Reliability Analysis of a Powerloom Plant Using Interval Valued Intuitionistic Fuzzy Sets

In this paper, we investigate the reliability analysis of a powerloom plant by using interval valued intuitionistic fuzzy sets (IVIFS). Herein, we modeled a powerloom plant as a gracefully degradable system having two units A(n) and B(m) connected in series. The reliability of n components of unit A and m components of unit B is assumed to be an IVIFS defined over the universe of discourse [0, 1]. Thus, the reliability of the system obtained is an IVIFS that covers the inherited uncertainty in data collection and reliability evaluation of a powerloom plant.


Introduction
It is well established that the classical set theoretic approach in reliability analysis fails to address the uncertainty up to a desired level.For systems with built-in multi-component parallel redundancy, graceful degradation [1] describes a smooth change to a lower performance level of the system.For such systems, it is therefore unrealistic to assign a number to the reliability of the different components of unit A and B. In 1965, Zadeh [2] introduced the concept of fuzzy sets.Since the inception of fuzzy set, it is widely used in reliability evaluation by many authors.Using it, Cai et al. [3] introduced the fuzzy state assumption and possibility assumption to give Profust and Possbist reliability respectively.Singer [4] presented fuzzy state theoretic approach to fault tree analysis.Chen [5] used arithmetic of fuzzy numbers to evaluate system reliability.Bowles and Palaez [6] ob-tained the Profust reliability estimates for a single unit gracefully degradable system.Fuzzy set theoretic approach for failure rate estimation developed by Pandey and Tyagi [7] provides comprehensive results in reliability estimation of a gracefully degradable system.
Since the introduction of fuzzy set by Zadeh [2] in 1965, the concept of fuzzy sets has been extended by various researchers such as L-fuzzy sets [8], fuzzy multisets [9], etc.But in the words of Gehrke et al. [10], many people believe that assigning an exact number to an expert's opinion is too restrictive, and that the assignment of an interval values is more realistic.Taking view of that apparently Sambuc [11] introduced interval valued fuzzy sets (IVFS).He called them φ -flou functions.Later, intuitionistic fuzzy set (IFS) was introduced by Atanassov [12] in 1983.IFS provides the opportunity to model hesitation and uncertainty by introducing a non-membership function in addition to membership function.Tyagi and Akram [13] used IFS to evaluate human reliability of a man-machine system.Wu and Mendel [14] made linguistic summarization using IF-THEN rules and interval type-2 fuzzy sets.Beliakov et al. [15] analyzed the median aggregation operator of IFS and IVFS.In 1989, Atanassov and Gargov [16] introduced the concept of IVIFS along with several of their properties.Unlike IFS, IVIFS maps the universe of discourse E to the set of intervals over [0, 1].
In our work, we use IVIFS to investigate the reliability evaluation of a powerloom plant.A powerloom plant is a small scale industrial setup that weaves threads of different varieties of colours to produce a long piece of cloth of standard size [17].In the present model, let A(n) and B(m) be two units having n and m ( ) n m ≥ components respectively in parallel redundancy.Units A(n) carries n threads of different colours through its n-identical parallel components, and after weaving threads of some specified colour combination together provides m threads to unit B(m).Unit B(m) receives these m threads through its m-identical parallel components and weaves them together to produce a long piece of cloth of some standard size.Failure of any component of either unit would result in missing a thread of that unit and deteriorate the quality of that product.Without any loss of generality, we can assume that at the most one component of these units may fail at any point of time.This shows that both the units of the system are gracefully degradable.Further, we may assume that owing to ( ) missing threads in unit A(n) and B(m) respectively, the quality of the product deteriorates beyond the acceptable limits.Therefore, system will be considered failed in view of reliability estimation, if k components of unit A(n) and l components of unit B(m) fail.Choice of k and l depends upon the tolerable limit to be prescribed by quality control.

Interval Valued Fuzzy Sets
An IVFS A over a basic set E is specified by a function [ ] ( )

Intuitionistic Fuzzy Sets
An IFS i A  [12] on X is given by The value is a lower bound on the degree of membership of x derived from the evidence for x and is a lower bound on the negation of x derived from the evidence against x.We refer to ( ) ( ) ( ) as the intuitionistic index of x.It is the hesitancy of x in i A  due to the lack of knowledge for x X ∈ or otherwise.When the universe of discourse X is discrete, an IFS i A  can be written as  .
An IFS i A  with continuous universe of discourse X as shown in Figure 1 can be written as

Interval Valued Intuitionistic Fuzzy Sets
An IVIFS [16] [18] is an extension of both, the IFS and IVFS.An IVIFS A over a universal set E is defined as an object of the form: where

Triangular Interval-Valued Intuitionistic Fuzzy Sets
A triangular interval valued intuitionistic fuzzy sets (TIVIFS) A over the universe E shown in Figure 2 is defined as: .

Arithmetic Operations on TIVIFS
Here, we introduce some basic arithmetic operations on TIVIFS.For this, let , , ; , , , ; , , ; , , , ; are two TIVIFS defined over a universal set E as shown in Figure 2, then following arithmetic operations between A and B can be defined [18].

Parallel Network
Let, , , ; , , , ; be n different IVIFS representing the reliability of n different components of a parallel network as shown in Figure 4, then the reliability R P of this parallel network can be given as: max inf ,inf , , , max sup ,sup

Reliability Analysis of a Powerloom Plant
As mentioned earlier, failure of any component of unit A(n) or B(m) results in missing a thread in either unit and causes the deterioration in the quality of the product.Herein, we considered that both the units in powerloom are gracefully degradable.Further, we may assume that if, ( ) threads of unit A(n) and ( ) l m ≤ ≤ threads of unit B(m) are missing, the quality of product deteriorates beyond the acceptable limit.In other words, we can say that the system is considered failed if k-components of unit A(n) and l-components of unit B(m) fail.Now, let, 1 2 , , , be the reliability of n identical components of unit A and 1 2 , , , be the reliability of m identical components of unit B respectively.for different values of k and l and obtained the reliability of powerloom.The reliability of powerloom plant results as an IVIFS in each case.Obviously, IVIFS are most generalized form of fuzzy sets.Thus, the new approach using IVIFS provides comprehensive results and rules out under/over estimation of reliability of the powerloom plant.

Figure 1 .
Figure 1.Intuitionistic fuzzy set explanation of a real number.
definition of IVIFS is analogous to the definition of IFS, where , j n =  be n different IVIFS representing the reliability of n different components of a series network as shown in Figure 3, then the reliability R S of this

Figure 2 .
Figure 2. Membership and non-membership function of TIVIFS.