Fuzzy reliability analysis of a pulping system in paper industry with general distributions for all random variables

This paper discussed the fuzzy reliability of pulping system in paper industry. Pulping system consists of four subsystems such as Digester, Knotter, Deckers, and Openers. These subsystems are working in series. In subsystems Knotter and Openers, one unit is operative and another unit is kept as cold standby. Fuzzy failure rates and fuzzy repair rates of the subsystems are taken as general. Trapezoidal fuzzy numbers are used to find out the reliability of the pulping system. Using Mehar’s method, the numerical results for reliability are obtained by considering exponential, Rayleigh, and Weibull distributions. On the basis of the results obtained in this study, it is suggested that paper industry management should involve Weibull distributed random variables for pulping system to attain higher reliability. Subjects: Science; Mathematics & Statistics; Statistics & Probability; Operations Research; SPC/Reliability/Quality Control

ABOUT THE AUTHOR Jitender Kumar, PhD, works as an assistant professor in Department of Statistics and O.R., Kurukshetra University, Kurukshetra, India. He specializes in Reliability Modeling and Analysis. His research papers appeared in different national and international repute journals. He has presented his research work in number of national and international conferences in India and Abroad. He is secretary of Indian Association of Reliability and Statistics (IARS). He is guiding a number of doctoral candidates. Kumar is a reviewer for reputed journals.

PUBLIC INTEREST STATEMENT
Reliability analysis of any industrial or mechanical system plays a vital role in optimization of efficiency and reduction of undesired factors like cost, failure, and any kind of hazard. It helps the manufacturer to make the system operative for a long interval of time. In most of the cases behavior of the system and system information need not be precise in actual practice. To overcome such issues analysis of system in fuzzy environment plays a significant role. Paper industry is one of the important industries in present time. To enhance the productivity, fuzzy analysis may be more effective rather than traditional one. This article contains fuzzy analysis of pulping system in paper industry. Most of the reliability analysis is done with exponential distribution. In this article we implemented general distributions. This opens new dimensions for researcher to work for betterment of other industrial systems. efficiency of any industrial system. Findings and outcomes of reliability analysis of industrial systems provide information about advantages and limitations of these systems. These information may be helpful in further designing and modifications of industrial systems. In the last few decades, reliability analysis of various industrial systems made a powerful impact on the growth of various industries such as paper sugar, fertilizers, oil and pipe industries, etc. Kumar, Pandey, and Singh (1991) studied the reliability of crystallization system of urea fertilizer industry having five subsystems in series. They discussed the system with constant failure rates and arbitrary repair rates. Kumar, Mehta, and Kumar (1997) discussed the steady-state behavior and maintenance of a desulphurization system in urea fertilizer plant. Gupta, Lal, Sharma, and Singh (2005) discussed the numerical analysis of reliability and availability of a butter oil processing plant using fourth-order Runga-Kutta method to solve the governing differential equations of the systems. Shakuntla, Lal, Bhatia, and Singh (2011) obtained availability of pipe manufacturing plant with simultaneous fail of two subsystems. Kadyan and Kumar (2015) discussed availability and profit analysis of a feeding system in sugar industry using constant failure and general repair rates. As far as commodities in daily use are concerned, paper is one of the most frequently used commodity in our life, but due to limitations of raw material and increasing consumption day by day, it is prominently realized that the production should be optimized under available resources and circumstances. Paper industry is one of the core industry linked to the basic human needs. The paper industries consists of systems or subsystems that use wood as raw material and produce pulp, paper, boards, and other cellulose-based products. The quality of these products generally depends on the pulp quality. But, due to lack of proper handling and information there is big variation in the quality and efficiency of the products of paper industries. Wood is the main raw material that is used by paper industries. Due to urbanization the green area is reducing day by day. In order to meet the requirements of daily consumption of paper and without harming the environment too much, every paper industry must focus on the law of maximum output with minimum hazard to the environment. To overcome these issues, reliability analysis of paper industry can play an important role. There are three major components of a paper industry namely washing system, feeding system, and pulping system. These subsystems inclusively make the paper industry a complex industry. Taking these facts in account many researchers paid an attention to this industry. Kumar, Singh, and Singh (1988) discussed the reliability analysis of the feeding system of paper industry. Having an attention to the problem faced by insufficient working of the feeding system, they derived useful information about the behavior of various parts of this subsystem. Kumar, Singh, and Pandey (1989a) discussed the availability of pulping system and derived some useful information to achieve optimum availability. Kumar, Singh, and Pandey (1989b) calculated the availability of washing system in paper industry with constant failure and repair rates. Satyavati (2011) obtained the long run availability of washing system of paper industry. All these authors deal with constant failure rates and repair rates. But in actual practice, mechanical and physical processes are not bound to follow such constraint.
In traditional reliability techniques, one has to restrict on crisp values, but actual behavior of any physical or mechanical process need not be mathematically exact or certain. This shortcoming can be overpowered by replacing the model with fuzzy model. On a fuzzy model derivatives involved are replaced by fuzzy derivatives. The concept of fuzzy derivatives was first introduced by Chang and Zadeh (1972). Since then a lot of work has been done by many researchers and scientists including Buckley and Feuring (1999, 2000, 2001, Buckley, Feuring, and Hayashi (2002), Abbasbandy and Viranloo (2002), Georgiou, Nieto, and Rodríguez-López (2005), and Allahviranloo, Ahmady, and Ahmady (2007) on fuzzy derivatives and their extensions. Lata and Kumar (2011) proposed an analytical method (Mehar's method with JMD representation) to evaluate fuzzy reliability of a piston manufacturing system. This method provides the resulting solution in terms of fuzzy numbers. Razak and Rajakumar (2013) discussed the reliability of a system based on fuzzy Markov model with fuzzy transitions. Vishwakarma and Sharma (2016) discussed uncertainty analysis of an industrial system using intutionistic fuzzy set (IFS) theory. In this paper reliability measures of a TAB manufacturing plant was discussed by vague lambda-tau technology based on IFS theory.
The conventional reliability techniques do not effectively handle vague and ambiguous terms. The fuzzy logic allows us to handle these things and to extract appropriate conclusion. The use of fuzzy numbers allowed us to include uncertainty or imprecision in the data caused by the facts like expert opinion operative conditions and other factors mentioned above. Motivated by these facts we used trapezoidal fuzzy numbers to represent failure and repair rates of the system. Furthermore, it has been observed that in most of the reliability models system parameters are restricted to follow a particular distribution but in actual practice the data related to the system is not bound to follow such constraints. Therefore to obtain more flexible information regarding the reliability of the system, the failure and repair rates are generally distributed. The fuzzy differential equations associated with the system are solved by Mehar's method with JMD representation.
The aim of present paper is twofold. For this, firstly we introduce fuzzy environment to failure rates and repair rates and secondly general distributions are used to analyze the pulping system of paper industry. This paper has been organized as follows: Section 1 is introductory in nature, Section 2 introduces the basic definitions and arithmetic operations related to traditional trapezoidal fuzzy numbers and JMD trapezoidal fuzzy numbers. In Section 3, a complete introduction about description of the system along with notations and assumptions of the system is given. Section 4 discussed the mathematical modeling of pulping system in paper industry. In Section 5, Mehar's method with JMD trapezoidal fuzzy number is presented. Section 6 deals with the results obtained by the numerical study along with the graphical representation of the numerical results. Conclusion drawn from analysis is discussed in Section 7.

Preliminaries
Fuzzy set theory is a generalization of set theory. The concept of fuzzy sets was introduced by Zadeh (1965). The motivation behind introducing the fuzzy set theory was to deal with the problems involving knowledge expressed in vague and linguistics terms. Much of human understanding is vague or imprecise. All these inexactness can be represented by a fuzzy set more efficiently rather than a crisp set. Now we present some concepts and basic definitions related to a fuzzy set along with JMD representation of trapezoidal fuzzy numbers, arithmetic operations between trapezoidal fuzzy numbers and arithmetic operations between JMD trapezoidal fuzzy numbers.

Basic definitions and formulae
In this section, some basic definitions and formulae associated with fuzzy set theory are presented.
Let X be the set of discourse. Then a fuzzy subset of X is a function :X → [0, 1] and it is denoted by μ X . A fuzzy subset μ A is called a fuzzy number if A is a subset of the set of real numbers and there exists at least one real number x such that μ A (x) = 1.

Trapezoidal fuzzy number
In this section, definitions of trapezoidal fuzzy number, zero trapezoidal fuzzy number, and quality of trapezoidal fuzzy numbers are presented.
Definition 2.2.1 A fuzzy number Ã = (a, b, c, d) is said to be a trapezoidal fuzzy number if its membership function is given by

JMD representation of trapezoidal fuzzy number
In the recent past, many researchers developed fuzzy Kolomogrov's differential equations for fuzzy Markov model and obtained fuzzy reliability on solving these fuzzy Kolomogrov's differential equations. Kumar and Kaur (2012) observed that solution thus obtained need not be a fuzzy number. To overcome this issue they proposed JMD representation of trapezoidal fuzzy number and proved that it is better to use the proposed representation of trapezoidal fuzzy numbers instead of usual representation of trapezoidal fuzzy numbers for finding the fuzzy optimal solution of fuzzy transportation problems. In this section, definitions of JMD trapezoidal fuzzy number, zero JMD trapezoidal fuzzy number, and equality of JMD trapezoidal fuzzy numbers are presented.
Observe that JMD representation of a trapezoidal fuzzy number is again a trapezoidal fuzzy number.

Arithmetic operations between trapezoidal fuzzy numbers
In this section, arithmetic operations between trapezoidal fuzzy numbers are presented.

Arithmetic operations between JMD trapezoidal fuzzy numbers
In this section, arithmetic operations between JMD trapezoidal fuzzy numbers are presented.

System description
Pulping system is one of the most important part of the paper industry which consists of four subsystems and the brief description of the subsystems and the operations performed by these subsystems are as follows: (1) Subsystem A (Digester): With the help of steam heating system this subsystem cooks the chips for several hours. The system fails completely with the failure of this subsystem.
(2) Subsystem B (Knotter): This subsystem removes the knots from the pulp produced by the digester. This subsystem contains two units out of which one unit is in standby mode and this subsystem fails completely on the failure of both units.
(3) Subsystem C (Deckers): The three deckers are working in series. The produce of the knotter is passed through these deckers by keeping the vacuum in the drums of the deckers. It removes the used liquor from the pulp. This process is repeated two or three times to attain optimum washing. The failure of any of these deckers causes the failure of this subsystem.
(4) Subsystem D (Opener): The opener is a subsystem rotating at high speed. This subsystem does the separation of fibers. The subsystem contains two units having one unit in standby. This subsystem fails only if both units fail.

Assumptions
(i) Pulping system consists of four subsystems as Digester, Knotter, Deckers, and Opener.
(ii) All subsystems work in series.
(iii) Subsystem B and subsystem D both have one unit in standby.
(iv) Repairmen always available with the system.
(v) Each subsystem has separate repair facility and there is no waiting time for repair in the system.
(vi) All the subsystems work as good as new after their repair.
(vii) Fuzzy failure and fuzzy repair rates are independent with each other.
(viii) Fuzzy failure and fuzzy repair rates follow general distributions.

Fuzzy Kolmogorov's differential equations associated with the system
In this section, the following sets of fuzzy differential equations are developed by using the system model ( Figure 1) of pulping system as follows:  Buckley and Feuring (2001) proposed two analytical methods to solve the fuzzy initial value problem for n th order ordinary differential equations given by  (i) for i = n, n − 1, … 1, ã n is a nonzero trapezoidal fuzzy number and ã n−1 ,ã n−2 , …ã 1 ,ã 0 are any type of trapezoidal fuzzy numbers. We solve the fuzzy differential Equation (E) by Mehar's method in following steps:

Results and discussion
The Weibull distribution is frequently used to model various engineering problems. The probability distribution function of Weibull distribution with two parameters is given as follows: From the above formulae, it is clear that if k = 1, it becomes the exponential distribution and when k = 2, it becomes the Rayleigh distribution.
Tables 1-3 show the solution of differential equations with JMD representation of trapezoidal fuzzy numbers at different times for different distributions such as Weibull, exponential, and Rayleigh. Table 5 shows the behavior of reliability for the Weibull, exponential, and Rayleigh distributions with respect to time and their graphical representation is shown in Figures 2-4. l i (t) = kl i (l i t) k−1 exp[−(l i t) k ], t ≥ 0, l i > 0 where i = 1, 2, 3, 4 g i (t) = kg i (g i t) k−1 exp[−(g i t) k ], t ≥ 0, g i > 0 Table 2. Solution of differential equations of pulping system using trapezoidal fuzzy numbers (with JMD representation) for exponential distributioñ P j (t) for t = 24P j (t) for t = 48P j (t) for t = 72P j (t) for t = 96P j (t) for t = 120  Table 5 and Figures 2-4 show that as the time increases, reliability of the system decreases for Weibull, exponential, and Rayleigh distribution. From Table 5 and Figures 2-4, it is found that reliability will be higher if all random variables associated with the system follow Weibull distribution rather than exponential and Rayleigh distribution. Table 3. Solution of differential equations of pulping system using trapezoidal fuzzy numbers (with JMD representation) for Rayleigh distributioñ P j (t) for t = 24P j (t) for t = 48P j (t) for t = 72P j (t) for t = 96P j (t) for t = 120 Using fuzzy failure rates and fuzzy repair rates (Table 4), solutions of fuzzy differential equations of pulping system using Mehar's method at different times for general distributions are shown in Tables 1-3.
The results of fuzzy differential equations obtained in Tables 1-3 are used to analyze the fuzzy reliability of pulping system.
Using the fuzzy probabilities for the pulping system shown in Tables 1-3 and using the formula � R(t) = � P 1 (t) ⊕ � P 2 (t) ⊕ � P 3 (t) ⊕ � P 4 (t), the corresponding fuzzy reliabilities for different distributions at different times are shown in Table 5.