Keywords

1 Introduction

This article has been indicated as a case study of fault tree analysis (FTA) method in marine engineering application. The two stroke engine is mainly used for the force of the ship in reliability testing facility. Due to the uncontrollable working conditions or weather conditions, it is difficult to get the failure value of the undesired event of FTA. In [6], Cai et al. Proposed system failure by use of fuzzy methodology. Fault tree analysis is used to get more reliable and reasonable results based on influencing factors of experts. Fault tree is built based on a top event with cause and effect presents the intermediate event with shows graphically as a rectangle. The subsequent intermediate events correlates lead to the top event. In [7], Singer approached FTA concept in reliability analysis. The main engine’s fault tree are faults of the multi cylinder, failure to the turbocharger and common components. Fuzzy set theory is applied in Fault tree method. In 1965, L.A Zadeh proposed degree of membership for fuzzy sets is a real numbers between [0, 1]. Determining the degree of membership in practical application is very difficult. Since approximate reasoning, many scholars have used the concept of interval valued fuzzy numbers and gave the expression of interval valued fuzzy numbers.

The fault tree of cylinder unit includes cylinder cover piston liner group failure, piston rod and combustion and start air components failure. The fault of piston cylinder is the main faults is the groups includes failures of cylinder head, piston, piston rings, liner and jacket. In [4], Rafat presented a Marine main engine’s fault tree to find the minimal cut set of first order. The faults of piston crank group are illustrated among the specific components are the piston rod, the stuffing box, crosshead bearings, a connecting rod and crankpin bearings. The starting and combustion process are highlighted includes faults to the exhaust valve, the air starting value, fuel injectors with high pressure pipes and fuel pump. The turbocharger is classified in to fault of turbine functioning components, compressor and common elements. The fault of crank system contains the damage to main bearings, thrust bearing and the shaft. The fault sub tree analyzed main engine is the tree of camshaft.

2 Preliminaries

2.1 Definition

An interval valued fuzzy set \( \tilde{R} \) on R is given by \( \tilde{R} = \left\{ {\left( {x,\left[ {\mu_{{\tilde{R}^{L} }} (x),\mu_{{\tilde{R}^{U} }} (x)} \right]} \right)} \right\} \) \( \forall x\varepsilon R \) Where \( 0 \le \mu_{{\tilde{R}^{L} }} (x) \le \mu_{{\tilde{R}^{U} }} (x) \le 1 \) and \( \mu_{{\tilde{R}^{L} }} (x),\mu_{{\tilde{R}^{U} }} (x)\varepsilon \left[ {0,1} \right] \) denoted by \( \mu_{{\tilde{R}}} (x) = \left[ {\mu_{{\tilde{R}^{L} }} (x),\mu_{{\tilde{R}^{U} }} (x)} \right] \), \( x\varepsilon R \) or \( \tilde{R} = \left[ {\tilde{R}_{L} ,\tilde{R}_{U} } \right] \)

2.2 Definition

An interval valued Triskaidecagonal fuzzy number \( \varpi \) on R is given \( \varpi = \left\{ {\left( {x,\left[ {\mu_{{\varpi^{L} }} (x),\mu_{{\varpi^{U} }} (x)} \right]} \right)} \right\}\,\forall x\varepsilon R \) Where \( 0 \le \mu_{{\varpi^{L} }} (x) \le \mu_{{\varpi^{U} }} (x) \le 1 \) and \( \mu_{{\varpi^{L} }} (x),\mu_{{\varpi^{U} }} (x)\varepsilon \left[ {0,1} \right] \) as \( \mu_{{\tilde{A}}} (x) = \left[ {\mu_{{\varpi^{L} }} (x),\mu_{{\varpi^{U} }} (x)} \right] \), \( x\varepsilon R \) or \( \varpi = \left[ {\varpi_{L} ,\varpi_{U} } \right] \)

Assume \( \varpi^{L} = \left( {\ell_{1} ,\ell_{2} ,\ell_{3} ,\ell_{4} ,\ell_{5} ,\ell_{6} ,\ell_{7} ,\ell_{8} ,\ell_{9} ,\ell_{10} ,\ell_{11} ,\ell_{12} ,\ell_{13} ;\delta } \right) \) and \( \varpi^{U} = \left( {\varsigma_{1} ,\varsigma_{2} ,\varsigma_{3} ,\varsigma_{4} ,\varsigma_{5} ,\varsigma_{6} ,\varsigma_{7} ,\varsigma_{8} ,\varsigma_{9} ,\varsigma_{10} ,\varsigma_{11} ,\varsigma_{12} ,\varsigma_{13} ;\gamma } \right) \) where \( 0 < \delta \le \gamma \le 1 \)

$$ \begin{aligned} & \varsigma_{1} < \ell_{1} < \varsigma_{2} < \ell_{2} < \varsigma_{3} < \ell_{3} < \varsigma_{4} < \ell_{4} < \varsigma_{5} < \ell_{5} < \varsigma_{6} < \ell_{6} < \varsigma_{7} < \ell_{8} < \varsigma_{8} < \ell_{9} < \varsigma_{9} < \ell_{10} < \varsigma_{10} < \ell_{11} < \varsigma_{11} \\ & < \ell_{12} < \varsigma_{12} < \ell_{13} < \varsigma_{13} \\ \end{aligned} $$

\( \ell_{1} ,\ell_{2} , \ldots ,\ell_{13} ,\varsigma_{1} ,\varsigma_{2} \ldots \varsigma_{13} \varepsilon R \). The membership function of \( \varpi_{L} ,\varpi_{U} \) is defined as follows

$$ \mu_{{\varpi_{{\overline{\text{TD}} }} }} L \left( x \right) = \left( {\begin{array}{*{20}l} {\delta \left( { \frac{1}{ 6}\frac{{x - \ell_{1} }}{{\ell_{2} - \ell_{1} }}} \right)} \hfill & {\ell_{1} \le {\text{x}} \le \ell_{2} } \hfill \\ {\delta \left( {\frac{1}{6} + \frac{1}{6}\frac{{x2 - \ell_{2} }}{{\ell_{3} - \ell_{2} }}} \right)} \hfill & {\ell_{2} \le {\text{x}} \le \ell_{3} } \hfill \\ {\delta \left( {\frac{2}{6} + \frac{1}{6}\frac{{x - \ell_{3} }}{{\ell_{4} - \ell_{3} }}} \right)} \hfill & {\ell_{3} \le {\text{x}} \le \ell_{4} } \hfill \\ {\delta \left( {\frac{3}{6} + \frac{1}{6} \frac{{x - \ell_{4} }}{{\ell_{5} - \ell_{4} }}} \right)} \hfill & {\ell_{4} \le {\text{x}} \le \ell_{5} } \hfill \\ {\delta \left( {\frac{4}{6} + \frac{1}{6} \frac{{x - \ell_{5} }}{{\ell_{6} - \ell_{5} }}} \right)} \hfill & {\ell_{5} \le {\text{x}} \le \ell_{6} } \hfill \\ {\delta \left( {\frac{5}{6} + \frac{1}{6} \frac{{x - \ell_{6} }}{{\ell_{7} - \ell_{6} }}} \right)} \hfill & {\ell_{6} \le {\text{x}} \le \ell_{7} } \hfill \\ {\delta \left( {1 - \frac{1}{6} \frac{{x - \ell_{7} }}{{\ell_{8} - \ell_{7} }}} \right)} \hfill & {\ell_{7} \le {\text{x}} \le \ell_{8} } \hfill \\ {\delta \left( {\frac{5}{6} - \frac{1}{6} \frac{{x - \ell_{8} }}{{\ell_{9} - \ell_{8} }}} \right)} \hfill & {\ell_{8} \le {\text{x}} \le \ell_{9} } \hfill \\ {\delta \left( {\frac{4}{6} - \frac{1}{6} \frac{{x - \ell_{9} }}{{\ell_{10} - \ell_{9} }}} \right)} \hfill & {\ell_{9} \le {\text{x}} \le \ell_{10} } \hfill \\ {\delta \left( { \frac{3}{6} - \frac{1}{6} \frac{{x - \ell_{10} }}{{\ell_{11} - \ell_{10} }}} \right)} \hfill & {\ell_{10} \le {\text{x}} \le \ell_{11} } \hfill \\ {\delta \left( {\frac{2}{6} - \frac{1}{6} \frac{{x - \ell_{11} }}{{\ell_{12} - \ell_{11} }}} \right)} \hfill & {\ell_{11} \le {\text{x}} \le \ell_{12} } \hfill \\ {\delta \left( {\frac{1}{6}\frac{{\ell_{13} - x}}{{\ell_{13} - \ell_{12} }}} \right)} \hfill & {\ell_{12} \le {\text{x}} \le \ell_{13} } \hfill \\ 0 \hfill & {{\text{x > }}\ell_{13} } \hfill \\ \end{array} } \right. $$
$$ \begin{aligned} \mu_{{\varpi_{{\overline{\text{TD}} }} }} U \left( x \right) = & \left( {\begin{array}{*{20}l} {\gamma \left( { \frac{1}{ 6}\frac{{x - \varsigma_{1} }}{{\varsigma_{2} - \varsigma_{1} }}} \right)} \hfill & {\varsigma_{1} \le {\text{x}} \le \varsigma_{2} } \hfill \\ {\gamma \left( {\frac{1}{6} + \frac{1}{6}\frac{{x2 - \varsigma_{2} }}{{\varsigma_{3} - \varsigma_{2} }}} \right)} \hfill & {\varsigma_{2} \le {\text{x}} \le \varsigma_{3} } \hfill \\ {\gamma \left( {\frac{2}{6} + \frac{1}{6}\frac{{x - \varsigma_{3} }}{{\varsigma_{4} - \varsigma_{3} }}} \right)} \hfill & {\varsigma_{3} \le {\text{x}} \le \varsigma_{4} } \hfill \\ {\gamma \left( {\frac{3}{6} + \frac{1}{6} \frac{{x - \varsigma_{4} }}{{\varsigma_{5} - \varsigma_{4} }}} \right)} \hfill & {\varsigma_{4} \le {\text{x}} \le \varsigma_{5} } \hfill \\ {\gamma \left( {\frac{4}{6} + \frac{1}{6} \frac{{x - \varsigma_{5} }}{{\varsigma_{6} - \varsigma_{5} }}} \right)} \hfill & {\varsigma_{5} \le {\text{x}} \le \varsigma_{6} } \hfill \\ \end{array} } \right. \\ & \left( {\begin{array}{*{20}l} {\gamma \left( {\frac{5}{6} + \frac{1}{6} \frac{{x - \varsigma_{6} }}{{\varsigma_{7} - \varsigma_{6} }}} \right)} \hfill & {\varsigma_{6} \le {\text{x}} \le \varsigma_{7} } \hfill \\ {\gamma \left( {1 - \frac{1}{6} \frac{{x - \varsigma_{7} }}{{\varsigma_{8} - \varsigma_{7} }}} \right)} \hfill & {\varsigma_{7} \le {\text{x}} \le \varsigma_{8} } \hfill \\ {\gamma \left( {\frac{5}{6} - \frac{1}{6} \frac{{x - \varsigma_{8} }}{{\varsigma_{9} - \varsigma_{8} }}} \right)} \hfill & {\varsigma_{8} \le {\text{x}} \le \varsigma_{9} } \hfill \\ {\gamma \left( {\frac{4}{6} - \frac{1}{6} \frac{{x - \varsigma_{9} }}{{\varsigma_{10} - \varsigma_{9} }}} \right)} \hfill & {\varsigma_{9} \le {\text{x}} \le \varsigma_{10} } \hfill \\ {\gamma \left( { \frac{3}{6} - \frac{1}{6} \frac{{x - \varsigma_{10} }}{{\varsigma_{11} - \varsigma_{10} }}} \right)} \hfill & {\varsigma_{10} \le {\text{x}} \le \varsigma_{11} } \hfill \\ {\gamma \left( {\frac{2}{6} - \frac{1}{6} \frac{{x - \varsigma_{11} }}{{\varsigma_{12} - \varsigma_{11} }}} \right)} \hfill & {\varsigma_{11} \le {\text{x}} \le \varsigma_{12} } \hfill \\ {\gamma \left( {\frac{1}{6} \frac{{\varsigma_{13} - x}}{{\varsigma_{13} - \varsigma_{12} }}} \right) } \hfill & {\varsigma_{12} \le {\text{x}} \le \varsigma_{13} } \hfill \\ 0 \hfill & {{\text{x > }}\varsigma_{13} } \hfill \\ \end{array} } \right. \\ \end{aligned} $$

Then

$$ \varpi = \left[ {\varpi^{L} ,\varpi^{U} } \right] = \left[ {\left[ {\begin{array}{*{20}l} {\left( {\ell_{1} ,\ell_{2} ,\ell_{3} ,\ell_{4} ,\ell_{5} ,\ell_{6} ,\ell_{7} ,\ell_{8} ,\ell_{9} ,\ell_{10} ,\ell_{11} ,\ell_{12} ,\ell_{13} ;\delta } \right),} \hfill \\ {\left( {\varsigma_{1} ,\varsigma_{2} ,\varsigma_{3} ,\varsigma_{4} ,\varsigma_{5} ,\varsigma_{6} ,\varsigma_{7} ,\varsigma_{8} ,\varsigma_{9} ,\varsigma_{10} ,\varsigma_{11} ,\varsigma_{12} ,\varsigma_{13} ;\gamma } \right)} \hfill \\ \end{array} } \right]} \right] $$

is called the level \( (\delta ,\gamma ) \) Interval valued Triskaidecagonal fuzzy number (Fig. 1).

Fig. 1.
figure 1

Interval valued Triskaidecagonal fuzzy number

Full size image

3 Application of Marine Main Engine Failure Using Interval Valued Triskaidecagonal Fuzzy Number

ʯ 1-Indicates the system failure to start of Main Engine.

ʯ 2-Indicates the failure to start of Engine due to cylinder unit 1 Failure

ʯ3-Indicates the failure to start of Engine due to cylinder unit 2 Failure. (Table 1)

Table 1. Components of marine main engine
Full size table
figure a

Parallel System:

The fuzzy reliability R ps of the parallel system shown below can be evaluated by using the expression as follows:

Step-1:

figure b

Step-2:

figure c

Step-3:

figure d

Similarly linguistic values can frame out for all the components. (Table 2)

Table 2. Linguistic values of marine main engine failure using Fault tree analysis method with Interval valued Triskaidecagonal fuzzy number
Full size table

Step-1:

figure e

Step-2:

figure f

Step-3:

figure g

Case:1 If \( 0 \le\Omega < \delta \), then

$$ \begin{aligned} \varpi_{l}^{L} (\alpha ) & = \frac{{6\Omega }}{\delta }(\ell_{2} - \ell_{1} ) + \ell_{1} \\ \varpi_{r}^{L} (\alpha ) & = \frac{{6\Omega }}{\delta }(\ell_{13} - \ell_{12} ) + \ell_{13} \\ \varpi_{l}^{U} (\alpha ) & = \frac{{6\Omega }}{\gamma }(\varsigma_{2} - \varsigma_{1} ) + \varsigma_{1} \\ \varpi_{r}^{U} (\alpha ) & = \frac{{6\Omega }}{\gamma }(\varsigma_{13} - \varsigma_{12} ) + \varsigma_{13} \\ \end{aligned} $$

Where \( \left\{ {\varpi_{l}^{L} (\Omega ),\varpi_{l}^{U} (\Omega )} \right\},\left\{ {\varpi_{r}^{L} (\Omega ),\varpi_{r}^{U} (\Omega )} \right\} \) are the \( \Omega - \) cuts of left and right side of level \( (\delta ,\gamma ) \) Interval valued Triskaidecagonal fuzzy number.

Case:2

If \( \delta \le\Omega \le \gamma \), then

$$ \begin{aligned} \varpi_{l}^{U} (\Omega ) & = 6\alpha (\varsigma_{7} - \varsigma_{6} ) - 5\varsigma_{7} + 6\varsigma_{6} \\ \varpi_{r}^{U} (\Omega ) & = 6\alpha (\varsigma_{9} - \varsigma_{8} ) - 4\varsigma_{8} + 5\varsigma_{9} \\ \end{aligned} $$

Definition: Signed distance

\( \kappa_{0} (h,0),h\varepsilon R \) is define as \( \kappa_{0} (h,0) = h \) and is called the signed distance from k to 0. \( \kappa_{0} \) meant is When (i) h > 0, \( \kappa_{0} (h,0) = h > 0 \) (i.e.) h is the right and distance from 0 is h. (ii) h < 0, \( \kappa_{0} (h,0) = h < 0 \) (i.e.) h is the left and distance from 0 is -h.

When \( 0 < \delta < \gamma \), we obtain the signed distance of from 0

$$ \varpi = \left[ {\begin{array}{*{20}l} {\left( {\ell_{1} ,\ell_{2} ,\ell_{3} ,\ell_{4} ,\ell_{5} ,\ell_{6} ,\ell_{7} ,\ell_{8} ,\ell_{9} ,\ell_{10} ,\ell_{11} ,\ell_{12} ,\ell_{13} ;\delta } \right),} \hfill \\ {\left( {\varsigma_{1} ,\varsigma_{2} ,\varsigma_{3} ,\varsigma_{4} ,\varsigma_{5} ,\varsigma_{6} ,\varsigma_{7} ,\varsigma_{8} ,\varsigma_{9} ,\varsigma_{10} ,\varsigma_{11} ,\varsigma_{12} ,\varsigma_{13} ;\gamma } \right)} \hfill \\ \end{array} } \right] $$

Case: 1 \( 0 \le\Omega \le \delta \)

$$ \kappa_{0} (\varpi_{l}^{U} (\Omega ),0) = \varpi_{l}^{U} (\Omega ),\kappa_{0} (\varpi_{l}^{L} (\Omega ),0) = \varpi_{l}^{L} (\Omega ),\kappa_{0} (\varpi_{r}^{L} (\Omega ),0) = \varpi_{r}^{L} (\Omega ),\kappa_{0} (\varpi_{r}^{U} (\Omega ),0) = \varpi_{r}^{U} (\Omega ) $$

The signed distance of the interval \( [\varpi^{L} (\Omega ),\varpi^{U} (\Omega )] \) from 0 is given by

$$ \kappa_{0} \left[ {(\varpi_{l}^{L} (\Omega ),\varpi_{l}^{U} (\Omega )),0} \right] = \frac{1}{2}\left[ {\kappa_{0} (\varpi_{l}^{U} (\Omega ),0) + \kappa_{0} (\varpi_{l}^{L} (\Omega ),0)} \right] = \frac{1}{2}\left[ {\varpi_{l}^{U} (\Omega ) + \varpi_{l}^{L} (\Omega )} \right] = \frac{1}{2}\left[ {\ell_{1} + \varsigma_{1} + \frac{{6\Omega }}{\gamma }(\varsigma_{2} - \varsigma_{1} ) + \frac{{6\Omega }}{\delta }(\ell_{2} - \ell_{1} )} \right] $$
(1)
$$ \begin{aligned} \kappa_{0} [(\varpi_{r}^{L} (\Omega ),\varpi_{r}^{U} (\Omega )),0] & = \frac{1}{2}[\kappa_{0} (\varpi_{r}^{U} (\Omega ),0) + \kappa_{0} (\varpi_{r}^{L} (\Omega ),0)] = \frac{1}{2}\left[ {\varpi_{r}^{U} (\Omega ) + \varpi_{r}^{L} (\Omega )} \right] \\ & = \frac{1}{2}\left[ {\ell_{13} + \varsigma_{13} + \frac{{7\Omega }}{\gamma }(\varsigma_{13} - \varsigma_{12} ) + \frac{{7\Omega }}{\delta }(\ell_{13} - \ell_{12} )} \right] \\ \end{aligned} $$
(2)

The signed distance of \( \left[ {\varpi_{l}^{L} (\Omega ),\varpi_{l}^{U} (\Omega )} \right] \cup [\varpi_{r}^{L} (\Omega ),\varpi_{r}^{U} (\Omega )] \) from 0 is given by

$$ \begin{aligned} & \kappa_{0} \left( {\left[ {\varpi_{l}^{L} (\Omega ),\varpi_{l}^{U} (\Omega )} \right] \cup [\varpi_{r}^{L} (\Omega ),\varpi_{r}^{U} (\Omega )],0} \right) = \frac{1}{2}\left[ {\kappa {}_{0}\left( {(\varpi_{l}^{L} (\Omega ),\varpi_{l}^{U} (\Omega )),0} \right) + \kappa {}_{0}\left( {(\varpi_{r}^{L} (\Omega ),\varpi_{r}^{U} (\Omega )),0} \right)} \right] \\ & \quad = \frac{1}{4}\left[ {\ell_{1} + \varsigma_{1} + \frac{{6\Omega }}{\gamma }(\varsigma_{2} - \varsigma_{1} ) + \frac{{6\Omega }}{\delta }(\ell_{2} - \ell_{1} ) + \ell_{13} + \varsigma_{13} + \frac{{6\Omega }}{\gamma }(\varsigma_{13} - \varsigma_{12} ) + \frac{{6\Omega }}{\delta }(\ell_{13} - \ell_{12} )} \right] \\ & \quad \quad \quad = \frac{1}{4}\left[ {\ell_{1} + \varsigma_{1} + \ell_{13} + \varsigma_{13} + \frac{{6\Omega }}{\gamma }(\varsigma_{2} - \varsigma_{1} + \varsigma_{13} - \varsigma_{12} ) + \frac{{6\Omega }}{\delta }(\ell_{2} - \ell_{1} + \ell_{13} - \ell_{12} )} \right] \\ \end{aligned} $$
(3)

The function in Eq. 3 is continuous on \( 0 \le\Omega \le \delta \) with respect to \( \Omega \), we can find the average value by integration

$$ \begin{aligned} & \frac{1}{\delta }\int\limits_{0}^{\lambda } {\kappa_{0} \left( {\left[ {\varpi_{l}^{L} (\Omega ),\varpi_{l}^{U} (\Omega )} \right] \cup [\varpi_{r}^{L} (\Omega ),\varpi_{r}^{U} (\Omega )],0} \right)d\alpha } \\ & = \frac{1}{8}( - 4\ell_{1} + 2\varsigma_{1} + 8\ell_{13} + 2\varsigma_{13} + 6\ell_{2} - 6\ell_{12} + \frac{6\delta }{\gamma }(\varsigma_{2} - \varsigma_{1} + \varsigma_{13} - \varsigma_{12} ) \\ \end{aligned} $$
$$ \kappa_{0} ({\mathop{P}\limits^{,}}_{1} ,0) = 0.999996205 $$

Case: 2 \( \delta \le\Omega \le \gamma \) The signed distance of \( [\varpi_{l}^{U} (\alpha ),\varpi_{r}^{U} (\alpha )] \) from 0 is given

$$ \begin{aligned} & d_{0} [(\varpi_{l}^{U} (\Omega ),\varpi_{l}^{U} (\Omega )),0] = \frac{1}{2}[d_{0} (\varpi_{l}^{U} (\Omega ),0) + d_{0} (\varpi_{l}^{U} (\Omega ),0)] = \frac{1}{2}\left[ {\varpi_{l}^{U} (\Omega ) + \varpi_{l}^{U} (\Omega )} \right] \\ & = \frac{1}{2}\left[ {6\Omega (\varsigma_{7} - \varsigma_{6} ) - 5\varsigma_{7} + 6\varsigma_{6} + 6\Omega (\varsigma_{8} - \varsigma_{7} ) + 5\varsigma_{9} - 4\varsigma_{8} } \right] = \frac{1}{2}\left[ {6\Omega (\varsigma_{7} - \varsigma_{6} + \varsigma_{8} - \varsigma_{7} ) - 5\varsigma_{7} + 6\varsigma_{6} + 5\varsigma_{9} - 4\varsigma_{8} } \right] \\ \end{aligned} $$

This function is also continuous function and the average value is calculated by integration.

$$ \frac{1}{\gamma - \delta }\int\limits_{\lambda }^{\rho } {(\kappa_{0} (\varpi_{l}^{U} (\Omega ),\varpi_{r}^{U} (\Omega ))} ,0)d\Omega = \frac{1}{\gamma - \delta }\frac{1}{4}\left[ {\begin{array}{*{20}l} {6\gamma^{2} (\varsigma_{7} - \varsigma_{6} + \varsigma_{9} - \varsigma_{8} ) + ( - 10\varsigma_{7} + 12\varsigma_{6} + 10\varsigma_{9} - 8\varsigma_{8} )\gamma } \hfill \\ { - 6\delta^{2} (\varsigma_{7} - \varsigma_{6} + \varsigma_{9} - \varsigma_{8} ) + ( - 10\varsigma_{7} + 12\varsigma_{6} + 14\varsigma_{9} - 14\varsigma_{8} )\delta } \hfill \\ \end{array} } \right] $$
$$ \kappa_{0} ({\mathop{P}\limits^{,}}_{1} ,0) = 0.99999837594 $$

Centroid Distance for lower bound:

$$ \frac{\delta }{12}[ - \ell_{1} + 2\ell_{7} - 2\ell_{12} + \ell_{13} ] + \frac{\delta }{6}[ - \ell_{3} - \ell_{2} - \ell_{4} - \ell_{5} - \ell_{6} + 5\ell_{7} + \ell_{9} - 5\ell_{8} + \ell_{10} + \ell_{11} + 2\ell_{12} ] + \delta [\ell_{8} - \ell_{7} ] $$
$$ \smallint \mu_{A} \left( x \right)dx = 0.00000112602\quad \quad \quad ({\text{i}}) $$
$$ \begin{aligned} & \frac{\delta }{12}[ - \ell_{3}^{2} - \ell_{2}^{2} - \ell_{4}^{2} - \ell_{5}^{2} - \ell_{6}^{2} - \ell_{7}^{2} + \ell_{8}^{2} + \ell_{9}^{2} + \ell_{10}^{2} + \ell_{11}^{2} + 2\ell_{12}^{2} ] + \frac{\delta }{36}[\frac{{2\ell_{2}^{3} - 3\ell_{1} \ell_{2}^{2} + \ell_{1}^{3} }}{{\ell_{2} - \ell_{1} }} + \frac{{2\ell_{3}^{3} - 3\ell_{2} \ell_{3}^{2} + \ell_{2}^{3} }}{{\ell_{3} - \ell_{2} }} + \frac{{2\ell_{4}^{3} - 3\ell_{3} \ell_{4}^{2} + \ell_{3}^{3} }}{{\ell_{4} - \ell_{3} }} \\ & + \frac{{2\ell_{5}^{3} - 3\ell_{4} \ell_{5}^{2} + \ell_{4}^{3} }}{{\ell_{5} - \ell_{4} }} + \frac{{2\ell_{6}^{3} - 3\ell_{5} \ell_{6}^{2} + \ell_{5}^{3} }}{{\ell_{6} - \ell_{5} }} + \frac{{2\ell_{7}^{3} - 3\ell_{6} \ell_{7}^{2} + \ell_{6}^{3} }}{{\ell_{7} - \ell_{6} }} - \frac{{2\ell_{8}^{3} - 3\ell_{7} \ell_{8}^{2} + \ell_{7}^{3} }}{{\ell_{8} - \ell_{7} }} - \frac{{2\ell_{9}^{3} - 3\ell_{8} \ell_{9}^{2} + \ell_{8}^{3} }}{{\ell_{9} - \ell_{8} }} - \frac{{2\ell_{10}^{3} - 3\ell_{9} \ell_{10}^{2} + \ell_{9}^{3} }}{{\ell_{10} - \ell_{9} }} \\ & - \frac{{2\ell_{11}^{3} - 3\ell_{10} \ell_{11}^{2} + \ell_{10}^{3} }}{{\ell_{11} - \ell_{10} }} - \frac{{2\ell_{12}^{3} - 3\ell_{11} \ell_{12}^{2} + \ell_{11}^{3} }}{{\ell_{12} - \ell_{11} }} + \frac{{2\ell_{12}^{3} - 3\ell_{13} \ell_{12}^{2} + \ell_{13}^{3} }}{{\ell_{13} - \ell_{12} }}] \\ \end{aligned} $$
$$ \int {X\mu_{A} \left( x \right)} dx = 0.00000182933\quad \quad ({\text{II)}} $$

Centroid distance for lower bound = 1.62459814

Centroid distance for upper bound:

$$ \frac{\gamma }{12}[ - \varsigma_{1} + 2\varsigma_{7} - 2\varsigma_{12} + \varsigma_{13} ] + \frac{\gamma }{6}[ - \varsigma_{3} - \varsigma_{2} - \varsigma_{4} - \varsigma_{5} - \varsigma_{6} + 5\varsigma_{7} + \varsigma_{9} - 5\varsigma_{8} + \varsigma_{10} + \varsigma_{11} + 2\varsigma_{12} ] + \gamma [\varsigma_{8} - \varsigma_{7} ] $$
$$ \int {\mu_{A} \left( x \right)dx = 0.0000027} \quad \quad \quad \left( {\text{i}} \right) $$
$$ \begin{aligned} & \frac{\gamma }{12}[ - \varsigma_{3}^{2} - \varsigma_{2}^{2} - \varsigma_{4}^{2} - \varsigma_{5}^{2} - \varsigma_{6}^{2} - \varsigma_{7}^{2} + \varsigma_{8}^{2} + \varsigma_{9}^{2} + \varsigma_{10}^{2} + \varsigma_{11}^{2} + 2\varsigma_{12}^{2} ] + \frac{\gamma }{36}[\frac{{2\varsigma_{2}^{3} - 3\varsigma_{1} \varsigma_{2}^{2} + \varsigma_{1}^{3} }}{{\varsigma_{2} - \varsigma_{1} }} + \frac{{2\varsigma_{3}^{3} - 3\varsigma_{2} \varsigma_{3}^{2} + \varsigma_{2}^{3} }}{{\varsigma_{3} - \varsigma_{2} }} + \frac{{2\varsigma_{4}^{3} - 3\varsigma_{3} \varsigma_{4}^{2} + \varsigma_{3}^{3} }}{{\varsigma_{4} - \varsigma_{3} }} \\ & + \frac{{2\varsigma_{5}^{3} - 3\varsigma_{4} \varsigma_{5}^{2} + \varsigma_{4}^{3} }}{{\varsigma_{5} - \varsigma_{4} }} + \frac{{2\varsigma_{6}^{3} - 3\varsigma_{5} \varsigma_{6}^{2} + \varsigma_{5}^{3} }}{{\varsigma_{6} - \varsigma_{5} }} + \frac{{2\varsigma_{7}^{3} - 3\varsigma_{6} \varsigma_{7}^{2} + \varsigma_{6}^{3} }}{{\varsigma_{7} - \varsigma_{6} }} - \frac{{2\varsigma_{8}^{3} - 3\varsigma_{7} \varsigma_{8}^{2} + \varsigma_{7}^{3} }}{{\varsigma_{8} - \varsigma_{7} }} - \frac{{2\varsigma_{9}^{3} - 3\varsigma_{8} \varsigma_{9}^{2} + \varsigma_{8}^{3} }}{{\varsigma_{9} - \varsigma_{8} }} - \frac{{2\varsigma_{10}^{3} - 3\varsigma_{9} \varsigma_{10}^{2} + \varsigma_{9}^{3} }}{{\varsigma_{10} - \varsigma_{9} }} \\ & - \frac{{2\varsigma_{11}^{3} - 3\varsigma_{10} \varsigma_{11}^{2} + \varsigma_{10}^{3} }}{{\varsigma_{11} - \varsigma_{10} }} - \frac{{2\varsigma_{12}^{3} - 3\varsigma_{11} \varsigma_{12}^{2} + \varsigma_{11}^{3} }}{{\varsigma_{12} - \varsigma_{11} }} + \frac{{2\varsigma_{12}^{3} - 3\varsigma_{13} \varsigma_{12}^{2} + \varsigma_{13}^{3} }}{{\varsigma_{13} - \varsigma_{12} }}] \\ \end{aligned} $$
$$ \int {X\mu_{A} \left( x \right)} dx = 0.00000258\quad \quad ({\text{II)}} $$

Centroid distance for upper bound = 0.955555556

Euclidean distance: \( \sqrt {\sum\limits_{i = 1}^{n} {\left| {x_{i} - y_{i} } \right|^{2} } } \), \( {\mathbf{ED}} = \sqrt {(x_{1} - y_{1} )^{2} + (x_{2} - y_{2} )^{2} + (x_{3} - y_{3} )^{2} + (x_{4} - y_{4} )^{2} + \ldots \ldots (x_{13} - y_{13} )^{2} } \)

\( {\mathbf{ED}} = \sqrt {0.00000000032} \) , where n = 13

Failure Modes & Effects Analysis of Marine Main engine:

  • Tabulation of components, consequences and safeguards are they associated with failure modes.

  • Identification/assessment of risk is derived from looking at each component in the case of multi-unit cylinder)

  • FTA- approach is commonly referred (Table 3).

    Table 3. Failure mode and Effect analysis
    Full size table

4 Conclusion

Thus Having the importance of distance for Interval valued Triskaidecagonal fuzzy number; a effective distance formula is presented for its computation. The performance of the Fault tree of marine main engine is analyzing using distance formula have compared with Centroid distance, Euclidean distance, and Signed distance to appeared as a appropriate condition for Interval valued Triskaidecagonal fuzzy number and conveniently distance methods presented in the approach of FTA in Marine Ship to find the effect and causes of failure modes. Further research is needed to propose distance formula for Interval valued Triskaidecagonal fuzzy number to find out more accuracy.