Jaya algorithm in estimation of P[X &gt; Y] for two parameter Weibull distribution

Saurabh L. Raikar; Dr. Rajesh S. Prabhu Gaonkar; Saurabh L. Raikar; Dr. Rajesh S. Prabhu Gaonkar

doi:10.3934/math.2022156

AIMS Mathematics

2022, Volume 7, Issue 2: 2820-2839. doi: 10.3934/math.2022156

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Research article Special Issues

Jaya algorithm in estimation of P[X > Y] for two parameter Weibull distribution

Saurabh L. Raikar ^{1
,
,},
Dr. Rajesh S. Prabhu Gaonkar ²

1.
Mechanical Engineering Department, Goa College of Engineering (affiliated to Goa University), Farmagudi, Ponda, Goa 403401, India
2.
Indian Institute of Technology Goa (IIT Goa), Farmagudi, Ponda, Goa 403401, India

Received: 15 September 2021 Accepted: 04 November 2021 Published: 19 November 2021
MSC : 62F10, 37N40

Jaya algorithm is a highly effective recent metaheuristic technique. This article presents a simple, precise, and faster method to estimate stress strength reliability for a two-parameter, Weibull distribution with common scale parameters but different shape parameters. The three most widely used estimation methods, namely the maximum likelihood estimation, least squares, and weighted least squares have been used, and their comparative analysis in estimating reliability has been presented. The simulation studies are carried out with different parameters and sample sizes to validate the proposed methodology. The technique is also applied to real-life data to demonstrate its implementation. The results show that the proposed methodology's reliability estimates are close to the actual values and proceeds closer as the sample size increases for all estimation methods. Jaya algorithm with maximum likelihood estimation outperforms the other methods regarding the bias and mean squared error.

Keywords:

Citation: Saurabh L. Raikar, Dr. Rajesh S. Prabhu Gaonkar. Jaya algorithm in estimation of P[X > Y] for two parameter Weibull distribution[J]. AIMS Mathematics, 2022, 7(2): 2820-2839. doi: 10.3934/math.2022156

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Abstract

1. Introduction

Reliability of the form P[X > Y] is used in cases of stress strength interference ^[1]. Stress and strength are important properties of a material. These properties do not have a fixed single value because of the uncertainties present in the environment like temperature, humidity, etc. So, they can be considered to follow a certain distribution. According to the interference theory, if stress and strength follow a certain distribution, then their interference area gives the probability of failure. The concept of stress strength interference in evaluating reliability has been used by many researchers in their studies. Liu et al. ^[2] evaluated the reliability of automotive seat adjuster by using the stress strength interference model. The finite element model of the seat-adjuster was constructed and the analysis was verified with the bench test. The theory has also been used in medical applications by Miller and Freivalds ^[3] to obtain the probability of failure of tendons in carpel tunnel syndrome.

Weibull distribution has been widely used by researchers in their study as it is capable of fitting large data types ^[4,5,6]. If x and y are the random variables following Weibull distribution W( $\sigma, {p}_{1}$ ) and W( $\sigma, {p}_{2}$ ) respectively then their pdf can be given as:

$f\left(x;\sigma , {p}_{1}\right) = \frac{{p}_{1}}{{\sigma }^{{p}_{1}}}{\left(x\right)}^{{p}_{1}-1}\mathrm{exp}\left\{-{\left(\frac{x}{\sigma }\right)}^{{p}_{1}}\right\}, x > 0, \sigma > 0, {p}_{1} > 0$

(1)

And

$f\left(y;\sigma , {p}_{2}\right) = \frac{{p}_{2}}{{\sigma }^{{p}_{2}}}{\left(y\right)}^{{p}_{2}-1}\mathrm{exp}\left\{-{\left(\frac{y}{\sigma }\right)}^{{p}_{2}}\right\}, y > 0, \sigma > 0, {p}_{2} > 0$

(2)

respectively. The corresponding cumulative distribution function (cdf) for strength and stress is given by

$F\left(x;\sigma , {p}_{1}\right) = 1-\mathrm{exp}\left\{-{\left(\frac{x}{\sigma }\right)}^{{p}_{1}}\right\}$

(3)

And

$F\left(y;\sigma , {p}_{2}\right) = 1-\mathrm{exp}\left\{-{\left(\frac{y}{\sigma }\right)}^{{p}_{2}}\right\}$

(4)

where ${p}_{1}$ & ${p}_{2}$ are shape parameters for strength and stress respectively and $\sigma$ is the common scale parameter. It is difficult to evaluate the stress strength reliability model when both parameters of Weibull distribution are different. The purpose of the present study is to show the effectiveness of Jaya algorithm in the estimation of reliability for Weibull distribution. Hence, it has been assumed that the scale parameter for stress and strength distribution remains the same. Stress strength Weibull distribution with common scale parameter has been used in estimating the reliability for strength of carbon fibers ^[7].

2. Estimation methods

In this study, some of the most widely used estimation methods are implemented namely maximum likelihood estimation, least squares estimation, and weighted least squares estimation. Louzada et al. ^[8] used these methods in estimating the parameters of extended exponential geometric distribution for medical data. Datsiou and Overend ^[9] presented a comparison of various methods including MLE and LSE in the estimation of parameters for Weibull distribution applied to a data of strength of glass fibers by evaluating the fitness of the parameters using Anderson Darling goodness of fit test. The above estimation methods are simple and easy to evaluate.

2.1. Maximum likelihood estimation (MLE)

Maximum likelihood estimation is one of the common and effective methods in the estimation of parameters ^[10,11,12]. Chacko and Mohan ^[13] used the MLE method in estimating the parameters of two-parameter Kumaraswamy-exponential distribution for progressive type-Ⅱ censored samples. Tzavelas ^[14] proposed estimation of parameters of three-parameter gamma distribution using MLE via reparameterization of function and predictor-corrector method. MLE method has also been used by Aggarwala and Balakrishnan ^[15] in the estimation of scale and location parameters of Laplace distribution. Ng et al. ^[16] discussed estimating the parameters of three-parameter Weibull distribution for type Ⅱ progressively censored samples using MLE and weighted MLE. Abushal ^[17] applied MLE technique to estimate the unknown parameters and reliability characteristics for Akash distribution.

Let x₁, x₂, x₃ …x_n be a random sample of size n drawn from W( $\sigma, {p}_{1}$ ) and y₁, y₂, y₃, …., y_n be the random sample of size m from W( $\sigma, {p}_{2}$ ). Then the likelihood function can be given as:

$L = \prod\limits _{i = 1}^{n}f\left({x}_{i}\right)\prod \limits_{j = 1}^{m}f\left({y}_{j}\right)$

(5)

$L = \prod \limits_{i = 1}^{n}\frac{{p}_{1}}{{\sigma }^{{p}_{1}}}{\left({x}_{i}\right)}^{{p}_{1}-1}\mathrm{exp}\left\{-{\left(\frac{{x}_{i}}{\sigma }\right)}^{{p}_{1}}\right\}.\prod\limits _{j = 1}^{m}\frac{{p}_{2}}{{\sigma }^{{p}_{2}}}{\left({y}_{j}\right)}^{{p}_{2}-1}\mathrm{exp}\left\{-{\left(\frac{{y}_{j}}{\sigma }\right)}^{{p}_{2}}\right\}$

(6)

$\mathrm{ln}L = n\mathrm{ln}{p}_{1}+m\mathrm{ln}{p}_{2}-n{p}_{1}\mathrm{ln}\sigma -m{p}_{2}\mathrm{ln}\sigma \\ +\left({p}_{1}-1\right)\sum \limits_{i = 1}^{n}\mathrm{l}\mathrm{n}\left({x}_{i}\right)+\left({p}_{2}-1\right)\sum \limits_{j = 1}^{m}\mathrm{l}\mathrm{n}\left({y}_{j}\right) \\ -\frac{1}{{\sigma }^{{p}_{1}}}\sum \limits_{i = 1}^{n}{\left({x}_{i}\right)}^{{p}_{1}}-\frac{1}{{\sigma }^{{p}_{2}}}\sum \limits_{\mathrm{j} = 1}^{\mathrm{m}}{\left({y}_{j}\right)}^{{p}_{2}}$

(7)

The log-likelihood function (7) is to be maximized in order to obtain the best estimates of parameters.

2.2. Least squares estimation (LSE) and weighted least squares estimation (WLSE)

The least squares estimation technique was used by Swain et al. ^[18] in Johnson's translation system for modeling glucose levels in diabetes, in the analysis of statistical models, and structural reliability. Ashour and Eltehiwy ^[19] proposed the application of the technique in estimation of parameters of exponentiated power Lindley distribution. Weighted least squares which is a modification of least squares estimation method has been used in many applications ^[20]. The method was used by Wu et al. ^[21] in moving identification and found the suitability of the application in time-varying systems. The technique has also been used in estimation of parameters of multiplicative generalization of binomial distribution ^[22]. Benchiha et al. ^[23] used LSE and WLSE techniques in estimating the parameters of weighted generalized Quasi Lindley distribution. The method of least square and weighted least square have a property of unbiased estimation for large number of observations and have been used in estimating stress-strength reliability for various distributions including Weibull distribution ^{[24,25,26,27]}.

Consider x₁, x₂, x₃, …, x_n is the random sample in ascending order of size n following Weibull distribution W(σ, p₁) and y₁, y₂, y₃, …, y_m is the random sample in ascending order of size m following Weibull distribution W(σ, p₂). Then the least squares criterion can be obtained as ^[28]

$Q = \sum \limits_{i = 1}^{n}{\left(\mathrm{ln}\left(\mathrm{ln}\left(\frac{1}{1-\widehat{F\left({x}_{i}\right)}}\right)\right)-{p}_{1}\mathrm{ln}\left({x}_{i}\right)+{p}_{1}\mathrm{ln}\left(\sigma \right)\right)}^{2} \\+\sum \limits_{j = 1}^{m}{\left(\mathrm{ln}\left(\mathrm{ln}\left(\frac{1}{1-\widehat{F\left({y}_{j}\right)}}\right)\right)-{p}_{2}\mathrm{ln}\left({y}_{j}\right)+{p}_{2}\mathrm{ln}\left(\sigma \right)\right)}^{2}$

(8)

The estimates of parameters can be obtained by minimizing function (8). The estimate values of F(x) and F(y) can be obtained by mean rank as

$\widehat{F\left({x}_{i}\right)} = \frac{i}{n+1}$ and

$\widehat{F\left({y}_{j}\right)} = \frac{j}{m+1} .$

Similarly, the criterion to be minimized for weighted least squares can be given as

$Q = \sum \limits_{i = 1}^{n}{{w}_{i}\left(\mathrm{ln}\left(\mathrm{ln}\left(\frac{1}{1-\widehat{F\left({x}_{i}\right)}}\right)\right)-{p}_{1}\mathrm{ln}\left({x}_{i}\right)+{p}_{1}\mathrm{ln}\left(\sigma \right)\right)}^{2} \\+\sum \limits_{j = 1}^{m}{w}_{j}{\left(\mathrm{ln}\left(\mathrm{ln}\left(\frac{1}{1-\widehat{F\left({y}_{j}\right)}}\right)\right)-{p}_{2}\mathrm{ln}\left({y}_{j}\right)+{p}_{2}\mathrm{ln}\left(\sigma \right)\right)}^{2} \\{\rm{where}} \; {w}_{i} = {\left(1-\widehat{F\left({x}_{i}\right)}\right)ln\left(1-\widehat{F\left({x}_{i}\right)}\right)}^{2} \;{\rm{and}} \; {w}_{j} = {\left(1-\widehat{F\left({y}_{j}\right)}\right)ln\left(1-\widehat{F\left({y}_{j}\right)}\right)}^{2}.$

(9)

The estimates of parameters using weighted least squares method can be obtained by minimizing Eq (9).

Equations (7)–(9) discussed above are optimization problems. Solutions to these equations using numerical computation do not yield precise results. It also has problems of slow convergence and non-convergence to real roots. So, these methods have to be assisted with a suitable optimization technique in order to improve their effectiveness. In this case, Jaya algorithm is used to optimize these functions.

3. Jaya algorithm

The application of metaheuristic techniques in optimization problems has seen increasing importance in modern times ^[29,30]. Jaya algorithm is a recent metaheuristic technique capable of solving a vast number of optimization problems with high effectiveness ^[31]. The researchers have used the technique in a number of applications and found satisfactory results. Meshram et al. ^[32] carried out electrical discharge machining with around eight control variables and two responses. Taguchi's L12 orthogonal array was used in designing the experiment. The regression equations were taken as objective functions and Jaya algorithm was used in optimization observing improvement in response variables. Caldeira and Gnanavelbabu ^[33] presented the implementation of Jaya algorithm for effectively solving the flexible job-shop scheduling problem. Gupta et al. ^[34] discussed the superiority of Jaya algorithm over other similar metaheuristic techniques in optimizing standard functions for the application of workflow scheduling in cloud computing. Jin et al. ^[35] identified the parameters of wind turbine power models using Jaya algorithm and monitoring with multivariate control charts. Du et al. ^[36] proposed a hybrid objective function for identifying the sites and extent of damage in a damage identification problem wherein Jaya algorithm was used to optimize the function in obtaining the parameters with good accuracy. Similarly, the algorithm has been used by many other researchers in such optimization problems ^{[37,38,39,40]}. In Jaya algorithm, the initial population with sets of parameters is randomly generated using upper and lower bounds. Then, each candidate in the population is updated based on the equation:

$\text{Z}_{\text{j}, \text{k}, \text{i}}^{\prime } = {{\text{Z}}_{\text{j}, \text{k}, \text{i}}}+{{\text{r}}_{1, \text{j}, \text{i}}}\left( {{\text{Z}}_{\text{j}, \text{best}, \text{i}}}\left| {{\text{Z}}_{\text{j}, \text{k}, \text{i}}} \right| \right)-{{\text{r}}_{2, \text{j}, \text{i}}}\left( {{\text{Z}}_{\text{j}, \text{ worst}, \text{i}}}-\left| {{\text{Z}}_{\text{j}, \text{k}, \text{i}}} \right| \right)$

(10)

where Z'_{j, k, i} is the updated value of variable k for candidate solution j, Z_{j, k, i} is the previous value of variable k for candidate solution j and i is the iteration number. r_{1, j, i} and r_{2, j, i} are the random variables between 0 and 1. The significance of the algorithm is that it continuously takes the candidate solution towards the best solution by the term (Z_{j, best, i}-│Z_{j, k, i}│) and away from the worst solution by the term (Z_{j, worst, i}-│Z_{j, k, i}│). Depending on the best and worst function value, the candidate solutions are updated. Figure 1 shows the flowchart of Jaya algorithm:

Figure 1. Flowchart of Jaya algorithm.

(p₁, p₂, σ)	R	(n, m)	$\widehat{R}$	Bias	MSE	T(s)
(1.5, 2, 1)	0.48063	(25, 25)	0.481730	0.001100	0.000214321	12.79
		(50, 50)	0.481418	0.000788	0.000103129	21.26
		(100,100)	0.480968	0.000338	0.000042815	39.67
		(500,500)	0.480770	0.000140	0.000009908	191.6
(2, 2, 1)	0.5	(25, 25)	0.498637	- 0.00136	0.000204486	12.38
		(50, 50)	0.500416	0.000416	0.000114916	21.64
		(100,100)	0.499798	- 0.00020	0.000050960	40.26
		(500,500)	0.500074	0.000074	0.000008967	198.1
(2.5, 2, 1)	0.5151	(25, 25)	0.516159	0.001059	0.000206911	12.48
		(50, 50)	0.515342	0.000242	0.000109094	23.14
		(100,100)	0.515160	0.000060	0.000048278	39.50
		(500,500)	0.515126	0.000026	0.000010192	188.5
(2.5, 2, 2)	0.515049	(25, 25)	0.515432	0.000383	0.000217130	12.64
		(50, 50)	0.514981	- 0.00007	0.000103620	21.85
		(100,100)	0.515108	0.000059	0.000051235	38.13
		(500,500)	0.515019	- 0.00003	0.000009259	186.9

0.495	0.496	0.558	0.585	0.641	0.68	0.702	0.704	0.733	0.739
0.742	0.753	0.757	0.762	0.765	0.775	0.778	0.791	0.807	0.822
0.839	0.845	0.85	0.856	0.857	0.858	0.868	0.868	0.89	0.899
0.899	0.915	0.918	0.919	0.935	0.939	0.947	0.948	0.956	0.963
0.968	0.97	0.976	0.992	0.993	0.997	0.999	1.013	1.017	1.028
1.045	1.046	1.056	1.06	1.063	1.064	1.074	1.086	1.114	1.136
1.157	1.163	1.166	1.168	1.18	1.22	1.295	1.352	1.352

0.573	0.643	0.665	0.672	0.681	0.709	0.712	0.723	0.723	0.738
0.74	0.746	0.76	0.761	0.762	0.764	0.777	0.789	0.789	0.79
0.792	0.802	0.807	0.826	0.827	0.862	0.88	0.883	0.886	0.886
0.898	0.904	0.914	0.943	0.947	0.949	0.971	0.972	0.976	0.978
0.985	0.987	0.994	1.005	1.009	1.019	1.028	1.036	1.054	1.056
1.067	1.072	1.074	1.094	1.162	1.168	1.172	1.198	1.214	1.215
1.274	1.326	1.514

Estimation method	$\widehat{{p}_{1}}$	$\widehat{{p}_{2}}$	$\widehat{\mathrm{\sigma }}$	K-S	p-value	Reliability	Compilation time(s)
MLE	5.5061	5.0514	0.9999	0.0564 (DS Ⅰ) 0.0881 (DS Ⅱ)	0.9773 (DS Ⅰ) 0.6929 (DS Ⅱ)	0.505824	0.079715
LSE	5.5647	5.7374	0.9982	0.0513 (DS Ⅰ) 0.1096 (DS Ⅱ)	0.9920 (DS Ⅰ) 0.4145 (DS Ⅱ)	0.497935	0.888459
WLSE	5.6584	4.9353	0.9880	0.0523 (DS Ⅰ) 0.1002 (DS Ⅱ)	0.9900 (DS Ⅰ) 0.5295 (DS Ⅱ)	0.509234	1.359188

	No. of estimated parameters	Log Likelihood	AIC	Delta AIC
MLE	3	-40861.12	81728.24	0
LSE	3	-81588.05	163182.1	81453.86
WLSE	3	-42321.46	84648.92	2920.68

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(p₁, p₂, σ)	R	(n, m)	$\widehat{R}$	Bias	MSE	T(s)
(1.5, 2, 1)	0.48063	(25, 25)	0.483988	0.003358	0.000374892	8.208
		(50, 50)	0.482671	0.002041	0.000296409	13.72
		(100,100)	0.480902	0.000272	0.000120576	24.11
		(500,500)	0.481391	0.000761	0.000033845	116.5
(2, 2, 1)	0.5	(25, 25)	0.501092	0.001092	0.000709956	8.773
		(50, 50)	0.500238	0.000238	0.000227603	13.44
		(100,100)	0.500839	0.000839	0.000243236	23.76
		(500,500)	0.500048	0.000048	0.000022361	107.9
(2.5, 2, 1)	0.5151	(25, 25)	0.515538	0.000438	0.000568429	8.163
		(50, 50)	0.517007	0.001907	0.000734081	13.48
		(100,100)	0.515227	0.000127	0.000108498	23.93
		(500,500)	0.514984	- 0.00012	0.000023476	118.8
(2.5, 2, 2)	0.515049	(25, 25)	0.517451	0.002402	0.001412757	8.760
		(50, 50)	0.517467	0.002418	0.000708432	13.92
		(100,100)	0.515638	0.000589	0.000117623	23.84
		(500,500)	0.515736	0.000687	0.000178473	119.2

(p₁, p₂, σ)	R	(n, m)	$\widehat{R}$	Bias	MSE	T(s)
(1.5, 2, 1)	0.48063	(25, 25)	0.484014	0.003384	0.001492711	10.32
		(50, 50)	0.481896	0.001266	0.000637294	18.11
		(100,100)	0.481325	0.000695	0.000666559	30.80
		(500,500)	0.482667	0.002037	0.000730472	143.5
(2, 2, 1)	0.5	(25, 25)	0.502422	0.002422	0.001433974	10.61
		(50, 50)	0.503042	0.003042	0.001274580	18.11
		(100,100)	0.502669	0.002669	0.001481735	31.15
		(500,500)	0.503678	0.003678	0.001585831	143.5
(2.5, 2, 1)	0.5151	(25, 25)	0.516737	0.001637	0.001899504	11.19
		(50, 50)	0.518774	0.003674	0.001496135	17.10
		(100,100)	0.517434	0.002334	0.001370110	31.35
		(500,500)	0.517283	0.002183	0.000599416	143.0
(2.5, 2, 2)	0.515049	(25, 25)	0.517838	0.002789	0.002217398	10.46
		(50, 50)	0.516697	0.001648	0.000536541	17.38
		(100,100)	0.519082	0.004033	0.001620499	32.25
		(500,500)	0.516500	0.001451	0.000401925	143.3

(p₁, p₂, σ)	R	(n, m)	$\widehat{R}$	Bias	MSE	T(s)
(1.5, 2, 1)	0.48063	(25, 25)	0.483988	0.003358	0.000374892	8.208
		(50, 50)	0.482671	0.002041	0.000296409	13.72
		(100,100)	0.480902	0.000272	0.000120576	24.11
		(500,500)	0.481391	0.000761	0.000033845	116.5
(2, 2, 1)	0.5	(25, 25)	0.501092	0.001092	0.000709956	8.773
		(50, 50)	0.500238	0.000238	0.000227603	13.44
		(100,100)	0.500839	0.000839	0.000243236	23.76
		(500,500)	0.500048	0.000048	0.000022361	107.9
(2.5, 2, 1)	0.5151	(25, 25)	0.515538	0.000438	0.000568429	8.163
		(50, 50)	0.517007	0.001907	0.000734081	13.48
		(100,100)	0.515227	0.000127	0.000108498	23.93
		(500,500)	0.514984	- 0.00012	0.000023476	118.8
(2.5, 2, 2)	0.515049	(25, 25)	0.517451	0.002402	0.001412757	8.760
		(50, 50)	0.517467	0.002418	0.000708432	13.92
		(100,100)	0.515638	0.000589	0.000117623	23.84
		(500,500)	0.515736	0.000687	0.000178473	119.2

(p₁, p₂, σ)	R	(n, m)	$\widehat{R}$	Bias	MSE	T(s)
(1.5, 2, 1)	0.48063	(25, 25)	0.484014	0.003384	0.001492711	10.32
		(50, 50)	0.481896	0.001266	0.000637294	18.11
		(100,100)	0.481325	0.000695	0.000666559	30.80
		(500,500)	0.482667	0.002037	0.000730472	143.5
(2, 2, 1)	0.5	(25, 25)	0.502422	0.002422	0.001433974	10.61
		(50, 50)	0.503042	0.003042	0.001274580	18.11
		(100,100)	0.502669	0.002669	0.001481735	31.15
		(500,500)	0.503678	0.003678	0.001585831	143.5
(2.5, 2, 1)	0.5151	(25, 25)	0.516737	0.001637	0.001899504	11.19
		(50, 50)	0.518774	0.003674	0.001496135	17.10
		(100,100)	0.517434	0.002334	0.001370110	31.35
		(500,500)	0.517283	0.002183	0.000599416	143.0
(2.5, 2, 2)	0.515049	(25, 25)	0.517838	0.002789	0.002217398	10.46
		(50, 50)	0.516697	0.001648	0.000536541	17.38
		(100,100)	0.519082	0.004033	0.001620499	32.25
		(500,500)	0.516500	0.001451	0.000401925	143.3

AIMS Mathematics

Jaya algorithm in estimation of P[X > Y] for two parameter Weibull distribution

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Abstract

1. Introduction

2. Estimation methods

2.1. Maximum likelihood estimation (MLE)

2.2. Least squares estimation (LSE) and weighted least squares estimation (WLSE)

3. Jaya algorithm

4. Reliability estimation

5. Simulation studies

6. Application to real life data

7. Conclusions

Conflict of interest

References

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