1 and go back to step 2. 

The MLE⁡ MLE �̂�𝑀𝐿(𝑡)can be calculated using maximum likelihood estimation invariance as 

follows: 

�̂�𝑀𝐿(𝑡) = 1 − exp (−�̂�𝑀𝐿𝑡
−�̂�𝑀𝐿)  

2.5 Bayesian Estimation 

In Bayesian statistical inference, the prior distribution is important. It indicates our prior knowledge 

or assumption about the parameters and can be used to better precisely predict the posterior 

distribution .Choosing an appropriate prior distribution is critical since it influences the final 

inference results. 

The gamma distribution is a versatile continuous probability distribution with numerous desirable 

qualities, making it a popular candidate for the prior distribution of parameters in Bayesian statistics.  

The parameters of the gamma distribution can be changed to fit various prior beliefs. Furthermore, 

the gamma distribution is conjugate, which means that when employed as a prior distribution, the 



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product with the likelihood function remains a gamma distribution; simplifying posterior distribution 

computations .The pdf for the gamma distribution is [11] 

𝜋(𝜔) =
𝑏𝑎

Γ(𝑎)
𝜔𝑎−1𝑒−𝑏𝜔, 𝜔 > 0, 𝑎, 𝑏 > 0 

In this section, we assume that λ and η are independent random variables. λ follows Gamma (𝑐1, 𝑑1) 

and η follows Gamma (𝑐2, 𝑑2). That means: 

𝜋1(𝜆) ∝ 𝜆
𝑑1−1𝑒−𝑐1𝜆𝜆 > 0, 𝑐1 > 0, 𝑑1 > 0

𝜋2(𝜂) ∝ 𝜂
𝑑2−1𝑒−𝑐2𝜂𝜂 > 0, 𝑐2 > 0, 𝑑2 > 0.

 

Thus, the joint prior distribution of 𝜆 and 𝜂 is: 

𝜋(𝜆, 𝜂) = 𝜋1(𝜆) × 𝜋2(𝜂) ∝ 𝜆
𝑑1−1𝜂𝑑2−1𝑒−𝑐1𝜆−𝑐2𝜂  

With reference to the Bayesian formulation, the posterior distribution of 𝜆 and 𝜂 is 

𝜋(𝜆, 𝜂 ∣ �̃�)∝ ℎ(𝜆, 𝜂 ∣ �̃�) × 𝜋(𝜆, 𝜂) 

 The posterior expectation of the function 𝑔(𝜆, 𝜂) of 𝜆 and 𝜂 is: 

𝐸[𝑔(𝜆, 𝜂) ∣ �̃�]= ∫  
+∞

0

 ∫  
+∞

0

 𝑔(𝜆, 𝜂)
𝜋(𝜆, 𝜂 ∣ �̃�)

∫  
+∞

0
 ∫  
+∞

0
 𝜋(𝜆, 𝜂 ∣ �̃�)𝑑𝜆𝑑𝜂

𝑑𝜆𝑑𝜂 

𝐸[𝑔(𝜆, 𝜂) ∣ �̃�] =
∫𝑔(𝜆, 𝜂)𝑒𝐻(𝜆,𝜂∣�̃�)+𝐺(𝜆,𝜂)𝑑(𝜆, 𝜂)

∫ 𝑒𝐻(𝜆,𝜂∣�̃�)+𝐺(𝜆,𝜂)𝑑(𝜆, 𝜂)
 

where 𝐺(𝜆, 𝜂) = ln 𝜋(𝜆, 𝜂).  

𝐸[𝑔(𝜆, 𝜂) ∣∣ �̃� ] = 𝑔(𝜆 ˆ𝑀𝐿 , 𝜂 ˆ𝑀𝐿) +
1

2(𝐴 + 𝐵 + 𝐶 + 𝐷)

𝐴 = (𝑔 ˆ𝜆𝜆 + 2𝑔 ˆ𝜆𝐺 ˆ𝜆)𝜙 ˆ𝜆𝜆 + (𝑔 ˆ𝜂𝜆 + 2𝑔 ˆ𝜂𝐺 ˆ𝜆)𝜙 ˆ𝜂𝜆

𝐵 = (𝑔 ˆ𝜆𝜂 + 2𝑔 ˆ𝜆𝐺 ˆ𝜂)𝜙 ˆ𝜆𝜂 + (𝑔 ˆ𝜂𝜂 + 2𝑔 ˆ𝜂𝐺 ˆ𝜂)𝜙 ˆ𝜂𝜂

𝐶 = (𝑔 ˆ𝜆𝜙 ˆ𝜆𝜆 + 𝑔 ˆ𝜂𝜙 ˆ𝜆𝜂)(𝐻 ˆ𝜆𝜆𝜆𝜙 ˆ𝜆𝜆 + 𝐻 ˆ𝜂𝜆𝜆𝜙 ˆ𝜂𝜆 + 𝐻 ˆ𝜆𝜂𝜆𝜙 ˆ𝜆𝜂 + 𝐻 ˆ𝜂𝜂𝜆𝜙 ˆ𝜂𝜂)

𝐷 = (𝑔 ˆ𝜆𝜙 ˆ𝜂𝜆 + 𝑔 ˆ𝜂𝜙 ˆ𝜂𝜂)(𝐻 ˆ𝜆𝜆𝜂𝜙 ˆ𝜆𝜆 +𝐻 ˆ𝜂𝜆𝜂𝜙 ˆ𝜂𝜆 + 𝐻 ˆ𝜆𝜂𝜂𝜙 ˆ𝜆𝜂 + 𝐻 ˆ𝜂𝜂𝜂𝜙 ˆ𝜂𝜂)

 

The element 𝜙𝑖𝑗(𝑖, 𝑗 = 𝜆, 𝜂)represents the inverse matrix of −𝐻𝑖𝑗.. The symbol �̂�𝜆𝜆 signifies taking 

the second derivative of 𝑔(𝜆, 𝜂) with regard to λ and putting �̂�𝑀𝐿 into it. Similarly, the rest can be 

illustrated as follows: 

𝐻𝜆𝜆𝜆 =∑  

𝑛

𝑖=1

  [2ℎ−3(𝜆, 𝜂 ∣ �̃�𝑖) (∫  
+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜆
𝑑𝑡)

3

− ℎ−1(𝜆, 𝜂 ∣ �̃�𝑖)∫  
+∞

0

 𝑠𝑖(𝑡)
∂3𝑦(𝑡; 𝜆, 𝜂)

∂𝜆3
𝑑𝑡] 

𝐻𝜆𝜆𝜂 = 𝐻𝜆𝜂𝜆 = 𝐻𝜂𝜆𝜆 



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=∑  

𝑛

𝑖=1

  [2ℎ−3(𝜆, 𝜂 ∣ �̃�𝑖)∫  
+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜂
𝑑𝑡 (∫  

+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜆
𝑑𝑡)

2

] 

−∑  

𝑛

𝑖=1

  [ℎ−2(𝜆, 𝜂 ∣ �̃�𝑖)∫  
+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜂
𝑑𝑡∫  

+∞

0

  𝑠𝑖(𝑡)
∂2𝑦(𝑡; 𝜆, 𝜂)

∂𝜆2
𝑑𝑡] 

+∑  

𝑛

𝑖=1

 ℎ−1(𝜆, 𝜂 ∣ �̃�𝑖)∫  
+∞

0

  𝑠𝑖(𝑡)
∂3𝑦(𝑡; 𝜆, 𝜂)

∂𝜆2 ∂𝜂
𝑑𝑡 

𝐻𝜂𝜂𝜆 = 𝐻𝜂𝜆𝜂 = 𝐻𝜆𝜂𝜂 

=∑  

𝑛

𝑖=1

  [2ℎ−3(𝜆, 𝜂 ∣ �̃�𝑖)∫  
+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜆
𝑑𝑡 (∫  

+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜂
𝑑𝑡)

2

] 

−∑  

𝑛

𝑖=1

  [ℎ−2(𝜆, 𝜂 ∣ �̃�𝑖)∫  
+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜆
𝑑𝑡∫  

+∞

0

  𝑠𝑖(𝑡)
∂2𝑦(𝑡; 𝜆, 𝜂)

∂𝜂2
𝑑𝑡] 

+∑  

𝑛

𝑖=1

  [ℎ−1(𝜆, 𝜂 ∣ �̃�𝑖)∫  
+∞

0

  𝑠𝑖(𝑡)
∂3𝑦(𝑡; 𝜆, 𝜂)

∂𝜂2 ∂𝜆
𝑑𝑡] 

𝐻𝜂𝜂𝜂 =∑  

𝑛

𝑖=1

  [2ℎ−3(𝜆, 𝜂 ∣ �̃�𝑖) (∫  
+∞

0

  𝑠𝑖(𝑡)
∂𝑦(𝑡; 𝜆, 𝜂)

∂𝜂
𝑑𝑡)

3

+ ℎ−1(𝜆, 𝜂 ∣ �̃�𝑖)∫  
+∞

0

  𝑠𝑖(𝑡)
∂3𝑦(𝑡, 𝜆, 𝜂)

∂𝜂3
𝑑𝑡] 

The loss function plays an important role in Bayesian statistical inference since it evaluates the 

difference between model predictions and actual outcomes. In the Bayesian framework, we used the 

posterior distribution to represent uncertainty and the loss function to select the best decision or 

prediction. Different loss functions produce different conclusions or predictions; thus, choosing an 

appropriate loss function is critical to the precision and reliability of Bayesian inference. Then, we 

investigate Bayesian estimate of unknown parameters using the SE and SSE loss functions. 

2.6 Bayesian Estimation under the SE Loss Function 

The SE loss function is defined as follows, 

𝐿1(𝜃, �̂�) =
�̂�

𝜃
+
𝜃

�̂�
− 2  

where �̂� is the estimator of unknown parameter 𝜃. 

3. Real Data Analysis  

In this application, we used real-time agriculture data from Kaggle.com [12] to analyze the Andra 

agriculture data set in 2019. We evaluated the data using an inverse Weibull distribution with 

intuitionistic fuzzy data, as shown in Tables 1 and 2. Mean Square Error for each crop 1.0551, 

0.0419, 0.2632, 28.5692, 0.0126, 0.0507, 0.0915, 0.0125, 1.0093, 0.0397 and Bayesian Estimates 

and Reliability (MSE) for each crop 0.0172, 0.0173, 0.0129, 0.0241, 0.0075, 0.0212, 0.0139, 0.0151, 



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0.0372, and 0.0073 respectively, Estimated fuzzy parameters for the 𝜆 are 2.5554, 2.8394 and 3.1233 

for the 𝜂 are 3.6139, 4.0155 and 4.4170. 

Table.1. Agriculture data of Andra in the year of 2019. 

Crop Season Membership Degree 

(μ\muμ) 

Non-Membership 

Degree (ν\nuν) 

Degree of 

Uncertainty (π\piπ) 

Arecanut Kharif 0.70 0.20 0.10 

Arhar/Tur Kharif 0.80 0.10 0.10 

Arhar/Tur Rabi 0.60 0.30 0.10 

Bajra Kharif 0.65 0.25 0.10 

Bajra Rabi 0.55 0.35 0.10 

Banana Whole 

Year 

0.90 0.05 0.05 

Black 

Pepper 

Whole 

Year 

0.85 0.10 0.05 

Cashewnut Kharif 0.75 0.20 0.05 

Castor Seed Kharif 0.60 0.30 0.10 

Castor Seed Rabi 0.50 0.40 0.10 

Table.2. intuitionistic fuzzy data values of Andra in the year of 2019 . 

Maximum Likelihood Estimator (MLE) Table 𝜆 = 4.3956 and 𝜂 = 7.3543. 

S.No Observed Yield(kg/ha) Predicted Yield(kg/ha) 

1 5.00 5.00 

2 4.67 4.67 

3 4.00 4.00 

4 3.60 3.60 

5 3.33 3.33 

 

 

 

Crop Season Area(ha) (MT) (kg/ha) Value(₹) (₹/kg) Crop 

Arecanut Kharif 1096 10418 899.2 188249 405.52 7.253333 

Arhar/Tur Kharif 237647 114451 899.2 40818249 87929.39 0.44 

Arhar/Tur Rabi 5940 3747 899.2 1020254 2197.8 0.43 

Bajra Kharif 20484 47045 899.2 3518332 7579.08 2.004545 

Bajra Rabi 4592 11322 899.2 788721.9 1699.04 2.48 

Banana Whole Year 97695 5861700 899.2 16780093 36147.15 60 

Black pepper Whole Year 17645 17645 899.2 3030705 6528.65 1 

Cashewnut Kharif 115785 115785 899.2 19887232 42840.45 1 

Castor seed Kharif 36827 25270 899.2 6325406 13625.99 0.7625 

Castor seed Rabi 1056 637 899.2 181378.6 390.72 0.684 



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Mathematical Results of the inverse Weibull distribution. 

      
Fig. 1: PDF and Empirical PDF of  inverse Weibull distribution using intuitionistic fuzzy set. 

         
Fig. 2: CDF and Empirical CDF of  inverse Weibull distribution using intuitionistic fuzzy set. 

 
Fig.3 : Reliability Estimation of inverse Weibull distribution with intuitionistic fuzzy set. 

4. Results and Discussion 

Fuzzy lifetime data can be used to estimate the reliability of IWD in real-life scenarios due to 

uncontrollable factors. However, fuzzy sets only have one membership degree parameter, resulting 

in a less precise description of the objective world. In contrast, intuitionistic fuzzy sets can better 

express uncertainty and fuzziness when dealing with fuzzy information. 

The EM algorithm was used to obtain MLEs, followed by the Lindley approximation to obtain BEs 

under SE and SSE loss functions. Results showed that changing the true values of multiple sets of 



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parameters resulted in a significantly smaller mean square error under Bayesian estimation compared 

to the maximum likelihood estimator.   

Kaggle.com supports the current research findings [11]. We discovered results for the Probability 

density function as well as PDF and Empirical PDF (Fig.1), Cumulative density function as well as 

CDF and Empirical CDF (Fig.2), and Reliability Estimation (Fig.3) in the Inverse Weibull 

distribution intuitionistic fuzzy set to the agriculture problem of Andra in the year of 2019 with fuzzy 

parameters and supporting MATLAB tool, which shows how well the suggested technique performs 

in predicting crop production for all seasons. 

5. Conclusion  

In this study, the Inverse Weibull distribution model is applied to real-time data in a fuzzy 

environment. This value is generated by combining the inverse Weibull distribution with the 

intuitionistic fuzzy set, and the resulting curve is used to assess the performance of the given data. 

The calculated Mean Squared Error (MSE), Maximum Likelihood Estimator (MLE)   and Fuzzy 

operating characteristic curves for several functions are examined. This includes the probability 

density function, empirical PDF, cumulative density function, empirical CDF, and reliability 

estimation for the inverse Weibull distribution. This study determined that the proposed approach 

can be implemented if the product's quality is uncertain. The study discovered that the strategy is 

quite efficient in unpredictable agriculture and situations. 

References 

[1]   Zadeh, Lotfi A. "Fuzzy sets." Information and control 8, no. 3 (1965): 338-353. 

[2]  Kumar, Pawan, and S. B. Singh. "Fuzzy system reliability using intuitionistic fuzzy Weibull lifetime distribution." 

International Journal of Reliability and Applications 16, no. 1 (2015): 15-26. 

 [3]  Ebrahimnejad, Ali, and Ezzatallah B. Jamkhaneh. "System reliability using generalized intuitionistic fuzzy Rayleigh 

lifetime distribution." Applications and Applied Mathematics: An International Journal (AAM) 13, no. 1 (2018): 7. 

[4] Akbari, Mohammad Ghasem, and Gholamreza Hesamian. "Time-dependent intuitionistic fuzzy system reliability 

analysis." Soft Computing 24 (2020): 14441-14448. 

[5] Roohanizadeh, Zahra, Ezzatallah Baloui Jamkhaneh, and Einolah Deiri. "A novel approach for analyzing system 

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(2021). 

[6]  Bhandari, Ashok Singh, Akshay Kumar, and Mangey Ram. "Fuzzy reliability evaluation of complex systems using 

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[7]  Taha, Taha Anwar, and Abbas Najim Salman. "Comparison Different Estimation Method for Reliability Function 

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[8] Bhadauria, Jaya, Sudha Rana, Anita Kumari, and Deepak Kumar. "Reliability analysis of different complex systems 

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[9] Chachra, Aayushi, Akshay Kumar, and Mangey Ram. "Intuitionistic fuzzy approach to reliability assessment of 

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[10] Patrai, Kalika, and Surbhi Gupta. "Reliability Estimation of a Degradable System using Intuitionistic Fuzzy Weibull 

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[11] Hu, Xue, and Haiping Ren. "Reliability Estimation of Inverse Weibull Distribution Based on Intuitionistic Fuzzy 

Lifetime Data." Axioms 12, no. 9 (2023): 838. 

[12] https://www.kaggle.com/datasets/akshatgupta7/crop-yield-in-indian-states-dataset