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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.20 No.3 pp.455-463
DOI : https://doi.org/10.7232/iems.2021.20.3.455

Estimating the Reliability for the Sequential System of Two Truncated Exponential Distribution

Aseel Nasser Sabti*, Ansseif A.latif Ansseif, Aseel Mahmood Shakir
College of Administration and Economics, Al-Iraqia University, Baghdad, Iraq
*Corresponding Author, E-mail: aseel.sabti@aliraqia.edu.iq
March 5, 2021 ; September 5, 2021 ; September 13, 2021

Abstract


The reliability of sequential system is one of the most important subjects in the current industrial system. In life testing, it has become a great challenge to assess the reliability with consecutive failure time, especially when some units of failure time are deleting. The majority of research in this field in oriented towards illustrating a unique truncated distribution, whereas our study identified two truncated exponential distributions where the failure time units are recorded before time T. In addition, we represent the new algorithm for iterative maximum likelihood (IMLE) to estimate the parameters and the reliability function. After that, a Visual Basic simulation program is successfully created to reflect on the efficiency of the used algorithm. An example and then a randomly data set for three models are chosen to find the values of reliability function. In conclusion, the paper shows that the IMLE estimator's performance is quite well and it gives a good fit of reliability function for the two truncated exponential distribution. The estimated parameters have also obtained a very good converge.



초록


    1. INTRODUCTION

    The reliability function and maximum likelihood estimation has attracted a lot of attention in its field, to the extent that several studies have been conducted about it in the most recent years. The most significant studies associated with reliability function and maximum likelihood can be summarized as follows:

    Andrews and Ridley (2001) The goal of this paper is to investigate the cause–consequence diagram method and its application on sequentially operating systems. It builds on previous work by suggesting new more restrictive guidelines that enable the proper construction of the diagram, as well as an analysis methodology that can be used in case dependencies exist between the events featured in the decision boxes. In their paper, they introduced a new symbol which can be used to differentiate between events that exist at a specified point in time and those that occur at that time in order to facilitate the analysis. On the other hand, the cause–consequence diagram method allows the accurate modeling of sequential or dependent systems with the retention of the failure logic.

    Adamyan and He (2002) In this paper, a methodology that can be used to detect the failure sequences and to assess the likelihood of them occurring in a manufacturing system is suggested. The method depends on Petri net modeling and reachability trees that are constructed based on the Petri nets. An automated machining and assembly system was used as an example to validate this method.

    Abdelkader (2006) offered an approach for employing right-truncated exponentially distributed random variables to model activity times in a stochastic activity network. A close estimation for the project's k-th moment completion time was achieved by truncating the exponential distribution from the right. The truncation point is influenced by on the choice of α. According to previously practiced methods, the ideal choices for α range between 0.01 and 0.05 inclusively.

    Rao et al. (2019) deliberated the exponentiated moment- based exponential distribution in the study of multicomponent stress-strength reliability. The reliability of the samples was determined by applying maximum likelihood (ML) on the computed distributions of strength and stress. In addition, Monte Carlo simulations were used to compute confidence intervals of the bulky samples. As a result, the sample size was inversely proportional to the bias and the mean squared error. The average bias is negative when α1α2 ; otherwise, the bias is positive for (s,k) = (1,3) and (s,k) = (2,4). As α1 increases for a fixed α2, the mean squared error decreases. Whereas it increases as α2 increases for a fixed α1 for (s,k) = (1,3) and (s,k) = (2,4). In addition, when the sample are large, the confidence intervals also shrink in range and the performance of the coverage probability is “satisfactory”.

    Bodhisuwan et al. (2017) established the zerotruncated negative binomial-Erlang distribution developed from negative binomial-Erlang distribution. In their paper, they predicted the parameters of the zero-truncated negative binomial-Erlang distribution via maximum likelihood estimation, compared the proposed algorithm with other distributions, and discussed some of the properties of the distribution. Furthermore, the parameter estimation method using the maximum likelihood estimation method is deliberated. In the application of ZTNB-EL distribution, the accuracy of the proposed distribution is compared with the ZTP, the ZTNB, ZTGNB and ZTPL distributions. The usefulness of the ZTNB-EL distribution is demonstrated by the number of methamphetamine in the Bangkok metropolitan region from the Office of the Narcotics Control Board (ONCB), Thailand. The log-likelihood, AIC, BIC and p-values of the K-S test for the goodness of fit are used to select the model. In conclusion, the study illustrated that ZTNBEL distribution provides a better fit when compared to the ZTP, ZTNB, ZTGNB and ZTPL distributions. Obviously, the ZTNB-EL distribution is an alternative distribution to analysis of count data with zero truncated.

    Mohie et al. (2013) presented the mid-truncated distributions, and derived some statistical properties of the mid- truncated distributions. In their results, a midtruncated Weibull distribution was obtained.

    Rather and Subramanian (2019) proposed the length biased Erlang-truncated exponential distribution, which is a special case of weighted distribution. The subject distribution was generated via length biased technique while the parameters were estimated by using the maximum likelihood method. The simulation experiment was carried out on different values of parameters, and real data were also analyzed to demonstrate the usefulness of the distribution. It was noticed that the bias, variance and MSE decreases when the sample size increases. Finally, the results indicated that the length biased erlangtruncated exponential distribution shows better fit than the Erlang-truncated exponential distribution.

    Kulshrestha et al. (2012) developed a marginal and joint moment generating functions of generalized order statistics from Erlang-truncated exponential distribution. The results for k-th record values and order statistics were deduced from the relations derived, and later they discussed using the conditional expectation of a function of generalized order statistics from this distribution.

    Okorie et al. (2016) suggested using the maximum likelihood method for estimating the Transmuted Erlang parameters Truncated Exponential distribution (TETE). As a result, the TETE distribution's hazard rate function is said to be constant, the TETE distribution was found to be a better candidate for the data, and the parameters of this distribution were very stable.

    Zaninetti and Ferraro (2008) compared between the usual Pareto distribution and its truncated version, which is a possible physical mechanism that uses Pareto tails for distributing the masses of stars.

    Kumar (2014) established some exact expression and recurrence relations satisfied by the quotient moments and conditional quotient moments of the upper record values from the Erlang-truncated exponential distribution.

    Rahmouni and Orabi (2018) offered a new lifetime data by relying on the compounding of exponential and generalized truncated logarithmic distributions. In this study, they also presented the failure rate functions and the reliability. Such a simulation study can have many applications, one of which is to demonstrate the maximum likelihood and other deferent methods.

    De Shyamal and Zacks (2015) proposed truncated sequential test procedures for testing the mean time between a system’s failures with exponential life distribution.

    Burkardt (2014) defined univariate and multivariate integrals in the truncated normal distribution. They also used the quadrature roles to estimate the integrals and combine product roles to create efficient sparse grids for the multivariate case. The authors argued that this sparse grid approach can be applied to stochastic quantities governed by a truncated normal distribution. A variety of information were collected in order to understand the truncated normal distribution in a simplified manner, as a method to define univariate and multivariate integrals where the truncated normal distribution plays the role of a PDF. They also demonstrated how to determine quadrature rules to estimate uni- and multi-variate integrals, by compounding the exponential and generalized truncated logarithmic distributions. As usual, they viewed the reliability and failure rate functions, and how to use maximum likelihood among other approaches.

    This paper’s organization as follows: Section 2 describes the definition of truncated exponential distribution. Section 3 considers the reliability of sequential system, Section 4 introduces the mathematical formulation and the Iterative Maximum Likelihood Estimation with the new algorithm used to find the estimators. In Section 5 a Visual Basic program executed with two groups of truncated exponential distributions generated randomly with different population sizes to determine the new estimators and the reliability function. Section 5 discusses an exact example with real life time data. Finally, the contributions and results of this paper are explained in Section 7.

    2. TRUNCATED EXPONENTIAL DISTRIBUTION

    A truncated distribution is a conditional distribution that results from restricting the domain of some probability distributions.

    • Definition: A random variable has a truncated exponential distribution with one parameter λ if its probability density function is: (Raschke, 2012)

      f ( t ; λ ) = e x p ( t λ ) / [ λ ( 1 e x p ( t λ ) ) ] 0 t T ,      0 < λ <
      (1)

    The cumulative distribution function is:

    F ( t   ; λ ) = 1 e x p ( t λ ) 1 e x p ( T λ ) 0 t T ,       λ > 0
    (2)

    Here λ is the scale parameter, T is the truncation point

    The survival functions are given by:

    S ( t ; λ ) = 1 F ( t ; λ )
    (3)

    = 1 1 e x p ( t λ ) 1 e x p ( T λ ) , 0 t T   ,       λ > 0
    (4)

    2.1 The Reliability of Sequential System

    In many complex industrial systems, where safety is very important, it is necessary to determine the reliability and safety assessment for sequential failures to handle the accident problems. Therefore, this paper represents the reliability assessment depending on the failure times of system components and the truncated failure time specified by a fixed time T.

    In general, the reliability function of the sequential system S with k non-symmetric independent component (C1, C2,…, Ck) for exponential distribution with one parameter and mean value λi (i= 1, 2, …, k) is: (Rao et al., 2019).

    R i ( s , k ) = i = 1 k R i ( t )
    (5)

    = e x p { t ( 1 λ 1 + 1 λ 2 + + 1 λ k ) }
    (6)

    In a more special case, let as introduce the reliability function of two exponential components (C1, C2) and with number of failure time (n1, n2 respectively. The failure time (t11, t12,…, t1r1) selected from the first component with number of units r1 and (t21, t22,…, t2r2 ) is the failure time from the second component with number of units r2 which failed before time T and (nr1r2) the remained units after time T then:

    r 1 t ¯ 1 = j = 1 r 1 t 1 j , r 2 t ¯ 2 = j = 1 r 2 t 2 j r = r 1 + r 2

    The maximum likelihood function can be written as:

    L ( t ¯ 1 , t ¯ 2 ,   r 1 , r 2 \ λ 1 , λ 2 ) = n ! r 1 ! r 2 ! ( n r ) ! 1 λ 1 r 1 e x p { r 1 t ¯ 1 λ } 1 λ 2 r 2 e x p { r 2 t ¯ 2 λ }
    (7)

    when,

      1 λ = 1 λ 1 + 1 λ 2

    The loglikelihood function is:

    l n L ( λ 1 , λ 2 ) = c o n s t a n t r 1 l n λ 1 r 2   l n λ 2       r 1 t ¯ 1 + r 2 t ¯ 2 + ( n r ) T λ
    (8)

    The first partial derivative of the previous function with respect to each parameter, as follows:

    λ 1   l n L ( λ 1 , λ 2 ) = r 1 λ 1 +     r 1 t ¯ 1 + r 2 t ¯ 2 + ( n r ) T λ 1 2
    (9)

    λ 2   l n L ( λ 1 , λ 2 ) = r 2 λ 2 +     r 1 t ¯ 1 + r 2 t ¯ 2 + ( n r ) T λ 2 2
    (10)

    If the last equation is equal to zero, we obtained M.L.E's estimations as follows: (Rather and Subramanian, 2019).

    λ ^ i =   r 1 t ¯ 1 + r 2 t ¯ 2 + ( n r ) T r i , i = 1 , 2
    (11)

    4. ITERATIVE MAXIMUM LIKELIHOOD ESTIMATION

    The iterative maximum likelihood method is the iterative manner used to find parameters of two groups of an exponential distribution, when the failure times are classified to m classes and the views (i) are recorded during the interval ( t j 1 , t j ) and before the truncated time T. then, the probability of failure is:

    f ( t ; λ ) = 1 λ i t j 1 t j e x p ( t λ ) d t
    (12)

    = λ λ i [ e x p ( t j 1 λ ) e x p ( t j λ ) ] j = 1 , 2 , , m , i = 1 , 2
    (13)

    The maximum likelihood can be written down as: (Mohie et al., 2013)

    L = c o n s t a n t λ r λ 1 r 1 λ 2 r 2 j = 1 m { e x p ( t j 1 λ ) e x p ( t j λ ) } r 1 j + r 2 j e x p { ( n r ) T λ }  
    (14)

    where;

    r i = j = 1 m r i j , i = 1 , 2 and r = r 1 + r 2 L n   L = c o n s t a n t + r 1 l n λ λ 1 + r 2 l n λ λ 2 + J = 1 m ( r 1 j + r 2 j ) l n { e x p ( t j 1 λ ) e x p ( t j λ ) } ( n r ) T λ
    (15)

    Suppose that,

    g j = t j + t j 1 2   a n d h j = ( t j t j 1 )
    (16)

    Substitute gj , hj in equation (15), the approximate estimation can be seen in equation (17)

    λ ^ i * = j ( r 1 j + r 2 j ) g j + ( n r ) T j ( r 1 j + r 2 j ) h j 2 12 λ ^ r i
    (17)

    As (Boardman and Kendell, 1970) suggested, the initial value of λ ^ is:

    λ ^ = 1 2 [ λ ^ 0 + { λ ^ 0 2 j ( r 1 j + r 2 j ) h j 2 3 r } 1 / 2 ]
    (18)

    λ ^ 0 = 1 r [ j ( r 1 j + r 2 j ) g j + ( n r ) T ]
    (19)

    So, to find the parameters of this model, a new algorithm was proposed to find the iterative maximum likelihood estimators as follows: (Sinha and Kale,1980)

    Algorithm of iterative maximum likelihood estimation:

    • Step 1: Generate two groups of exponential distribution as random variables.

    • Step 2: Assign the initial value λ1, λ2 , the sample value n, the failure time T, the two subpopulation n1, n2 and the number of classes' m.

    • Step 3: Test the failure time ( t 11 , t 12 , , t 1 n 1 ) , ( t 21 , t 22 , , t 2 n 2 ) during each interval ( t j t j 1 ) , where j =1, 2,…, m, then determine r1, r2 which failed before the time, T.

    Note that; ( n r 1 r 2 ) the remained units after time T.

    • Step 4: Compute the functions g j = t j + t j 1 2 and h j = ( t j t j 1 )

    • Step 5: Compute λ ^ 0 from equation (19).

    • Step 6: Estimate λ ^ from equation (18).

    • Step 7: Estimate the two parameters λ ^ 1 * , λ ^ 2 * from equation (17)

    • Step 8: Use the new equation to find the new value of λ ^ as follows:

      λ ^ = λ ^ 1 * . λ ^ 2 * λ ^ 1 * + λ ^ 2 *
      (20)

    • Step 9: Put the new value λ ^ which is obtained from the previous step in equation (17).

    • Step 10: Repeat steps 7 to 9 until the two parameters are converging.

    5. SIMULATION RESULTS

    In this section we used simulation tests to solve the iterative maximum likelihood algorithm and determines the new estimators of two truncated exponential distribution and reliability. Table 1 gives the initial parameters randomly selected with population size (32 and 120) for three models. Moreover, we suppose the first model when λ1 <λ2, the second model when λ1>λ2 and the third model when λ1 =λ2 . The failure time T and the number of classes m are assumed to demonstrate algorithms' working, with repetition equal to 50000.

    As shown in Table 1, three different models with m classes equal to 10, failure time 5 and length of time interval equal to 0.5.

    Table 2 represents the parameters obtained by using an iterative maximum likelihood estimator for each model at iteration equal to 50000. The converges of the estimated parameter λ ^ with the initial parameter for three models can be seen. Also, we observed that, the new estimated parameter is located between the default values in the first and in the second model and for all sample space, for example; in the first model λ1 =2,λ2 =4 and the obtained parameter λ ^ is equal to 2.843 at sample size 32, and equal to 3.085 at sample space 120 similarly with the second model. Also, we observe that λ 1 * increase if the sample size increases, whereas λ 2 * decreases if the sample size increases. Furthermore, the reliability functions for each model are reported in Tables 3-5.

    Tables 3-5 summarize the computational experience to estimate the reliability functions. During the algorithm’s performance, it is clear that the estimated reliability in the first model increased when λ1 <λ2 . In the second model the estimated reliability decreased when λ1>λ2 . In the third model the estimated reliability increases when λ1 =λ2 . For more clarification, see Figures 1-3.

    Figures 1-3 represent the starting and ending values of true and estimated parameter to each model respectively. Furthermore, the closest model from true reliability function is the third model when λ 1 * = λ 2 * , followed by the second model when λ 1 * > λ 2 * , then the first model when λ 1 * < λ 2 * .

    6. DATA ANALYSIS

    This section considers one real data set from Sinha (1979 p.100). A company supplied 150 electronic units where each unit has two exponential component A and B. The unit fails before time 550. The number of failures and failure times were recorded as follows:

    Identifying the values as bellow:

    n=150 , T=550 , r1 =95, r2 =72 , r =131 , hj =50 , j =1, 2,…,11

    At the first step we should find the midpoint

    g j = t j + t j 1 2 and specify the number of failures for two components as shown in Table 7.

    At the second step we obtain IMLE’s estimators as previously described λ ^ 0 = 386.573 from equation (16),

    λ ^ = 386.045 , from equation (15).

    Substituting λ ^ 0 , λ ^ in (14) then, λ ^ 1 * = 857.151 , λ ^ 2 * = 702.388

    Hence,

    λ ^ = 857.151 * 702.388 857.151 + 702.388 = 386.045

    In view of this example, we have convergence in obtained parameter.

    7. CONCLUSIONS

    This study reveals a more complex contribution in the industrial filed. The novel feature of our approach is the ability of iterative algorithms to integrate failure time from two exponential distributions. In our study, we estimated the reliability function, and identified an example as well as three simulation models. The simulation results according to Visual Basic program (Version 6.0) show that, the estimated reliability increase when the sample size increases for the first and the third model, the good convergence with true reliability is valid for all sample size, and that the third model is the closest to true reliability where the default parameters are equal. We also obtained the converges of estimated parameters at each sample size for all models. Overall, since the performance of IMLE algorithm’s is quite well and gives a good fit reliability function, thus it can be applied in the industrial system. In Future work, the iterative algorithms can be used with more complicated distributions and with identical or different truncated failures.

    Aseel Nasser Sabti received her Ph.D. degree in Operations Research Science from Anadolu University, Turkey in the year 2018. Her area of interests includes Operations Research, Optimization, Analysis & Decisionmaking.

    Ansseif A.latif Ansseif received his M.Sc. degree in Operations Research from the University of Baghdad, Iraq in the year 2010. Currently a Ph.D. student at the University of Sfax, Tunisa. His research interests focus on the areas of Operations Research, Optimization, Analysis & Decision-making.

    Aseel Mahmood Shakir received her Ph.D. degree in Statistic from the University of Baghdad, Iraq in the year 2017. Her research interests focus on the fields of Analysis, Statistical Programs, and Applied Statistics.

    Figures

    IEMS-20-3-455_F1.gif

    The reliability function for the first model.

    IEMS-20-3-455_F2.gif

    The reliability function for the second model.

    IEMS-20-3-455_F3.gif

    The reliability function for the third model.

    Tables

    The set of parameters for selected models

    The estimated parameters for selected models

    The reliability results of the first model at generation 50000

    The reliability results of the second model at generation 50000

    The reliability results of the third model at generation 50000

    Number of failures for two components:

    Number of failures for two components

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