A new method to solve time-dependent intuitionistic fuzzy differential equations and its application to analyze the intuitionistic fuzzy reliability of industrial systems

Abstract

To the best of our knowledge, there is no method in the literature for solving such differential equations in which all the parameters except independent variables are represented by intuitionistic fuzzy numbers. In this article, a new method is proposed for solving such nth-order time-dependent intuitionistic fuzzy linear differential equations. To show the application of the proposed method in real-life problems, the time-dependent intuitionistic fuzzy Kolmogorov’s differential equations, obtained by Markov model of condensate system, are solved by the proposed method, and the obtained solution is used to analyze the intuitionistic fuzzy reliability of condensate system.

Keywords

Time-dependent intuitionistic fuzzy differential equations intuitionistic fuzzy reliability time-dependent trapezoidal intuitionistic fuzzy number

Introduction

In real-life situations, usually experts are not able to estimate the exact number of failures and repairs of components of industrial systems with respect to time. Therefore, fuzzy set theory (Zadeh, 1965) is used to evaluate the fuzzy reliability of industrial system. However, in some situations, there may be hesitation about the number of failures or repairs. The fuzzy set considers only the degree of belongingness and degree of nonbelongingness. Fuzzy set theory does not incorporate the hesitation degree. To handle such situations, Atanassov (1986) extended the concept of fuzzy set theory by intuitionistic fuzzy set (IFS) theory.

The IFS is being studied and used in different fields of science. Since, in the IFS proposed by Atanassov (1986), all the elements of IFS are independent from time. So, it cannot be used to represent parameters that depend upon time, for example, the reliability of industrial systems depends upon time, which cannot be represented by IFS proposed by Atanassov (1986). To handle such situations, Kumar et al. (2011) extended the concept of time-dependent fuzzy set (Aliev and Kara, 2004) by time-dependent IFS.

Since then, for analyzing the intuitionistic fuzzy reliability (IFR) by using Markov model, there is a need to solve time-dependent intuitionistic fuzzy Kolmogorov’s differential equations (IFKDEs), but till now, there is no method in the literature for solving time-dependent intuitionistic fuzzy differential equations (IFDEs). So, several authors (Cheng et al., 2009; Mahapatra and Mahapatra, 2010; Mahapatra and Roy, 2009; Manjit Verma et al., 2012; Shu et al., 2006) have extended the concept of fuzzy fault tree by intuitionistic fuzzy fault tree, but till now, no one has extended the concept of fuzzy Markov model by time-dependent intuitionistic fuzzy Markov model for analyzing the IFR of industrial systems.

In this article, a new method is proposed for solving nth-order time-dependent intuitionistic fuzzy linear differential equations. To show the application of the proposed method in real-life problems, the time-dependent IFKDEs, obtained by Markov model of condensate system, are solved by the proposed method, and the obtained solution is used to analyze the IFR of condensate system.

This article is organized as follows. In section “Preliminaries,” some basic definitions and arithmetic operations between time-dependent trapezoidal intuitionistic fuzzy numbers (TrIFNs) are presented. In section “Proposed method for solving time-dependent IFDEs,” a new method is proposed for solving time-dependent IFDEs. In section “Case study,” to show the application of the proposed method in real-life problems, the time-dependent IFKDEs, obtained by using time-dependent intuitionistic fuzzy Markov model of condensate system, are solved by using the proposed method. The conclusion is discussed in section “Conclusion.”

Preliminaries

In this section, some basic definitions and arithmetic operations between time-dependent TrIFNs are presented (Kumar et al., 2011; Mahapatra and Mahapatra, 2010; Mahapatra and Roy, 2009).

Basic definitions

In this section, some basic definitions are presented (Kumar et al., 2011).

Definition 1

Let $X$ be a universe of discourse and $T$ be a nonempty set. Elements of $T$ are called time moments. A time-dependent IFS ${\tilde{A}}^{i} (t)$ in $X$ is an object having the following form

{\tilde{A}}^{i} (t) = {〈 x, μ_{{\tilde{A}}^{i} (t)} (x), ν_{{\tilde{A}}^{i} (t)} (x) 〉 : x \in X, t \in T}

where $μ_{{\tilde{A}}^{i} (t)} : X \to [0, 1]$ and $ν_{{\tilde{A}}^{i} (t)} : X \to [0, 1]$ define the degrees of membership and nonmembership, respectively, of the element $x \in X$ at time moment $t$ to ${\tilde{A}}^{i} (t) \subseteq X$ . For every $x \in X$ and $t \in T$ , $0 \leq μ_{{\tilde{A}}^{i} (t)} (x) + ν_{{\tilde{A}}^{i} (t)} (x) \leq 1$ .

Definition 2

A time-dependent IFS ${\tilde{A}}^{i} (t)$ in the universe of discourse $X$ is convex if and only if

Membership function $μ_{{\tilde{A}}^{i} (t)} (x)$ of ${\tilde{A}}^{i} (t)$ is fuzzy convex, that is

μ_{{\tilde{A}}^{i} (t)} (λ x_{1} + (1 - λ) x_{2}) \geq \min (μ_{{\tilde{A}}^{i} (t)} (x_{1}), μ_{{\tilde{A}}^{i} (t)} (x_{2})) \forall x_{1}, x_{2} \in X, λ \in [0, 1]

Nonmembership function $ν_{{\tilde{A}}^{i} (t)} (x)$ of ${\tilde{A}}^{i} (t)$ is fuzzy concave, that is

ν_{{\tilde{A}}^{i} (t)} (λ x_{1} + (1 - λ) x_{2}) \leq \max (ν_{{\tilde{A}}^{i} (t)} (x_{1}), ν_{{\tilde{A}}^{i} (t)} (x_{2})) \forall x_{1}, x_{2} \in X, λ \in [0, 1]

Definition 3

A time-dependent intuitionistic fuzzy subset ${\tilde{A}}^{i} (t) = {〈 x, μ_{{\tilde{A}}^{i} (t)} (x), ν_{{\tilde{A}}^{i} (t)} (x) : x \in R, t \in T 〉}$ of the real line $R$ is called a time-dependent intuitionistic fuzzy number (IFN) if

${\tilde{A}}^{i} (t)$ is convex and normal.

$μ_{{\tilde{A}}^{i} (t)}$ is upper semicontinuous and $ν_{{\tilde{A}}^{i} (t)}$ is lower semicontinuous.

Supremum ${\tilde{A}}^{i} (t) = {x \in R, t \in T : ν_{{\tilde{A}}^{i} (t)} (x) < 1}$ is bounded.

Definition 4

A time-dependent IFN ${\tilde{A}}^{i} (t)$ , defined on the universal set of real numbers $R$ , denoted as ${\tilde{A}}_{TrIFN}^{i} (t) = (a_{1} (t), a_{2} (t), a_{3} (t), a_{4} (t); {a'}_{1} (t), a_{2} (t), a_{3} (t), {a'}_{4} (t))$ is said to be a time-dependent TrIFN if degree of membership, $μ_{{\tilde{A}}^{i} (t)} (x)$ , and degree of nonmembership, $ν_{{\tilde{A}}^{i} (t)} (x)$ , are given by

μ_{{\tilde{A}}^{i} (t)} (x) = {\begin{matrix} \frac{x - a_{1} (t)}{a_{2} (t) - a_{1} (t)}, & a_{1} (t) \leq x < a_{2} (t) \\ 1, & a_{2} (t) \leq x < a_{3} (t) \\ \frac{a_{4} (t) - x}{a_{4} (t) - a_{3} (t)}, & a_{3} (t) \leq x < a_{4} (t) \\ 0, & otherwise \end{matrix}

ν_{{\tilde{A}}^{i} (t)} (x) = {\begin{matrix} \frac{a_{2} (t) - x}{a_{2} (t) - {a'}_{1} (t)}, & {a'}_{1} (t) \leq x < a_{2} (t) \\ 0, & a_{2} (t) \leq x < a_{3} (t) \\ \frac{x - a_{3} (t)}{{a'}_{4} (t) - a_{3} (t)}, & a_{3} (t) \leq x < {a'}_{4} (t) \\ 1, & otherwise \end{matrix}

where ${a'}_{1} (t) \leq a_{1} (t) \leq a_{2} (t) \leq a_{3} (t) \leq a_{4} (t) \leq {a'}_{4} (t)$ .

Definition 5

Two time-dependent TrIFNs ${\tilde{A}}_{TrIFN}^{i} (t) = (a_{1} (t), a_{2} (t), a_{3} (t), a_{4} (t); {a'}_{1} (t), a_{2} (t), a_{3} (t), {a'}_{4} (t))$ and ${\tilde{B}}_{TrIFN}^{i} (t) = (b_{1} (t), b_{2} (t), b_{3} (t), b_{4} (t); {b'}_{1} (t), b_{2} (t), b_{3} (t), {b'}_{4} (t))$ are said to be equal, that is, ${\tilde{A}}_{TrIFN}^{i} (t) = {\tilde{B}}_{TrIFN}^{i} (t)$ if and only if $a_{1} (t) = b_{1} (t)$ , $a_{2} (t) = b_{2} (t)$ , $a_{3} (t) = b_{3} (t)$ , $a_{4} (t) = b_{4} (t)$ , ${a'}_{1} (t) = {b'}_{1} (t)$ , and ${a'}_{4} (t) = {b'}_{4} (t)$ .

Arithmetic operations between time-dependent TrIFNs

In this section, some arithmetic operations between time-dependent TrIFNs, defined on a universal set of real numbers $R$ , are presented (Kumar et al., 2011; Mahapatra and Mahapatra, 2010; Mahapatra and Roy, 2009).

Let ${\tilde{A}}_{TrIFN}^{i} (t) = (a_{1} (t), a_{2} (t), a_{3} (t), a_{4} (t); {a'}_{1} (t), a_{2} (t), a_{3} (t), {a'}_{4} (t))$ and ${\tilde{B}}_{TrIFN}^{i} (t) = (b_{1} (t), b_{2} (t), b_{3} (t), b_{4} (t); {b'}_{1} (t), b_{2} (t), b_{3} (t), {b'}_{4} (t))$ be two time-dependent TrIFNs then

{\tilde{A}}_{T r I F N}^{i} (t) ⊖ {\tilde{B}}_{T r I F N}^{i} (t) = (a_{1} (t) + b_{1} (t), a_{2} (t) + b_{2} (t), a_{3} (t) + b_{3} (t), a_{4} (t) + b_{4} (t); {a^{'}}_{1} (t) + {b^{'}}_{1} (t), a_{2} (t) + b_{2} (t), a_{3} (t) + b_{3} (t), {a^{'}}_{4} (t) + {b^{'}}_{4} (t))

{\tilde{A}}_{TrIFN}^{i} (t) \oplus {\tilde{B}}_{TrIFN}^{i} (t) = (a_{1} (t) - b_{4} (t), a_{2} (t) - b_{3} (t), a_{3} (t) - b_{2} (t), a_{4} (t) - b_{1} (t); {a'}_{1} (t) - {b'}_{4} (t), a_{2} (t) - b_{3} (t), a_{3} (t) - b_{2} (t), {a'}_{4} (t) - {b'}_{1} (t))

λ {\tilde{A}}_{TrIFN}^{i} (t) = {\begin{matrix} (λ a_{1} (t), λ a_{2} (t), λ a_{3} (t), λ a_{4} (t); λ {a'}_{1} (t), λ a_{2} (t), λ a_{3} (t), λ {a'}_{4} (t)), λ \geq 0 \\ (λ a_{4} (t), λ a_{3} (t), λ a_{2} (t), λ a_{1} (t); λ {a'}_{4} (t), λ a_{3} (t), λ a_{2} (t), λ {a'}_{1} (t)), λ \leq 0 \end{matrix}

{\tilde{A}}_{TrIFN}^{i} (t) \otimes {\tilde{B}}_{TrIFN}^{i} (t) \approx (minimum (a (t)), minimum (b (t)), maximum (b (t)), maximum (a (t))); (minimum (a' (t)), minimum (b (t)), maximum (b (t)), maximum (a' (t)))

where

a (t) = (a_{1} (t) b_{1} (t), a_{1} (t) b_{4} (t), a_{4} (t) b_{1} (t), a_{4} (t) b_{4} (t))

b (t) = (a_{2} (t) b_{2} (t), a_{2} (t) b_{3} (t), a_{3} (t) b_{2} (t), a_{3} (t) b_{3} (t))

and

a' (t) = ({a'}_{1} (t) {b'}_{1} (t), {a'}_{1} (t) {b'}_{4} (t), {a'}_{4} (t) b' {(t)}_{1}, {a'}_{4} (t) {b'}_{4} (t))

Proposed method for solving time-dependent IFDEs

In this section, a new method is proposed for solving nth-order time-dependent intuitionistic fuzzy linear differential equation with initial conditions.

Any nth-order time-dependent intuitionistic fuzzy linear differential equation with initial conditions can be written as

\sum_{j = 0}^{n} {\tilde{a}}_{j} (t) \otimes {\tilde{y}}^{(j)} (t) = \tilde{g} (t), {\tilde{y}}^{(j)} (0) = {\tilde{γ}}_{j}, j = 0, 1, \dots, n - 1

(1)

where

{\tilde{y}}^{(j)} (t) = \frac{d^{j} \tilde{y} (t)}{d x^{j}}, {\tilde{a}}_{j} (t)

and $\tilde{g} (t)$ are time-dependent TrIFNs.

The solution of equation (1) can be obtained as follows:

Step 1. Assuming $a_{j} (t) = (a_{j (1)} (t), a_{j (2)} (t), a_{j (3)} (t), a_{j (4)} (t); {a^{'}}_{j (1)} (t), a_{j (2)} (t), a_{j (3)} (t), {a^{'}}_{j (4)} (t))$ , ${\tilde{y}}_{j} (t) = (y_{1}^{(j)} (t), y_{2}^{(j)} (t), y_{3}^{(j)} (t), y_{4}^{(j)} (t); {y^{'}}_{1}^{(j)} (t), y_{2}^{(j)} (t), y_{3}^{(j)} (t), {y^{'}}_{4}^{(j)} (t))$ , and $\tilde{g} (t) = (g_{1} (t), g_{2} (t), g_{3} (t), g_{4} (t); {g^{'}}_{1} (t), g_{2} (t), g_{3} (t), {g^{'}}_{4} (t))$ , the time-dependent intuitionistic fuzzy linear differential equation (1) can be written as

\begin{array}{l} \sum_{j = 0}^{n} (a_{j (1)} (t), a_{j (2)} (t), a_{j (3)} (t), a_{j (4)} (t); {a^{'}}_{j (1)} (t), a_{j (2)} (t), a_{j (3)} (t), {a^{'}}_{j (4)} (t)) \otimes (y_{1}^{(j)} (t), y_{2}^{(j)} (t), y_{3}^{(j)} (t), y_{4}^{(j)} (t); {y^{'}}_{1}^{(j)} (t), y_{2}^{(j)} (t), y_{3}^{(j)} (t), {y^{'}}_{4}^{(j)} (t)) \\ = (g_{1} (t), g_{2} (t), g_{3} (t), g_{4} (t); {g^{'}}_{1} (t), g_{2} (t), g_{3} (t), {g^{'}}_{4} (t)) (y_{1}^{(j)} (t), y_{2}^{(j)} (t), y_{3}^{(j)} (t), y_{4}^{(j)} (t); {y^{'}}_{1}^{(j)} (t), y_{2}^{(j)} (t), y_{3}^{(j)} (t), {y^{'}}_{4}^{(j)} (t)) = (γ_{j (1)}, γ_{j (2)}, γ_{j (3)}, γ_{j (4)}; {γ^{'}}_{j (1)}, γ_{j (2)}, γ_{j (3)}, {γ^{'}}_{j (4)}) j = 0, 1, \dots, n - 1 \end{array}

(2)

Step 2. Using Definition 5 and section “Arithmetic operations between time-dependent TrIFNs,” the time-dependent IFDE equation (2) can be transformed into following ordinary differential equations

\sum_{j = 0}^{n} minimum (b_{j} (t) z^{(j)} (t)) = g_{1} (t), y_{1}^{(j)} (0) = γ_{j (1)}, j = 0, 1, \dots, n - 1

\sum_{j = 0}^{n} minimum (c_{j} (t) z^{(j)} (t)) = g_{2} (t), y_{2}^{(j)} (0) = γ_{j (2)}, j = 0, 1, \dots, n - 1

\sum_{j = 0}^{n} maximum (c_{j} (t) z^{(j)} (t)) = g_{3} (t), y_{3}^{(j)} (0) = γ_{j (3)}, j = 0, 1, \dots, n - 1

\sum_{j = 0}^{n} maximum (b_{j} (t) z^{(j)} (t)) = g_{4} (t), y_{4}^{(j)} (0) = γ_{j (4)}, j = 0, 1, \dots, n - 1

\sum_{j = 0}^{n} minimum ({b'}_{j} (t) {z'}^{(j)} (t)) = {g'}_{1} (t), y'_{1}^{(j)} (0) = γ'_{j (1)}, j = 0, 1, \dots, n - 1

\sum_{j = 0}^{n} maximum ({b'}_{j} (t) {z'}^{(j)} (t)) = {g'}_{4} (t), y'_{4}^{(j)} (0) = γ'_{j (4)}, j = 0, 1, \dots, n - 1

where

b_{j} (t) z^{(j)} (t) = (a_{j (1)} (t) y_{1}^{(j)} (t), a_{j (1)} (t) y_{4}^{(j)} (t), a_{j (4)} (t) y_{1}^{(j)} (t), a_{j (4)} (t) y_{4}^{(j)} (t))

c_{j} (t) z^{(j)} (t) = (a_{j (2)} (t) y_{2}^{(j)} (t), a_{j (2)} (t) y_{3}^{(j)} (t), a_{j (3)} (t) y_{2}^{(j)} (t), a_{j (3)} (t) y_{3}^{(j)} (t))

b'_{j} (t) z'^{(j)} (t) = ({a'}_{j (1)} (t) y'_{1}^{(j)} (t), {a'}_{j (1)} (t) y'_{4}^{(j)} (t), {a'}_{j (4)} (t) y'_{1}^{(j)} (t), {a'}_{j (4)} (t) y'_{4}^{(j)} (t))

Step 3. Solve the ordinary differential equations, obtained in Step 2, to find the values of $y_{1} (t), y_{2} (t), y_{3} (t), y_{4} (t), {y'}_{1} (t), and {y'}_{4} (t)$ .

Step 4. Put the values of $y_{1} (t), y_{2} (t), y_{3} (t), y_{4} (t), {y'}_{1} (t), and {y'}_{4} (t)$ obtained from Step 3, in $\tilde{y} (t) = (y_{1} (t), y_{2} (t), y_{3} (t), y_{4} (t); {y'}_{1} (t), y_{2} (t), y_{3} (t), {y'}_{4} (t))$ to find the solution of time-dependent intuitionistic fuzzy linear differential equation (1).

Case study

The condensate system helps the power plants to function efficiently and keeps them in continuous operation for optimal performance. The condensate system consists of six subsystems, namely, $A, B, C, D, E, and F$ in series. Gupta and Tewari (2009) used crisp Markov model to analyze the crisp reliability of condensate system. In this section, to show the application of proposed method, the time-dependent intuitionistic fuzzy Markov model (shown in Figure 1) is obtained by replacing the crisp parameters of the same Markov model with time-dependent TrIFNs. Further, the time-dependent IFKDEs, obtained by time-dependent intuitionistic fuzzy Markov model, are solved by applying the proposed method.

Figure 1.

Time-dependent intuitionistic fuzzy Markov model of condensate system.

Subsystem $A$ consists of condenser. It is single unit arranged in series. Failure of this unit causes the complete failure of the system.

Subsystem $B$ consists of gland steam condenser arranged in series. Failure of this unit causes the complete failure of the system.

Subsystem $C$ consists of one drain cooler arranged in series. Failure of this unit causes the complete failure of the system.

Subsystem $D$ consists of three low-pressure heaters arranged in series. Failure of any one unit causes the complete failure of the system.

Subsystem $E$ consists of deaerator arranged in series. Failure of this unit causes the complete failure of the system.

Subsystem $F$ consists of two condensate extraction pumps arranged in parallel; one in operation and the other in cold standby. Complete failure of the system will occur when both failed at a time.

Notation

In this section, notation that is used for analyzing the IFR of condensate system is presented.

$◯$ : indicates that the system is in full working state.

$▭$ : indicates that the system is in failed state.

$A, B, C, D, E, and F$ : represent full working states of subsystems.

$F_{1}$ : denotes that the subsystem $F$ is working on standby unit.

$a, b, c, d, e, f$ : represent failed states of subsystems.

${\tilde{P}}_{0} (t)$ : time-dependent intuitionistic fuzzy probability of the system working with full capacity at time $t$ .

${\tilde{P}}_{1} (t)$ : time-dependent intuitionistic fuzzy probability of the system in cold standby state.

${\tilde{P}}_{2} (t) to P_{12} (t)$ : time-dependent intuitionistic fuzzy probability of the system in failed state.

${\tilde{ϕ}}_{i} (t), i = 1 to 6$ : time-dependent intuitionistic fuzzy failure rates of subsystems $A, B, C, D, E, and F$ , respectively.

${\tilde{λ}}_{i} (t), i = 1 to 6$ : time-dependent intuitionistic fuzzy repair rates of subsystems $A, B, C, D, E, and F$ , respectively.

Data

The time-dependent trapezoidal intuitionistic fuzzy failure rates and time-dependent trapezoidal intuitionistic fuzzy repair rates that are assumed for analyzing the IFR of condensate system are shown in Table 1.

Table 1.

Time-dependent trapezoidal intuitionistic fuzzy failure rates and time-dependent trapezoidal intuitionistic fuzzy repair rates for the different sub-systems of condensate system

Time-dependent trapezoidal intuitionistic fuzzy failure rate per day	Time-dependent trapezoidal intuitionistic fuzzy repair rate per day
${\tilde{ϕ}}_{1} (t) = (0.00615, 0.00684, 0.00836, 0.00919; 0.00553, 0.00684, 0.00836, 0.01011)$	${\tilde{λ}}_{1} (t) = (0.243, 0.27, 0.33, 0.363; 0.218, 0.27, 0.33, 0.399)$
${\tilde{ϕ}}_{2} (t) = (0.00818, 0.00909, 0.01111, 0.01222; 0.00736, 0.00909, 0.01111, 0.01344)$	${\tilde{λ}}_{2} (t) = (0.122, 0.135, 0.165, 0.182; 0.109, 0.135, 0.165, 0.2)$
${\tilde{ϕ}}_{3} (t) = (0.00332, 0.00369, 0.00451, 0.00496; 0.003, 0.00369, 0.00451, 0.00545)$	${\tilde{λ}}_{3} (t) = (0.284, 0.315, 0.385, 0.424; 0.256, 0.315, 0.385, 0.466)$
${\tilde{ϕ}}_{4} (t) = (0.00616, 0.00684, 0.00836, 0.00919; 0.00554, 0.00684, 0.00836, 0.01011)$	${\tilde{λ}}_{4} (t) = (0.203, 0.225, 0.275, 0.303; 0.183, 0.225, 0.275, 0.333)$
${\tilde{ϕ}}_{5} (t) = (0.00267, 0.00297, 0.00363, 0.00399; 0.00241, 0.00297, 0.00363, 0.00439)$	${\tilde{λ}}_{5} (t) = (0.151, 0.168, 0.206, 0.226; 0.136, 0.225, 0.275, 0.333)$
${\tilde{ϕ}}_{6} (t) = (0.0243, 0.027, 0.033, 0.0363; 0.02187, 0.027, 0.033, 0.0399)$	${\tilde{λ}}_{6} (t) = (0.223, 0.248, 0.303, 0.333; 0.2, 0.248, 0.303, 0.366)$

Time-dependent IFKDEs for the condensate system

In this section, time-dependent IFKDEs are developed by using time-dependent intuitionistic fuzzy Markov model of condensate system. Time-dependent IFKDEs for the condensate system, developed by using intuitionistic fuzzy Markov model, are given as follows

\frac{d {\tilde{P}}_{0} (t)}{dt} \oplus {\tilde{δ}}_{1} (t) {\tilde{P}}_{0} (t) = {\tilde{λ}}_{1} (t) {\tilde{P}}_{2} (t) \oplus {\tilde{λ}}_{2} (t) {\tilde{P}}_{3} (t) \oplus {\tilde{λ}}_{3} (t) {\tilde{P}}_{4} (t) \oplus {\tilde{λ}}_{4} (t) {\tilde{P}}_{5} (t) \oplus {\tilde{λ}}_{5} (t) {\tilde{P}}_{6} (t) \oplus {\tilde{λ}}_{6} (t) {\tilde{P}}_{1} (t)

(3)

no \frac{d {\tilde{P}}_{1} (t)}{dt} \oplus {\tilde{δ}}_{2} (t) {\tilde{P}}_{1} (t) = {\tilde{λ}}_{1} (t) {\tilde{P}}_{7} (t) \oplus {\tilde{λ}}_{2} (t) {\tilde{P}}_{8} (t) \oplus {\tilde{λ}}_{3} (t) {\tilde{P}}_{9} (t) \oplus {\tilde{λ}}_{4} (t) {\tilde{P}}_{10} (t) \oplus {\tilde{λ}}_{5} (t) {\tilde{P}}_{11} (t) \oplus {\tilde{λ}}_{6} (t) {\tilde{P}}_{12} (t) \oplus {\tilde{ϕ}}_{6} (t) {\tilde{P}}_{0} (t)

(4)

\frac{d {\tilde{P}}_{1 + i} (t)}{dt} \oplus {\tilde{λ}}_{i} (t) {\tilde{P}}_{1 + i} (t) = {\tilde{ϕ}}_{i} (t) {\tilde{P}}_{0} (t), i = 1, 2, 3, 4, 5

(5)

\frac{d {\tilde{P}}_{6 + i} (t)}{dt} \oplus {\tilde{λ}}_{i} (t) {\tilde{P}}_{6 + i} (t) = {\tilde{ϕ}}_{i} (t) {\tilde{P}}_{1} (t), i = 1, 2, 3, 4, 5, 6

(6)

where

{\tilde{δ}}_{1} (t) = {\tilde{ϕ}}_{1} (t) \oplus {\tilde{ϕ}}_{2} (t) \oplus {\tilde{ϕ}}_{3} (t) \oplus {\tilde{ϕ}}_{4} (t) \oplus {\tilde{ϕ}}_{5} (t) \oplus {\tilde{ϕ}}_{6} (t)

{\tilde{δ}}_{2} (t) = {\tilde{ϕ}}_{1} (t) \oplus {\tilde{ϕ}}_{2} (t) \oplus {\tilde{ϕ}}_{3} (t) \oplus {\tilde{ϕ}}_{4} (t) \oplus {\tilde{ϕ}}_{5} (t) \oplus {\tilde{ϕ}}_{6} (t) \oplus {\tilde{λ}}_{6} (t)

with time-dependent intuitionistic fuzzy initial conditions

{\tilde{P}}_{0} (0) = (0.95, 0.955, 0.965, 0.97; 0.945, 0.955, 0.965, 0.975)

{\tilde{P}}_{1} (0) = (0.004, 0.0045, 0.0055, 0.006; 0.0035, 0.0045, 0.0055, 0.0065)

and

{\tilde{P}}_{j} (0) = (0.004, 0.0045, 0.0055, 0.006; 0.0035, 0.0045, 0.0055, 0.0065), j = 2 to 12

Solution of time-dependent IFKDEs for the condensate system

Since, there is no method in the literature that can be used for solving time-dependent IFKDEs for the condensate system, developed in section “Time-dependent IFKDEs for the condensate system,” So, the proposed method is used to find the solution of these equations. The results obtained by using the proposed method at = 12 days, = 24 days, = 36 days and = 48 days is shown in Table 2.

Table 2.

Solution of time-dependent IFKDEs for the condensate system obtained by using proposed method

	${\tilde{p}}_{j} (t)$ for t = 12 days	${\tilde{p}}_{j} (t)$ for t = 24 days	${\tilde{p}}_{j} (t)$ for t = 36 days	${\tilde{p}}_{j} (t)$ for t = 48 days
$j$	$(p_{j 1} (t), p_{j 2} (t), p_{j 3} (t), p_{j 4} (t); {p'}_{j 1} (t), p_{j 2} (t), p_{j 3} (t), {p'}_{j 4} (t))$	$(p_{j 1} (t), p_{j 2} (t), p_{j 3} (t), p_{j 4} (t); {p'}_{j 1} (t), p_{j 2} (t), p_{j 3} (t), {p'}_{j 4} (t))$	$(p_{j 1} (t), p_{j 2} (t), p_{j 3} (t), p_{j 4} (t); {p'}_{j 1} (t), p_{j 2} (t), p_{j 3} (t), {p'}_{j 4} (t))$	$(p_{j 1} (t), p_{j 2} (t), p_{j 3} (t), p_{j 4} (t); {p'}_{j 1} (t), p_{j 2} (t), p_{j 3} (t), {p'}_{j 4} (t))$
0	(0.760896,0.760971,0.763181,0.765165; 0.760884,0.760971,0.763181,0.767427)	(0.743159,0.746228,0.753479,0.757468; 0.739729,0.746228,0.753479,0.761427)	(0.740458,0.7443,0.752577,0.756875; 0.735982,0.7443,0.752577,0.76105)	(0.739958,0.743991,0.752473,0.756819; 0.735182,0.743991,0.752473,0.76102)
1	(0.075375,0.076831,0.079356,0.080464; 0.073834,0.076831,0.079356,0.081398)	(0.079772,0.08041,0.081685,0.082323; 0.079163,0.08041,0.081685,0.082851)	(0.080445,0.080888,0.081915,0.082479; 0.080086,0.080888,0.081915,0.082952)	(0.080578,0.08097,0.081945,0.082496; 0.080293,0.08097,0.081945,0.082962)
2	(0.018806,0.019039,0.019328,0.019416; 0.018556,0.019039,0.019328,0.019514)	(0.018876,0.018959,0.019117,0.019195; 0.01884,0.018959,0.019117,0.019305)	(0.018758,0.018866,0.019068,0.019163; 0.018699,0.018866,0.019068,0.019284)	(0.018731,0.018849,0.019063,0.01916; 0.018656,0.018849,0.019063,0.019283)
3	(0.041181,0.042988,0.04592,0.047051; 0.039518,0.042988,0.04592,0.048181)	(0.048045,0.048984,0.050171,0.050503; 0.047397,0.048984,0.050171,0.050948)	(0.049312,0.049916,0.050614,0.050789; 0.049124,0.049916,0.050614,0.051128)	(0.04955,0.050064,0.05066,0.050812; 0.049513,0.050064,0.05066,0.051139)
4	(0.008842,0.008917,0.00899,0.009; 0.00878,0.008917,0.00899,0.009025)	(0.008721,0.008765,0.008837,0.008868; 0.008712,0.008765,0.008837,0.008909)	(0.008662,0.008722,0.008817,0.008854; 0.008636,0.008722,0.008817,0.008901)	(0.008651,0.008715,0.008814,0.008853; 0.008617,0.008715,0.008814,0.0089)
5	(0.021859,0.022289,0.022874,0.023035; 0.021332,0.022289,0.022874,0.023227)	(0.02258,0.022726,0.022939,0.022999; 0.022391,0.022726,0.022939,0.023134)	(0.022493,0.022643,0.022884,0.022959; 0.022314,0.022643,0.022884,0.023107)	(0.02246,0.02262,0.022875,0.022954; 0.022267,0.02262,0.022875,0.023104)
6	(0.011766,0.012135,0.012697,0.012948; 0.011383,0.012135,0.012697,0.013134)	(0.012972,0.013095,0.013252,0.013362; 0.012842,0.013095,0.013252,0.013421)	(0.013082,0.013156,0.013263,0.013363; 0.013016,0.013156,0.013263,0.013418)	(0.013085,0.013154,0.01326,0.013361; 0.013028,0.013154,0.01326,0.013417)
7	(0.001622,0.001717,0.001871,0.001931; 0.001524,0.001717,0.001871,0.001983)	(0.001981,0.002012,0.002059,0.002078; 0.001951,0.002012,0.002059,0.002095)	(0.00203,0.002045,0.002074,0.002087; 0.002021,0.002045,0.002074,0.002101)	(0.002038,0.00205,0.002075,0.002088; 0.002034,0.00205,0.002075,0.002102)
8	(0.003258,0.003564,0.004124,0.00437; 0.002972,0.003564,0.004124,0.004606)	(0.004812,0.005006,0.005285,0.005379; 0.004637,0.005006,0.005285,0.005467)	(0.00525,0.005353,0.005481,0.005518; 0.005186,0.005353,0.005481,0.005563)	(0.005365,0.005431,0.005512,0.005536; 0.005355,0.005431,0.005512,0.005573)
9	(0.000778,0.000819,0.000882,0.000906; 0.000738,0.000819,0.000882,0.000926)	(0.000921,0.000934,0.000953,0.000961; 0.000909,0.000934,0.000953,0.000967)	(0.000938,0.000946,0.000959,0.000964; 0.000935,0.000946,0.000959,0.00097)	(0.000941,0.000948,0.000959,0.000965; 0.00094,0.000948,0.000959,0.00097)
10	(0.001841,0.001965,0.002172,0.002253; 0.001711,0.001965,0.002172,0.002328)	(0.002348,0.002395,0.002463,0.002484; 0.002291,0.002395,0.002463,0.002508)	(0.002429,0.002452,0.002488,0.0025; 0.002404,0.002452,0.002488,0.002517)	(0.002442,0.00246,0.00249,0.0025; 0.002426,0.00246,0.00249,0.002518)
11	(0.000954,0.001031,0.001168,0.00123; 0.000878,0.001031,0.001168,0.001282)	(0.001322,0.001358,0.001411,0.001434; 0.001282,0.001358,0.001411,0.001448)	(0.001403,0.001418,0.00144,0.001454; 0.001389,0.001418,0.00144,0.001461)	(0.00142,0.001429,0.001443,0.001456; 0.001415,0.001429,0.001443,0.001462)
12	(0.006815,0.007229,0.007931,0.008221; 0.006385,0.007229,0.007931,0.008461)	(0.008488,0.008621,0.008843,0.008941; 0.008351,0.008621,0.008843,0.009012)	(0.008734,0.008788,0.008916,0.008988; 0.008701,0.008788,0.008916,0.009041)	(0.008774,0.008812,0.008924,0.008992; 0.008768,0.008812,0.008924,0.009044)

IFKDEs: intuitionistic fuzzy Kolmogorov’s differential equations.

IFR evaluation of condensate system

In this section, the results of time-dependent IFKDEs, shown in Table 2, are used to analyze the IFR of condensate system. Using the time-dependent intuitionistic fuzzy probabilities for the subsystems of condensate system, shown in Table 2, IFR $\tilde{R} (t) = (R_{11} (t), R_{12} (t), R_{13} (t), R_{14} (t); R'_{11} (t), R_{12} (t), R_{13} (t), R'_{14} (t)) = {\tilde{P}}_{0} (t) \oplus {\tilde{P}}_{1} (t)$ of condensate system is evaluated. The variation in reliability of condensate system at = 12 days and = 24 days is shown in Figure 2.

Figure 2.

TrIFN representing IFR of condensate system.

Also, the variation in reliability of condensate system with time is shown in Figure 3.

Figure 3.

Variation in reliability of condensate system with respect to time.

Conclusion

A new method is proposed for solving time-dependent IFDEs. To illustrate the proposed method, time-dependent IFKDEs, obtained by using time-dependent intuitionistic fuzzy Markov model of condensate system, are solved and the obtained solution is used to evaluate the IFR of condensate system. The proposed method cannot be used for solving nonlinear IFDEs. So, in future, it may be tried to modify the proposed method for the same.

Footnotes

Acknowledgements

The authors would like to thank the Editor-in-Chief “Biren Prasad” and anonymous referees for various suggestions that have led to an improvement in both the quality and clarity of the article. Dr Amit Kumar would also like to acknowledge the adolescent inner blessings of Mehar. Amit Kumar believes that Mehar is an angel for him and without Mehar’s blessing it was not possible to think the idea proposed in this article. Mehar is the lovely daughter of Parmpreet Kaur (Research Scholar under his supervision).

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Author Biographies

Sneh Lata is a PhD student under the supervision of Dr. Amit Kumar. She has completed MSc (Mathematics and Computing) from Thapar University, Patiala (Punjab), India. She has published 7 papers in journals and conference proceedings. Her area of research is Fuzzy Reliabilty Analysis.

Amit Kumar is an Assistant Professor in School of Mathematics and Computer Applications, Thapar University, Patiala (Punjab), India. He has published more than 70 papers in journals and conference proceedings. His current area of research is Fuzzy Optimization, Fuzzy Reliability Analysis and Vague Set Theory.

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