Performance analysis of “Injection Moulding Machine” under fuzzy environment through contemporary arithmetic operations on right triangular generalized fuzzy numbers (RTrGFN)

Abstract

The reliability of an industrial system plays a significant role in new technological era where everyone is concerned about the performance of associated industry. With the frequent demands of the customers, the job of the production industries becomes more tedious to produce the required products at a high rate. To fulfill the customer’s demand, the initial focus of the industries is to work well without any interruption/failure in the entire production process. In this paper, Reliability, Availability, MTTF, MTTR, MTBF, ENOF is proposed for an industrial system namely “Injection moulding machine”. For this, the membership function for right triangular generalized fuzzy numbers (RTrGFN) is proposed with the certain level of confidence. The real data of the Injection moulding machine is taken to validate the proposed methodology. The input data is extracted from the records/maintenance sheets of several years and found uncertainty due to the loss of any information or human mistakes. The parameters of the system like failure rate and repair time is retrieved. Based on data, AND-OR combination for the system is constructed. The lambda-tau methodology along with RTrGFN and its corresponding arithmetic operations is used for the performance analysis of the considered “Injection moulding machine”. For better understanding the results are discussed with the aid of graphs. Also after seeing the result one can conclude that this methodology is one of the best methodology for the performance analysis of a conventional system. Also authors have added the highlights and future scope of the research work at the end of the manuscript.

Keywords

Reliability indicators right triangular generalized fuzzy number alpha cuts injection moulding machine

1 Introduction

Reliability can be defined according to the perception of individuals in several ways. Reliability is the probability that when the system performs its specified task for a given period of time under defined conditions; the system will work without failure. In everyday life, the reliability plays vital role, like longtime use of computers, television, mobile phones, factories, power stations, thermal plants etc. Zio [35] stated the role of the reliability in scientific era. It is rising in both qualitative and quantitative manner in this growing world. The main objective of this paper was to focus on the addition of reliability to the safety of the system and risk analysis. In past, several authors had contributed towards the evaluation of various performance measures especially in reliability for different industrial systems as well as in other systems using different approaches. The structure system is made up of subparts/components/units/subsystems. Thus each and every subsystem plays an important role for the endurance of any structure system. As long as the human design the systems, their efforts are to design the reliable one. But failure is an unavoidable phenomenon and it leads to either failure of complete system or degradation of the system. Hence every component has the contribution in the entire working of the system to authorize the required level of reliability. Many researchers focused on finding the critical component of the structure system. Mishra [27] proposed the concepts of organization and analysis of the data, fault-tree analysis reliability modeling and its assessment methods, under environmental conditions. Dhiman and Kumar [13] calculated the reliability of the thermal power plant and sensitivity analysis to find out the most sensitive component of the system. Tsarouhas [32] calculated RAM for ice-cream industry to plan the maintenance strategy. It helped to increase the system performance. The author also found the critical components in the system. Emidu et al. [14] calculated the reliability indices for a composite power system by utilizing Monte Carlo simulation. It mainly used the probability distributions to obtain the failure rates and repair times. Choudhary et al. [11] analyzed RAM of the cement plant and found the critical components from reliability point of view and availability point of view. The authors also claimed that the study was useful for deciding the maintenance program. Aggarwal et al. [5] computed RAMD with the help of Markov birth-death process and showed the importance of the method in finding the reliability indices for each component of the system. Chinnaiyan and Somasundaram [10] found the reliability component based system using software simulation method. I showed that when reliability of subpart increases and it leads to increase in the reliability of entire system. Chen et al. [8] designed Phased-mission structures which were displayed by the different conditions like structure phases and workload condition in different phases, environment conditions which may results in the changes in structure behavior. He introduced a new methodology for fuzzy risk examination using generalized fuzzy numbers by taking distinct spreads and distinct heights. The authors calculated the reliability of a system by Binary Decision Diagram (BDD). Fodor and Bede [15] represented the overview of new approaches and compared the methods for closure. Kumar et al. [24] used the Markov method for modeling and obtained reliability and availability for the considered system. In this paper, the authors considered failure rate and repair times followed as exponential distribution. Monte-Carlo simulation was used to find the solution of the model. Levitin et al. [26] proposed an algorithm for calculating the reliability of non-repairable systems in mixed configuration and with common cause failure by using Universal generating function (UGF) and generalized reliability block diagram technique. All these methods are considered as traditional approaches.

In persistence to the study of RAM analysis and examine the other parameters of the complex structures. The future behavior of the system can be predicted with the help of available data. Generally, the data collected from the records are inaccurate and uncertain. The main reason is human errors and many researchers used the extended approach. Zadeh [33] extended the classical sets and introduced fuzzy sets and its membership grades. It was a great contribution in industrial systems. This concept added the logics to traditional approaches. Zimmermann [34] introduced the various arithmetic operations on fuzzy numbers, which can be used for various applications in different fields. Chang et al. [6] proposed vague fault tree method to compute fault interval and vague reliability of the components of the automatic gun system. Chen [7] calculated the reliability of weapon systems by fuzzy arithmetic operations and also used defuzzification method to find the crisp value for the fuzzy result. Dhiman and Garg [12] introduced some upgraded arithmetic operations on generalized trapezoidal fuzzy number with different degree of confidence level to find out the reliability indices and to preserve the flatness of fuzzy numbers. It helped to obtain the better results rather than the results obtained from existing approach. Garg [16] introduced the lambda-tau approach to analyze the performance of the repairable structure system. Petri net model is proposed instead of fault tree. The drawbacks of the existing traditional methods were covered with the help of this methodology. Garg [17] shows the importance of the score function has been presented in the interval-valued intuitionistic fuzzy sets (IVIFSs) and multi criteria decision making (MCDM) in decision making process. The author also presented the sensitivity of decision maker preferences under fuzzy environment. Garg and Ansha [18] used the distribution functions and complementary of studied distribution functions to study the arithmetic operations for two generalized parabolic fuzzy numbers. This gave the advantage over the alpha-cut method and also showed the results with comparison. Kabir et al. [19] proposed a methodology which works for systems with uncertain data. It included intuitionistic fuzzy set theory as well as expert judgment and it was used to calculate the failure probability of temporal fault trees (TFTs). Knezevic and Odoom [20] used Petri nets rather than fault tree methodology to calculate the reliability parameters using fuzzy Lambda-Tau technique under fuzzy environment. Komal [21] focused to analyze the reliability of the system namely compressor house unit (CHU), which is basically a component of coal fired thermal power plant. To carry out this study, the author used different fuzzy numbers/membership functions and T_ω-based generalized fuzzy Lambda-Tau (TBGFLT) method. The importance of this methodology was shown with the comparison with traditional methods. Komal et al. [22] showed the importance of reliability, availability and maintainability (RAM) analysis of a complex repairable industrial system. In this paper, genetic algorithms based lambda– tau (GABLT) technique was used by the authors to estimate the RAM by utilizing the uncertain data. The efficiency of this method was shown to improve the system performance. Kundu [25] considered the multi-objective reliability-redundancy allocation problem (MORRAP) of mixed configuration, which was used to compromise the solution of maximum probability of maximum probability and minimize cost of the system. To recoup the impreciseness and accuracy in the data, type-2 fuzzy numbers were used to model the interval reliabilities of the components and defuzzification technique was used to convert the fuzzy values to defuzzify values. Mon and Cheng [28] computed the fuzzy reliability of the system for its components with different membership grades. The authors used the fuzzy distribution functions in place of classical probability. Sharma [29] showed the importance of the fuzzy sets over the classical sets. The failure rates were calculated by sugeno’s fuzzy failure rate evaluation. A vague inference engine was taken to illustrate the methodology and results were compared with standard approaches. Sharma et al. [30] computed the reliability of a series parallel network. In this paper, the reliability of each component was considered as trapezoidal fuzzy number and its arithmetic operations were performed to find the fuzzy reliability of the entire system. Singhal and Sharma [31] analyzed the reliability and availability in transient state and steady state of butter oil processing machine by Markov-process and uncertainty in the data was covered by using generalized fuzzy numbers. Abu Arqub [1] presented the replicating kernel Hilbert space method for obtaining precise and numerical solutions to fuzzy Fredholm-Volterra integrodifferential equations. The results suggest that the current approach and simulated annealing give an effective scheduling mechanism for such fuzzy equations. Kumar and Dhiman [23] offer an application of improved arithmetic operations on trapezoidal fuzzy numbers to extend the dependability from point estimation to interval estimation. It can assist the decision maker in analysing the system’s behaviour and making smart decisions to lessen the likelihood of a mishap. Abu Arqub et al. [2] established and computed a recent characterization theorem with the utilization of a fuzzy strong generalized differentiability form alongside two fractional solutions. In order to combat the behaviour of fuzzy fractional numerical solutions, the conduct of error outside of the reproducing kernel theory is investigated and discussed. Abu Arqub et al. [3] utilized the fuzzy highly generalized differentiability form, a novel fuzzy characterization theorem is created and computed alongside two fuzzy fractional solutions. Beyond the reproducing kernel theory, the attitude of fuzzy fractional numerical solutions, convergence analysis, and error behaviour are investigated and argued. Alshammari et al. [4] presented a method for optimising the approximation solution by minimizing a residual function to obtain an r-level representation with a rapidly convergent series solution. The method’s influence, capacity, and feasibility are demonstrated by evaluating some applications.

It is understood after literature review that failure can’t be ignored and it is the phenomenon that can’t be avoided while analyzing the performance of the system. If the past and present record of the system is available, then, on this basis future behavior of the system can be predicted. But there is uncertainty in the existing records due to the loss of information and human errors. In literature review, many authors discussed the uncertainty with different illustrations. But the case of conventional systems has not been discussed. There exist many repairable industrial systems which has been established many years ago and still in use. The reliability parameters are the key to analyze the performance of the system. This kind of system requires maintenance in regular interval of time. Maintainability is the key for long-run of system and continuous production without any interruption.

The main objective of this paper is to analyze the performance of the conventional system under fuzzy environment. A repairable industrial system namely ‘Injection moulding machine’ is extensively studied. It was observed that system experiences a continuous failure after the initial break-in period or failure stage. But, if the collected data is referred from the state of the conventional structure system, then parameters of the subsystems/components exhibits the continuous failure rates and repair times in one direction after the failure is observed in the system. It motivates to invent a new membership number right triangular generalized fuzzy number (RTrGFN) for uncertain parameters of the system. On this basis, arithmetic operations are regenerated. With the help of these improved arithmetic operations, the performance of the conventional system is calculated under fuzzy environment. This methodology is more practical than the traditional approach. The importance of the proposed work is shown with the help of graphs.

The next section 2 describes the assumptions which are taken initially for the system and notations which are followed in the whole paper. The section 3 is showing the Numerical Illustration. Section 4 gives the proposed methodology and section 5 gives the mathematical formulation of the problem. The section 6 gives the Result and discussion. The section 7 gives the conclusion of the paper. Section 8 gives the future directions.

2 Assumptions and notations

The following assumptions have been taken for the considered system.

The structure system is conventional.

The failure rate and breakdown hours are independent to each component.

There is no statistically dependent component in the system structure.

Every component exhibits continuous failure since the failure observed for the first time.

The components of the structure system are repairable.

The components are to be assumed in perfectly working state after the repair.

The notations given in Table 1 have used in the present paper.

Table 1
Notations

λ _i Failure rate of ith component

τ _i Repair time of ith component

ℓ Level of satisfaction/ acceptance

ℓ_min Minimum of all ℓ_i, i=1,2, \dots n

λ _SS Failure rate of the structure system

τ _SS Repair time of the structure system

R _SS Reliability of the system structure

MTTF Mean time to failures

MTBF Mean time between failures

MTBF Mean time between failures

A _SS Availability of the structure system

ENOF Expected number of failures

AND Gate

OR Gate

3 Numerical illustration

The data is extracted from the Injection moulding machine-SIGMA 50-1 from Polyplastics Company situated in Haryana. It had been installed in year 2000. The record of the maintenance sheet is studied to carry out the required parameters that contain faults, failure rates and repair rates and its types for each major components of the system. The mission time is assumed to be 8640 hrs. (Approximately one year).

3.1 System description

Injection moulding machine is used for manufacturing of thermoplastic materials. It helps to give heat to the raw material/plastics and pouring it into the mould cavity by applying pressure at a given temperature without any alteration in the chemical composition. Thus required shape is obtained to set the shape of the material. It is a vast technique in industries. Primarily it is performed on glasses, elastomers and most often on thermoplastic and thermoplastic polymers, but this method is also used in the manufacturing of thermoplastic materials. This method is achieved by heating the raw material/plastics and pouring it into the mould cavity by applying pressure at a given temperature without any alteration in the chemical composition. Thus required shape is provided and cool down to set the shape of the material. The flow diagram of the process is shown in Fig. 1.

Fig. 1

Flow diagram of Injection Moulding Machine.

The main parts or subsystems of Injection Moulding Machine are:

Electric motor

Material Hopper

Injection Ram

Heating Device(Heaters)

Movable pattern

A mould cavity

Ejectors

All the parts are working in series as shown in Fig. 2 below.

Fig. 2

The systematic diagram of Injection moulding structure.

Generally, injection moulding devices work horizontally i.e. in series. The injection moulding system consists of a barrel (a cylindrical pipe). The hopper is positioned at the end of the vessel. The hydraulic ram or revolving screw is mounted within the barrel by the electric motor used to provide power. The heating device (heater) is connected to the barrel used to melt the moulding material that comes down from the hopper. A mould cavity was added to the other side of the barrel. The mould is placed within the mould cavity and a shifting pattern is used in the development process. In addition, the mould consists of copper, aluminum and machine steel.

The life cycle of the various substance moulds is special. It can be selected as per necessity. The use of injection moulding is similar to extrusion which acts as an injection as the name implies. The moulding material/raw material is pumped into the hopper by means of the feeding tool. Before the moulding material goes down to the container (barrel) under the pressure of gravity, the substance is heated beforehand with a circumferential heater on the tube. As the powder of the moulding enters the barrel from the hopper, it begins to melt and the hydraulic ram or spinning screw moves the medium forward into the mould by applying some friction. The liquid plastic substance is poured into the closed mould attached to the other side of the barrel; it is included in this broken mould. The moulding material tends to inject continuously into a rotating screw. Pressure shall refer to the hydraulic system. After injection, the pressure is exerted for some time or held in the same place with some power. Once the entire cycle has been finished, the parts created are then cooled. Then the mould is removed and some ejectors are used to detach the part properly without injury. Upon extracting the portion of the container, it is closed again. This cycle (refer to Fig. 3) is performed very rapidly and automatically.

Fig. 3

One Complete cycle of the process.

3.2 Working conditions of the system

The parameters of this procedure vary according to the circumstances and specifications.

The cycle time for the production of a single part is generally between 5 and 60 seconds depending on the manufacture of the parts.

The temperature of the moulding medium ranges from 150–350 degrees centigrade.

The injection pressure ranges from 100–150 MPa.

The weight of the components produced by this process is usually between 100 g to 500 g.

The output capacity of injection moulding shall be 12–16 thousand parts per cycle. Block diagram of the structure has been given more closely in Fig. 4 more in which ‘AND gate’ is showing that the structure will carry out its intended function properly without an interruption of a single unit.

Fig. 4

Block Diagram of Injection moulding structure.

Block diagram of the structure has been given a closer look in Fig. 5 in which ‘OR gate’ is showing that the structure may stop to carry out its intended function properly with an interruption due to any one of subparts.

Fig. 5

Block Diagram major units of Injection moulding structure.

4 Proposed methodology

4.1 Right Triangular Generalized Fuzzy Number (RTrGFN) and Alpha-cuts

A fuzzy numbers $\tilde{F} = 〈 (f_{1}, f_{2}; ℓ) | f_{1}, f_{2} \in R 〉$ , is said to be right triangular generalized fuzzy number (RTrGFN) if its membership function $μ_{\tilde{F}} (x) : R \to [0, ℓ]$ satisfies the following conditions:

$μ_{\tilde{F}} (x)$ is continuous.

If x ∈ (- ∞ , f₁] ∪ [f₂, ∞), $μ_{\tilde{F}} (x) = 0$ .

If x = f₁, $μ_{\tilde{F}} (x) = ℓ$ where 0 < ℓ ⩽1.

If x ∈ [f₁, f₂], the membership function sharply declines.

The membership function μ_A (x) : R → [0, l] is defined as

$μ_{\tilde{F}} (x) = {\begin{matrix} ℓ; x = f_{1} \\ ℓ (\frac{f_{2} - x}{f_{2} - f_{1}}); f_{1} < x ⩽ f_{2}) \\ 0; otherwise \end{matrix}$ (1)

And pictorial representation of the membership function is represented in Fig. 6.

Fig. 6

Pictorial representation of Right Triangular Generalized Fuzzy Number (RTrGFN).

Alpha cuts: Alpha cuts are the most important and often used notion in fuzzy set theory. When displaying an element x ∈ U (Universal set) that generally belongs to a fuzzy set, we may require that its membership value be greater than a certain threshold α ∈ [0, 1].

The alpha cut is given by the following equation (2) and same is represented in Fig. 7 ${\tilde{F}}_{α} = [{\tilde{F}}_{α}^{L}, {\tilde{F}}_{α}^{R}] = [α, f_{2} - \frac{α}{ℓ} (f_{2} - f_{1}); α \in (0, ℓ]]$ (2)

Fig. 7

Pictorial representation of alpha-cut.

4.2 Contemporary arithmetic operations on Right Triangular Generalized Fuzzy Number (RTrGFN)

The practical use of fuzzified operations is shown to be straightforward, requiring no more computation than in the classical tolerance analysis when dealing with error intervals.

Let F = (f₁, f₂ ; l_F) and G = (g₁, g₂ ; l_G) be the two right triangular fuzzy numbers. Their membership functions are defined as $\begin{matrix} μ_{F} = {\begin{matrix} l_{F}; x = f_{1} \\ l_{F} (\frac{f_{2} - x}{f_{2} - f_{1}}); f_{1} < x ⩽ f_{2} \\ 0; otherwise \end{matrix} \\ μ_{G} = {\begin{matrix} l_{G}; x = g_{1} \\ l_{G} (\frac{g_{2} - x}{g_{2} - g_{1}}); g_{1} < x ⩽ g_{2} \\ 0; otherwise \end{matrix} \end{matrix}$ (3)

Where f₁, f₂, g₁, g₂ ∈ R and 0 ⩽ l_F, l_G ⩽ 1

The right triangular fuzzy numbers can be transformed to right triangular generalized fuzzy numbers as follows. $\begin{matrix} F \to F^{*} and G \to G^{*}; l = l_{min} = min (l_{F}, l_{G}) \\ F^{*} = F = (f_{1}, f_{1}, f_{2}; l_{F}); G^{*} = (g_{1}, g_{2}^{*}, g_{2}; l_{G}) \\ where g_{2}^{*} = g_{1} + l (\frac{g_{2} - g_{1}}{L_{G}}) \end{matrix}$ (4)

Operation 1: Scalar multiplication of two numbers

Scalar multiplication of right triangular generalized fuzzy numbers F = (f₁, f₂ ; l_F) with confidence level generates a right triangular generalized fuzzy number.

K = kF = (K₁, K₂ ; l_F) ; k ⩾ 0.

where $\begin{matrix} K_{1} = {kf}_{1} \\ K_{2} = {kf}_{2} \end{matrix}$

$K^{*} = kF = (K_{1}^{*}; K_{2}^{*}; l_{F}); k ⩽ 0, k = - n (say)$ .

where $\begin{matrix} K_{1}^{*} = - {nf}_{2} \\ K_{2}^{*} = - {nf}_{1} \end{matrix}$

Operation 2: Addition of two numbers

Addition of two right triangular generalized fuzzy numbers F = (f₁, f₂ ; l_F) and G = (g₁, g₂ ; l_G) confidence levels generates a right triangular generalized fuzzy number A = F + G = (A₁, A₂, A₃ ; l) where

$\begin{matrix} A_{1} = f_{1} + g_{1} \\ A_{2} = f_{1} + (g_{1} + l (\frac{g_{2} - g_{1}}{l_{G}})) \\ A_{3} = f_{2} + g_{2} \end{matrix}$

Operation 3: Subtraction of two numbers

Subtraction of two right triangular generalized fuzzy numbers F = (f₁, f₂ ; l_F) and G = (g₁, g₂ ; l_G) with two different confidence levels generates a right triangular generalized fuzzy number S = F - G = (S₁, S₂, S₃ ; l) where

$\begin{matrix} S_{1} = f_{1} - g_{2} \\ S_{2} = f_{1} - (g_{1} + θ (\frac{g_{2} - g_{1}}{l_{G}})) \\ S_{3} = f_{2} - g_{1} \end{matrix}$

Operation 4: Multiplication of two numbers

Multiplication of two right triangular generalized fuzzy numbers F = (f₁, f₂ ; l_F) and G = (g₁, g₂ ; l_G) with two different confidence levels generates a right triangular generalized fuzzy number M = F ⊗ G = (M₁, M₂, M₃ ; l) where

$\begin{matrix} M_{1} = f_{1} * g_{1} \\ M_{2} = f_{1} * (g_{1} + l (\frac{g_{2} - g_{1}}{l_{G}})) \\ M_{3} = f_{2} * g_{2} \end{matrix}$

Operation 5: Division of two numbers

Division of two right triangular generalized fuzzy numbers F = (f₁, f₂ ; l_F) and G = (g₁, g₂ ; l_G) with two different confidence levels generates a right triangular generalized fuzzy number D = F/G = (D₁, D₂, D₃ ; l)

where

$\begin{matrix} D_{1} = \frac{f_{1}}{g_{2}} \\ D_{2} = \frac{f_{1}}{(g_{1} + l (\frac{g_{2} - g_{1}}{l_{G}}))} \\ D_{3} = \frac{f_{2}}{g_{1}} \end{matrix}$ .

Operation 6: Complement of a number

Complement of a right triangular generalized fuzzy number F = (f₁, f₂ ; l_F) is

C = (C₁, C₂ ; l) where

$\begin{matrix} C_{1} = 1 - f_{2} \\ C_{2} = 1 - f_{1} \end{matrix}$

4.2.1 Cases for operations

Let F and G are two right triangular generalized fuzzy number on which the operations to be applied. Based upon the satisfaction level, three cases arise and which are discussed in this section.

Case 1: Let F = (f₁, f₂ ; ℓ ₁) and G = (g₁, g₂ ; ℓ ₂) be two Right Triangular Generalized Fuzzy Numbers with same satisfaction level as shown in Fig. 8.

Fig. 8

Two distinct Right Triangular Generalized Fuzzy Number (RTrGFN).

It is the case when level of satisfaction for F and G are equal.

The corresponding membership functions of F and G are given as: $\begin{matrix} μ_{F} = {\begin{matrix} ℓ; x = f_{1} \\ ℓ \cdot (\frac{f_{2} - x}{f_{2} - f_{1}}); x \in (f_{1}, f_{2}) \\ 0; otherwise \end{matrix} \\ μ_{G} = {\begin{matrix} ℓ; x = g_{1} \\ ℓ \cdot (\frac{g_{2} - x}{g_{2} - g_{1}}); x \in (g_{1}, g_{2}) \\ 0; otherwise \end{matrix} \end{matrix}$ (5)

The following arithmetic operations can be carried out:

•Addition of two numbers: $A = F + G = (f_{1} + g_{1}, f_{2} + g_{2}; ℓ)$ (6)

•Subtraction of two numbers: $S = F - G = (f_{1} - g_{2}, f_{2} - g_{1}; ℓ)$ (7)

•Multiplication of two numbers: $M = F * G = (f_{1} * g_{1}, f_{2} * g_{2}; ℓ)$ (8)

•Division of two numbers: $D = F / G = (f_{1} / g_{2}, f_{2} / g_{1}; ℓ)$ (9)

Alpha cut of Right Triangular Generalized Fuzzy Number (RTrGFN)

Let F = (f₁, f₂ ; l) be any number, then alpha cut of F is given by $A_{α} = [A_{α}^{L}, A_{α}^{R}] = (f_{1}, f_{2} - \frac{α}{ℓ} (f_{2} - f_{1}))$ (10)

Case 2: ℓ₁ ≠ ℓ ₂ ℓ₁ < ℓ ₂ ; ℓ _min = min(ℓ ₁, ℓ ₂) = ℓ ₁.

Therefore, G = (g₁, g₂ ; ℓ ₂) can be modified as $G^{*} = (g_{1}^{*}, g_{2}^{*}, g_{3}^{*}; ℓ_{1})$ according to the minimum weight and the same is shown in Fig. 9.

Fig. 9

Generalization of G and transformation of G to G^*.

Where $g_{1}^{*} = g_{1}, g_{2}^{*} = g_{1} + ℓ_{min} (\frac{g_{2} - g_{1}}{ℓ_{2}}), g_{3}^{*} = g_{2}$ (11)

Also F can be transformed by using [15] as $F = (f_{1}, f_{2}, f_{2}; ℓ_{1})$ (12)

After the transformation of F and G as shown in Fig. 9, their membership functions are given by $\begin{matrix} μ_{F} = {\begin{matrix} ℓ_{1}; x = f_{1} \\ ℓ_{1} \cdot (\frac{f_{2} - x}{f_{2} - f_{1}}); (f_{1}, f_{2}) \\ 0; otherwise \end{matrix} \\ μ_{G}^{*} = {\begin{matrix} ℓ_{1}; (g_{1}^{*}, g_{2}^{*}) \\ ℓ_{1} \cdot (\frac{g_{3}^{*} - x}{g_{3}^{*} - g_{2}^{*}}); (g_{2}^{*}, g_{3}^{*}) \\ 0; otherwise \end{matrix} \end{matrix}$ (13)

The following arithmetic operations can be carried out:

•Addition of two numbers $A = F + G * = (f_{1} + g_{1}^{*}, f_{2} + g_{2}^{*}, f_{2} + g_{3}^{*}; ℓ_{1})$ (14)

•Subtraction of two numbers: $S = F - G^{*} = (f_{1} - g_{3}^{*}, f_{2} - g_{2}^{*}, f_{2} - g_{1}^{*}; ℓ_{1})$ (15)

•Multiplication of two numbers: $M = F * G^{*} = (f_{1} * g_{1}^{*}, f_{2} * g_{2}^{*}, f_{2} * g_{3}^{*}; ℓ_{1})$ (16)

•Division of two numbers: $D = F / G^{*} = (f_{1} / g_{3}^{*}, f_{2} / g_{2}^{*}, f_{2} / g_{1}^{*}; ℓ_{1})$ (17)

Case-3: If the two generalized fuzzy numbers F and G are obtained after ℓ - cut the level as shown in Fig. 10. $\begin{matrix} μ_{F} = {\begin{matrix} ℓ; (f_{1}, f_{2}) \\ ℓ (\frac{f_{3} - x}{f_{3} - f_{2}}); (f_{2}, f_{3}) \\ 0; otherwise \end{matrix} \\ μ_{G} = {\begin{matrix} ℓ; (g_{1}, g_{2}) \\ ℓ (\frac{g_{3} - x}{g_{3} - g_{2}}); (g_{2}, g_{3}) \\ 0; otherwise \end{matrix} \end{matrix}$ (18)

Fig. 10

Two transformed generalized fuzzy numbers F and G are obtained after l - cut.

The following arithmetic operations can be carried out:

•Addition of two numbers: $A = F + G = (f_{1} + g_{1}, f_{2} + g_{2}, f_{3} + g_{3}; ℓ)$ (19)

•Subtraction of two numbers: $S = F - G = (f_{1} - g_{3}, f_{2} - g_{2}, f_{3} - g_{1}; ℓ)$ (20)

•Multiplication of two numbers: $M = F * G = (f_{1} * g_{1}, f_{2} * g_{2}, f_{3} * g_{3}; ℓ)$ (21)

•Division of two numbers: $D = F / G = (f_{1} / g_{3}, f_{2} / g_{2}, f_{3} / g_{1}; ℓ)$ (22)

Alpha cut of Right Triangular Generalized Fuzzy Number (RTrGFN)

Let F = (f₁, f₂, f₃ ; l) be any number, and then alpha cut of F is given by $A_{α} = [A_{α}^{L}, A_{α}^{R}] = (f_{1}, f_{3} - \frac{α}{ℓ} (f_{3} - f_{2}))$ (23)

5 Mathematical Formulation of the problem

Firstly, the data from the record was collected in a tabular form. From the maintenance sheets, the breakdown hours were extracted. The complete record is shown with the help of Table 2.

Table 2
Data of maintenance sheets records regarding failure rates and repair rates

Faults observed in Electric Motor(A) Maintenance carried out Failure rate Breakdown hours (B/D hrs) Type of maintenance

Motor not start (A₁) Fuse replaced 0.0001 1 Corrective Electrical

Discontinuation of M/C’s heating due to power cut(A₂) Input Problem of module rectified 0.0001 0.33 Corrective Electrical

Motor not run(A₃) Plc module cleaned and tightened 0.0001 1.25 Corrective Electrical

Motor stop working time to time(A₄) Modules cleaned and resetting carried out 0.0013 1 Corrective Electrical

Main motor tripping problem(A₅) Modules cleaned and refitted 0.0001 0.75 Corrective Electrical

Faults observed in Material Hopper(B) Maintenance carried out Failure rate Breakdown hours (B/D hrs) Type of maintenance

Material stick problem in machine hopper glass(B₁) Hopper loader cleaned and refitted glass 0.0004 3.2 Corrective Electrical

Refilling not working (B₂) Lumps formation removed from hopper 0.0004 2.33 Corrective Electrical

Refilling pressure not working(B₃) All modules cleaned and factory setup setting carried out 0.0004 2.50 Corrective Electrical

Faults observed in Injection Ram(C) Maintenance carried out Failure rate Breakdown hours(B/D hrs) Type of maintenance

Injection not working(C₁) Clamp transducer setting carried out 0.0013 1 Corrective Electrical

Injection not working(C₂) Injection relay changed and module cleaned 0.0013 2.67 Corrective Electrical

Injection unit and Injection not working(C₃) Purge shield changed 0.0013 1.67 Corrective Electrical

Pressure Problem in machine(C₄) Pressure settings carried out 0.0013 3 Corrective Electrical

Pressure not working(C₅) DC valve connector tightened 0.0013 1 Corrective Electrical

Locking Pressure not working(C₆) Refilling DC valve interchanged 0.0013 2 Corrective Electrical

Locking Pressure not working(C₇) Toggle tightened to increase valve 0.0013 5.42 Corrective Mechanical

Faults observed in Heaters (each zone) Z (i) , i = 1, 2, 3 Maintenance carried out Failure rate Breakdown hours (B/D hrs) Type of maintenance

Discontinuation of heating (D₁ in Z1, D₄ in Z2, D₇ in Z3) Heating wiring fault rectified 0.0013 0.67 Corrective Electrical

Heating off after power Cut(D₂ in Z1, D₅ in Z2, D₈ in Z3) Module bases replaced 0.0013 1.17 Corrective Electrical

Temperature variation(D₃ in Z1, D₆ in Z2, D₉ in Z3) Temperature setting carried out 0.0013 1.5 Corrective Electrical

Faults observed in Movable pattern(E) Maintenance carried out Failure rate Breakdown hours (B/D hrs) Type of maintenance

Clamp force not working(E₁) Valve of wire repair and connector change 0.0013 3 Corrective Electrical

Clamp opening pressure not working(E₂) Clamping transducer wire fault rectified 0.0013 2 Corrective Electrical

Faults observed in a mould cavity(F) Maintenance carried out Failure rate Breakdown hours (B/D hrs) Type of maintenance

Mould not closed(F₁) Software loaded and transducer calibration carried out 0.0013 17 Corrective Electrical

Mould not closed(F₂) Mould open close valve cleaned 0.0013 2.50 Corrective Electrical

Mould opening problem(F₃) Module cleaned and resetting carried out 0.0013 1.17 Corrective Electrical

Mould opening problem(F₄) Rest and transducer checking carried out 0.0013 1.17 Corrective Electrical

Faults observed in Ejectors(G) Maintenance carried out Failure rate Breakdown hours (B/D hrs) Type of maintenance

Ejector stay forward not working(G₁) Ejector valve, relief valve and check valve cleaned 0.0002 2.50 Corrective Electrical

Ejector not working and oil leakage(G₂) Hydraulic hose pipe replaced 0.0002 1.42 Corrective Mechanical

Oil leakage from barrel unit(G₃) Fitting of drain pipe 0.0002 0.25 Corrective Mechanical

Faults observed in Electric Motor(A)	Maintenance carried out	Failure rate	Breakdown hours (B/D hrs)	Type of maintenance
Motor not start (A₁)	Fuse replaced	0.0001	1	Corrective	Electrical
Discontinuation of M/C’s heating due to power cut(A₂)	Input Problem of module rectified	0.0001	0.33	Corrective	Electrical
Motor not run(A₃)	Plc module cleaned and tightened	0.0001	1.25	Corrective	Electrical
Motor stop working time to time(A₄)	Modules cleaned and resetting carried out	0.0013	1	Corrective	Electrical
Main motor tripping problem(A₅)	Modules cleaned and refitted	0.0001	0.75	Corrective	Electrical
Faults observed in Material Hopper(B)	Maintenance carried out	Failure rate	Breakdown hours (B/D hrs)	Type of maintenance
Material stick problem in machine hopper glass(B₁)	Hopper loader cleaned and refitted glass	0.0004	3.2	Corrective	Electrical
Refilling not working (B₂)	Lumps formation removed from hopper	0.0004	2.33	Corrective	Electrical
Refilling pressure not working(B₃)	All modules cleaned and factory setup setting carried out	0.0004	2.50	Corrective	Electrical
Faults observed in Injection Ram(C)	Maintenance carried out	Failure rate	Breakdown hours(B/D hrs)	Type of maintenance
Injection not working(C₁)	Clamp transducer setting carried out	0.0013	1	Corrective	Electrical
Injection not working(C₂)	Injection relay changed and module cleaned	0.0013	2.67	Corrective	Electrical
Injection unit and Injection not working(C₃)	Purge shield changed	0.0013	1.67	Corrective	Electrical
Pressure Problem in machine(C₄)	Pressure settings carried out	0.0013	3	Corrective	Electrical
Pressure not working(C₅)	DC valve connector tightened	0.0013	1	Corrective	Electrical
Locking Pressure not working(C₆)	Refilling DC valve interchanged	0.0013	2	Corrective	Electrical
Locking Pressure not working(C₇)	Toggle tightened to increase valve	0.0013	5.42	Corrective	Mechanical
Faults observed in Heaters (each zone) Z (i) , i = 1, 2, 3	Maintenance carried out	Failure rate	Breakdown hours (B/D hrs)	Type of maintenance
Discontinuation of heating (D₁ in Z1, D₄ in Z2, D₇ in Z3)	Heating wiring fault rectified	0.0013	0.67	Corrective	Electrical
Heating off after power Cut(D₂ in Z1, D₅ in Z2, D₈ in Z3)	Module bases replaced	0.0013	1.17	Corrective	Electrical
Temperature variation(D₃ in Z1, D₆ in Z2, D₉ in Z3)	Temperature setting carried out	0.0013	1.5	Corrective	Electrical
Faults observed in Movable pattern(E)	Maintenance carried out	Failure rate	Breakdown hours (B/D hrs)	Type of maintenance
Clamp force not working(E₁)	Valve of wire repair and connector change	0.0013	3	Corrective	Electrical
Clamp opening pressure not working(E₂)	Clamping transducer wire fault rectified	0.0013	2	Corrective	Electrical
Faults observed in a mould cavity(F)	Maintenance carried out	Failure rate	Breakdown hours (B/D hrs)	Type of maintenance
Mould not closed(F₁)	Software loaded and transducer calibration carried out	0.0013	17	Corrective	Electrical
Mould not closed(F₂)	Mould open close valve cleaned	0.0013	2.50	Corrective	Electrical
Mould opening problem(F₃)	Module cleaned and resetting carried out	0.0013	1.17	Corrective	Electrical
Mould opening problem(F₄)	Rest and transducer checking carried out	0.0013	1.17	Corrective	Electrical
Faults observed in Ejectors(G)	Maintenance carried out	Failure rate	Breakdown hours (B/D hrs)	Type of maintenance
Ejector stay forward not working(G₁)	Ejector valve, relief valve and check valve cleaned	0.0002	2.50	Corrective	Electrical
Ejector not working and oil leakage(G₂)	Hydraulic hose pipe replaced	0.0002	1.42	Corrective	Mechanical
Oil leakage from barrel unit(G₃)	Fitting of drain pipe	0.0002	0.25	Corrective	Mechanical

Step 2: Conversion of crisp number into fuzzy number

Fuzzy numerical data can be viewed using real-line fuzzy subsets, defined as fuzzy numbers. Here the failure rate is represented by lambda (λ) and repair time is represented by tau(τ). Take 15% spread on right side for both the parameters as given in Fig. 11. Now the fuzzified values of λ and τ are represented as (λ, λ + 15 %) and (τ, τ + 15 %) respectively.

Fig. 11

15% spread of failure rate and repair time.

Step 3: Generalized the fuzzified data to accurate level of satisfaction

In the following Table 6, failure rate and repair times are given in right triangular generalized fuzzy numbers up to same level of satisfaction as defined in section 4.2 (Case 2).

Generalizing of data given in Table 3.

Table 3

Data represented as Right Triangular Fuzzy number (RTrFN)

Components	Failure rates	Repair times
A₁	(0.0001, 0.000115; 0.9)	(1, 1.15; 0.9)
A₂	(0.0001, 0.000115; 0.9)	(0.33, 0.3795; 0.9)
A₃	(0.0001, 0.000115; 0.9)	(1.25, 1.4375; 0.9)
A₄	(0.0013, 0.001495; 0.8)	(1, 1.15; 0.8)
A₅	(0.0001, 0.000115; 0.8)	(0.75, 0.8625; 0.8)
B₁	(0.0004, 0.00046; 0.8)	(3.2, 3.68; 0.8]
B₂	(0.0004, 0.00046; 0.9)	(2.33, 2.6795; 0.9)
B₃	(0.0004, 0.00046; 0.8)	(2.5, 2.875; 0.8)
C₁	(0.0013, 0.001495; 0.9)	(1, 1.15; 0.9)
C₂	(0.0013, 0.001495; 0.9)	(2.67, 3.0705; 0.9)
C₃	(0.0013, 0.001495; 0.9)	(1.67, 1.9205; 0.9)
C₄	(0.0013, 0.001495; 0.9)	(3, 3.45; 0.9)
C₅	(0.0013, 0.001495; 0.8)	(1, 1.15; 0.8)
C₆	(0.0013, 0.001495; 0.8)	(2, 2.3; 0.8)
C₇	(0.0013, 0.001495; 0.8)	(5.42, 6.233; 0.8)
D₁ =D₄ =D₇	(0.0013, 0.001495; 0.8)	(0.67, 0.7705; 0.8)
D₂ =D₅ =D₈	(0.0013, 0.001495; 0.8)	(1.17, 1.3455; 0.8)
D₃ =D₆ =D₉	(0.0013, 0.001495; 0.9)	(1.5, 1.725; 0.9)
E₁	(0.0013, 0.001495; 0.9)	(3, 3.45; 0.9)
E₂	(0.0013, 0.001495; 0.8)	(2, 2.3; 0.8)
F₁	(0.0013, 0.001495; 0.8)	(17, 19.55; 0.8)
F₂	(0.0013, 0.001495; 0.8)	(2.5, 2.875; 0.8)
F₃	(0.0013, 0.001495; 0.9)	(1.17, 1.3455; 0.9)
F₄	(0.0013, 0.001495; 0.9)	(1.17, 1.3455; 0.9)
G₁	(0.0002, 0.00023; 0.9)	(2.5, 2.875; 0.9)
G₂	(0.0002, 0.00023; 0.8)	(1.42, 1.633; 0.8)
G₃	(0.0002, 0.00023; 0.8)	(0.25, 0.2875; 0.8)

Step 4: Obtaining failure rate and repair time of the structure system using lambda tau methodology and fault tree analysis.

Table 4

Data represented as Right Triangular Generalized Fuzzy number (RTrGFN)

Components	Failure rates	Repair times
A₁^*	(0.0001, 0.000113333, 0.000115; 0.8)	(1, 1.133333333, 1.15; 0.8)
A₂^*	(0.0001, 0.000113333, 0.000115; 0.8)	(0.33, 0.374, 0.3795; 0.8)
A₃^*	(0.0001, 0.000113333, 0.000115; 0.8)	(1.25, 1.416666667, 1.4375; 0.8)
A₄^*	(0.0013, 0.001495, 0.001495; 0.8)	(1, 1.15, 1.15; 0.8)
A₅^*	(0.0001, 0.000115, 0.000115; 0.8)	(0.75, 0.8625, 0.8625; 0.8)
B₁^*	(0.0004, 0.00046, 0.00046; 0.8)	(3.2, 3.68, 3.68; 0.8]
B₂^*	(0.0004, 0.000453333, 0.00046; 0.8)	(2.33, 2.640666667, 2.6795; 0.8)
B₃^*	(0.0004, 0.00046, 0.00046; 0.8)	(2.5, 2.875, 2.875; 0.8)
C₁^*	(0.0013, 0.001473333, 0.001495; 0.8)	(1, 1.133333333, 1.15; 0.8)
C₂^*	(0.0013, 0.001473333, 0.001495; 0.8)	(2.67, 3.026, 3.0705; 0.8)
C₃^*	(0.0013, 0.001473333, 0.001495; 0.8)	(1.67, 1.892666667, 1.9205; 0.8)
C₄^*	(0.0013, 0.001473333, 0.001495; 0.8)	(3, 3.4, 3.45; 0.8)
C₅^*	(0.0013, 0.001473333, 0.001495; 0.8)	(1, 1.15, 1.15; 0.8)
C₆^*	(0.0013, 0.001495, 0.001495; 0.8)	(2, 2.3, 2.3; 0.8)
C₇	(0.0013, 0.001495, 0.001495; 0.8)	(5.42, 6.233, 6.233; 0.8)
D₁^* =D₄^=D₇^	(0.0013, 0.001495, 0.001495; 0.8)	(0.67, 0.7705, 0.7705; 0.8)
D₂^* =D₅^* =D₈^*	(0.0013, 0.001495, 0.001495; 0.8)	(1.17, 1.3455, 1.3455; 0.8)
D₃^* =D₆^=D₉^	(0.0013, 0.001473333, 0.001495; 0.8)	(1.5, 1.7, 1.725; 0.8)
E₁^*	(0.0013, 0.001473333, 0.001495; 0.8)	(3, 3.4, 3.45; 0.8)
E₂^*	(0.0013, 0.001495, 0.001495; 0.8)	(2, 2.3, 2.3; 0.8)
F₁^*	(0.0013, 0.001495, 0.001495; 0.8)	(17, 19.55, 19.55; 0.8)
F₂^*	(0.0013, 0.001495, 0.001495; 0.8)	(2.5, 2.875, 2.875; 0.8)
F₃^*	(0.0013, 0.001473333, 0.001495; 0.8)	(1.17, 1.326, 1.3455; 0.8)
F₄^*	(0.0013, 0.001473333, 0.001495; 0.8)	(1.17, 1.326, 1.3455; 0.8)
G₁^*	(0.0002, 0.000226667, 0.00023; 0.8)	(2.5, 2.833333333, 2.875; 0.8)
G₂^*	(0.0002, 0.00023, 0.00023; 0.8)	(1.42, 1.633, 1.633; 0.8)
G₃^*	(0.0002, 0.00023, 0.00023; 0.8)	(0.25, 0.2875, 0.2875; 0.8)

The Lambda-Tau methodology is a solution strategy for dealing with repairable systems in the fault tree model. The approach demands redundant-free expressions from the model, which means that the basic events of the model must not be repeated. The basic formulas for failure rate and repair time related with the fault tree model’s logical AND- and OR-gates are derived.

With the help of AND-OR gates in Fig. 5, we can obtain the resultant expressions for the resultant values of failure rates and repair time individually for each parameter by using Table 5.

Table 5

Basic lambda-tau expressions

Gate	AND		OR
	λ _AND	τ _AND	λ _OR	τ _OR
Expressions	$\prod_{j = 1}^{n} λ_{j} [\sum_{i = 1}^{n} \prod_{\begin{array}{l} j = 1 \\ i \neq j \end{array}}^{n} τ_{j}]$	$\frac{\prod_{i = 1}^{n} τ_{j}}{\sum_{j = 1}^{n} [\prod_{\begin{array}{l} j = 1 \\ i \neq j \end{array}}^{n} τ_{i}]}$	$\sum_{i = 1}^{n} λ_{i}$	$\frac{\sum_{i = 1}^{n} λ_{i} τ_{i}}{\sum_{i = 1}^{n} λ_{i}}$

Based on the above Table 5, the expressions for the failure rate and repair time of the structure system can be obtained as: $λ_{SS} = λ A + λ B + λ C + λ Z + λ E + λ F + λ G$ (24)

$τ_{SS} = \frac{λ A * τ A + λ B * τ B + λ C * τ C + λ Z * τ Z + λ E * τ E + λ F * τ F + λ G * τ G}{λ_{SS}}$ (25)

Therefore, based on the above Table 6 and using the arithmetic operations defined in section 4.2 (Case 3), the failure rate and repair time of the structure system can be obtained and shown in Table 7.

Table 6

Failure rates and Repair time of the major components represented as RTrGFN

S. No.	Component	Failure rate	Repair time
1.	Electric motor	(0.0288 × 10^-17, 0.0938 × 10^-17, 0.1014 × 10^-17 ; 0.8)	(0.0798 0.1590 0.2808; 0.8)
2.	Material Hopper	(0.2128 × 10^-10, 0.4060 × 10^-10, 0.4280 × 10^-10 ; 0.8)	(0.6623 1.0017 1.3321; 0.8)
3.	Injection Ram	(0.0279 × 10^-13, 0.1361 × 10^-13, 0.1495 × 10^-13 ; 0.8)	(0.1083 0.2857 0.6665; 0.8)
4.	Heating Device(Heater)	(0.2337 × 10^-7, 0.4578 × 10^-7, 0.4698 × 10^-7 ; 0.8)	(0.1248 0.3803 1.0144; 0.8)
5.	Movable pattern	(0.0845 × 10^-4, 0.1256 × 10^-4, 0.1285 × 10^-4 ; 0.8)	(1.0435 1.3719 1.5870; 0.8)
6.	A mould cavity	(0.3603 × 10^-9, 0.9145 × 10^-9, 0.9583 × 10^-9 ; 0.8)	(0.3033 0.5243 0.8067; 0.8)
7.	Ejectors	(0.3624 × 10^-10, 0.7088 × 10^-10, 0.7289 × 10^-10 ; 0.8)	(0.1481 0.2250 0.2980; 0.8)

Table 7

Failure rates and Repair time of structure system represented as RTrGFN

λ _SS	(0.0847 × 10^-4, 0.1261 × 10^-4, 0.1290 × 10^-4)
τ _SS	(0.6838, 1.3679, 2.4134)

Now expressions for reliability parameters are given (Table 8) as

Table 8

Expressions of Reliability indices

Parameters	Expressions
MTTF	1/λ_SS
MTTR	1/μ_SS = τ_SS
MTBF	MTTF+MTTR = 1/λ_SS + τ_SS
R _SS	e ^-λ_SS*t
A _SS	$\frac{μ_{SS}}{μ_{SS} + λ_{SS}} + \frac{λ_{SS}}{μ_{SS} + λ_{SS}} e^{- (μ_{SS} + λ_{SS}) * t}$
ENOF	$\frac{μ_{SS} \cdot λ_{SS} \cdot t}{μ_{SS} + λ_{SS}} + \frac{λ_{SS}^{2}}{{(μ_{SS} + λ_{SS})}^{2}} [1 - e^{- (μ_{SS} + λ_{SS}) * t}]$

Resultant values of fuzzy reliability parameters are calculated and given in Table 9

Table 9

Fuzzified values of Reliability indices

S. No	Parameters	Fuzzy values of parameters
1.	MTTF	(0.7752 × 10⁵, 0.7930 × 10⁵, 1.1806 × 10⁵)
2.	MTTR	(0.6838, 1.3679, 2.4134)
3.	MTBF	(0.7752 × 10⁵, 0.7930 × 10⁵, 1.1807 × 10⁵)
4.	R _SS	(0.8945,0.8968, 0.9296)
5.	A _SS	(0.2833,0.9999,3.52903)
6.	ENOF	(0.0207, 0.1089, 0.3933)

6 Results and discussion

The value of each reliability parameter is calculated up to a confidence level 80% which means that authors are 80% confident that the true value of the parameter lies within the given range. The Confidence interval is generally based on standard error of the mean and obtained by multiplying the critical value with standard error. It provides more transparency for readers to better understand the reliability of the parameter values presented in the study. The reliability parameters with alpha-cuts from 0 to 0.8 have been shown in the following Table 10.

Table 10
Alpha-cuts of reliability parameters

Alpha MTTF MTTR MTBF

Left Spread Right Spread Left Spread Right Spread Left Spread Right Spread

0.8 77520 79300 0.6838 1.3679 77520 79300

0.7 77520 84145 0.6838 1.498588 77520 84146.25

0.6 77520 88990 0.6838 1.629275 77520 88992.5

0.5 77520 93835 0.6838 1.759963 77520 93838.75

0.4 77520 98680 0.6838 1.89065 77520 98685

0.3 77520 103525 0.6838 2.021338 77520 103531.3

0.2 77520 108370 0.6838 2.152025 77520 108377.5

0.1 77520 113215 0.6838 2.282713 77520 113223.8

0 77520 118060 0.6838 2.4134 77520 118070

Alpha Reliability Availability ENOF

Left Spread Right Spread Left Spread Right Spread Left Spread Right Spread

0.8 0.8945 0.8968 0.2833 0.9999 0.0207 0.1089

0.7 0.8945 0.9009 0.2833 1.316041 0.0207 0.14445

0.6 0.8945 0.905 0.2833 1.632183 0.0207 0.18

0.5 0.8945 0.9091 0.2833 1.948324 0.0207 0.21555

0.4 0.8945 0.9132 0.2833 2.264465 0.0207 0.2511

0.3 0.8945 0.9173 0.2833 2.580606 0.0207 0.28665

0.2 0.8945 0.9214 0.2833 2.896748 0.0207 0.3222

0.1 0.8945 0.9255 0.2833 3.212889 0.0207 0.35775

0 0.8945 0.9296 0.2833 3.52903 0.0207 0.3933

Alpha	MTTF	MTTR	MTBF
0.8	77520	79300	0.6838	1.3679	77520	79300
0.7	77520	84145	0.6838	1.498588	77520	84146.25
0.6	77520	88990	0.6838	1.629275	77520	88992.5
0.5	77520	93835	0.6838	1.759963	77520	93838.75
0.4	77520	98680	0.6838	1.89065	77520	98685
0.3	77520	103525	0.6838	2.021338	77520	103531.3
0.2	77520	108370	0.6838	2.152025	77520	108377.5
0.1	77520	113215	0.6838	2.282713	77520	113223.8
0	77520	118060	0.6838	2.4134	77520	118070
Alpha	Reliability	Availability	ENOF
	Left Spread	Right Spread	Left Spread	Right Spread	Left Spread	Right Spread
0.8	0.8945	0.8968	0.2833	0.9999	0.0207	0.1089
0.7	0.8945	0.9009	0.2833	1.316041	0.0207	0.14445
0.6	0.8945	0.905	0.2833	1.632183	0.0207	0.18
0.5	0.8945	0.9091	0.2833	1.948324	0.0207	0.21555
0.4	0.8945	0.9132	0.2833	2.264465	0.0207	0.2511
0.3	0.8945	0.9173	0.2833	2.580606	0.0207	0.28665
0.2	0.8945	0.9214	0.2833	2.896748	0.0207	0.3222
0.1	0.8945	0.9255	0.2833	3.212889	0.0207	0.35775
0	0.8945	0.9296	0.2833	3.52903	0.0207	0.3933

Here the interval values in the Table 10 have been given with the different satisfaction level. The value of MTTF lies in the interval (77520, 79300) with the confidence level 0.8 and below this confidence level, the lower value of every alpha cut remains same and upper value of alpha cut increases as displayed in Fig. 12. And similarly for other parameters, this nature has been observed. The value of MTTR lies in the interval (0.6838, 1.3679) as displayed in Fig. 13, the value of MTBF lies in the interval (77520, 79300) as displayed in Fig. 14, the value of reliability lies in the interval (0.8945, 0.8968) as displayed in Fig. 15, the value of availability lies in the interval (2.833, 0.9999) as displayed in Fig. 16, the value of ENOF lies in the interval (0.0207, 0.1089) as displayed in Fig. 17 with the confidence level 0.8 and lower value remains unchanged for all confidence level below this. The graphs have clearly followed the effects. The main significance of the these values are that after achieving the required degree of satisfaction, the structure’s reliability and availability drop consistently with increasing time. It remains the same for the Critical interval; with the desired level of satisfaction, and then the graph begins to fall rapidly on \hfilneg \hrule \hrule the right side of the triangle. Eventually these graphs shows that with maximum satisfaction level that is 0.8, the reliability parameters attain the most accurate range values and after that these are accomplishing the range with same lower value in the range of reliability parameters but different highest values which is increasing along with decreasing in level of satisfaction. The range values with 0.8 satisfaction level is taken into consideration for practical results.

Fig. 12

MTTF.

Fig. 13

MTTR.

Fig. 14

MTBF.

Fig. 15

Reliability.

Fig. 16

Availability.

Fig. 17

ENOF.

7 Conclusion

Present research reflects the importance of Right Triangular Generalized fuzzy number along with the alpha cuts in context to performance analysis of a conventional industrial structure under the fuzzy environment. The proposed membership function is more convenient to use in practice for this kind of system in which system experiences a continuous failure after the initial break-in period or failure stage. The consequences have been followed by the graphs evidently. After attaining desired level of satisfaction, reliability and availability of the structure decreases uniformly with increase in time. From each graph (Figs. 12–17), one can see that the minimum value of reliability parameter is fix with 80% satisfaction level. It remains same for the Critical interval; with desired level of satisfaction, and then the graph starts falling rapidly on right side of triangle. Basically, it shows that the value of the parameter is increasing with the increase in time. Although, resultant value of each parameter is increasing but its satisfaction level is low. Undoubtedly, performance analysis under fuzzy environment provides more flexibility for making decisions. This experiment helps the plant personal to plan the maintenance of subparts in accordance to attain the reliability and availability of the structure system with highest level of satisfaction that is 80%. So that structure system works for the expected time for disaffected production without any interruption of failure. The next section defines the future directions for the research.

8 Future direction/research

In future, the working and failure culture of more systems will be focused. The new membership functions will be defined to analyze the system’s future performance based upon the available information.

Footnotes

Acknowledgments

This work is supported by Symbiosis Institute of Technology, Symbiosis International (Deemed University) (SIU), Pune, Maharashtra, India.

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λ _i	Failure rate of ith component
τ _i	Repair time of ith component
ℓ	Level of satisfaction/ acceptance
ℓ_min	Minimum of all ℓ_i, i=1,2, \dots n
λ _SS	Failure rate of the structure system
τ _SS	Repair time of the structure system
R _SS	Reliability of the system structure
MTTF	Mean time to failures
MTBF	Mean time between failures
MTBF	Mean time between failures
A _SS	Availability of the structure system
ENOF	Expected number of failures
	AND Gate
	OR Gate