A new approach for solving flow shop scheduling problems with generalized intuitionistic fuzzy numbers

Abstract

In this paper a new centroid based ranking grade for generalized intuitionistic fuzzy numbers is proposed. The centroid point of membership function and non membership function of generalized intuitionistic fuzzy numbers in term of its parametric form is used for grading. The parametric representation of generalized intuitionistic fuzzy numbers involves left fuzziness index, right fuzziness index and modal value of membership and non membership functions. To reveal the performance of the proposed ranking grade, a comparison study has been made over the existing methods. Furthermore the proposed ranking method has been used for estimating the minimum total elapsed time to a flow shop scheduling problem involving generalized intuitionistic fuzzy number. An improved result for flow shop scheduling problem has been attained using the proposed ranking grade and has been illustrated through an example.

Keywords

Fuzzy set generalized intuitionistic fuzzy numbers centroid point ranking grade flow shop scheduling problem

1 Introduction

As a generalization of fuzzy set theory [16], Attanasov introduced intuitionistic fuzzy set (IFS) [15] which handles ill known quantities more flexibly. Because of its flexibility and the nature of expressing vagueness more precisely, intuitionistic fuzzy numbers (IFNs) has been studied and applied in different areas including decision making process. In many situations comparison of IFNs are difficult because of its overlapping nature. In this regard, many authors have studied on ranking methods for IFNs. Li [5, 6] developed a new ranking methodology based on value index and ambiguity index for generalized triangular intuitionistic fuzzy numbers (GTIFNs) and applied to multi-attribute decision making problems. For comparison of IFNs, Wang and Zang [12] proposed ranking method based on expected values, score function and accuracy function of IFNs. Dubey and mehra [4] obtained an optimal solution for intuitionistic fuzzy linear programming problem using Li’s ranking method. Followed by Li, Rezvani [31] proposed ranking method for both membership and non membership functions of trapezoidal intuitionistic fuzzy numbers. Wei and tan [3] have proposed a unique possibility degree ranking method for ‘n’ IFNs. Arun prakash [13] discussed about ranking of IFNs using centroid point of its membership function and non membership functions.

As motivated by the above mentioned ranking procedures, in this paper we have developed a new ranking grade using centroid point of membership and non membership functions of generalized intuitionistic fuzzy numbers (GIFNs) in terms of its parametric form is used for grading. The proposed ranking grade produces better result when compared with other ranking methods mentioned in literature.

In our day to day life flow shop sequencing methodology plays a vibrant role in manufacturing industries, processing units in computer environment, management etc., In most of the production sectors processed and semi processed goods are transferred from one machine to next machine for obtaining the final output. Under such situations the transportation durations are important in allocating the resources optimally. In today’s competitive business world, efficient scheduling has become mandatory to obtain minimum total completion time and optimal rental cost to improve their productions. In real world situations estimating the processing time and setup time are quite difficult. Hence the processing time and transportation time are represented as GTIFNs to deal with unreliable situations more effectively and it might help decision makers to obtain better result. Many researchers have contributed to flow shop scheduling problem under fuzzy environment. But there are still some gaps in flow shop scheduling problem involving fuzzy numbers as it is not adequate to deal with real life situations. Hence this paper attempts at considering a flow shop scheduling problem under IFS. Angelov [21] introduced the concept of intuitionistic fuzzy optimization problem. Parvathi and Malathi [25] presented a methodology for solving intuitionistic fuzzy linear programming problem, by constructing membership and non membership function for objective function and its constraints. Mukherjee and Basu [30] introduced a solution procedure for intuitionistic fuzzy assignment problem with and without restrictions on job cost and person cost based on his/her competence. The difficulty of the existence of negative objective function which arises while solving intuitionistic fuzzy assignment problem is overcomed by Ali Mahmoodirad et al. [1].

Johnson [28] introduced the renowned Johnson’s rule for ‘n’ jobs, ‘m’ machines flow shop scheduling problems to obtain optimal makespan. Maggu and Das [23] proposed the concept of equivalent jobs for job-blocks criteria in flow shop scheduling problem which includes the transportation time. Henri Prade [10] had implemented fuzzy set theory in a flow shop scheduling problem. Ishibuchi and Lee [9] formulated fuzzy flow shop scheduling problems whose processing times are fuzzy numbers and defined the concept of non dominated solutions based on an inequality relation between fuzzy numbers. Jing-Shing Yao and FrengTsc Lin [11] incorporated the concept of statistics with fuzzy set in flow shop scheduling problem. Martin and Roberto [17], studied fuzzy scheduling models on real time system whose tasks are periodic with execution times and deadlines as fuzzy numbers. Aggarwal et al. [26] obtained optimal completion time and rental cost of fuzzy flow shop scheduling problem under a specified rental policy with job block criteria. Uthra et al. [8] did a comparative study on flow shop scheduling problems whose processing times are GTIFNs and obtained a deterministic optimal solution.

In this paper we have also attempted to obtain an optimal mean completion time and minimized rental cost for flow shop scheduling problems involving generalized intuitionistic fuzzy processing time and transportation time without converting to crisp equivalent form by using the proposed ranking grade.

2 Basic concepts

In this section definitions and basic concepts of generalized intuitionistic fuzzy number used in this study are discussed.

Definition 1. [15] Let X be a universe of discourse. An Intuitionistic Fuzzy Set (IFS) ${\tilde{a}}^{I}$ in X is given by ${\tilde{a}}^{I} = {(x, μ_{{\tilde{a}}^{I} (x)}, γ_{{\tilde{a}}^{I}} (x)) / x \in X}$ , where the function $μ_{{\tilde{a}}^{I}} : X \to [0, 1]$ determines the degree of membership and the function $γ_{{\tilde{a}}^{I}} : X \to [0, 1]$ determines the degree of non membership of every element x ∈ X. Also for all x ∈ X, we have $0 \leq μ_{{\tilde{a}}^{I}} (x) + γ_{{\tilde{a}}^{I}} (x) \leq 1 .$

Definition 2. [15] For every common fuzzy subset ${\tilde{a}}^{I}$ on X, Intuitionistic Fuzzy Index of an element $x \in {\tilde{a}}^{I}$ is defined as $π_{{\tilde{a}}^{I}} (x) = 1 - μ_{{\tilde{a}}^{I}} (x) - γ_{{\tilde{a}}^{I}} (x)$ . For every x ∈ X, we have $0 \leq π_{{\tilde{a}}^{I}} (x) \leq 1 .$ This is called as degree of hesitancy or degree of uncertainty of the element x ∈ X. For each element $x \in {\tilde{a}}^{I}, \partial_{{\tilde{a}}^{I}} (x) = μ_{{\tilde{a}}^{I}} (x) (1 + π_{{\tilde{a}}^{I}} (x))$ represents the degree of favor and $η_{{\tilde{a}}^{I}} (x) = ν_{{\tilde{a}}^{I}} (x) (1 + π_{{\tilde{a}}^{I}} (x))$ represents the degree of against.

For example, let us consider an intuitionistic fuzzy set whose membership grade $μ_{{\tilde{a}}^{I}} (x)$ (the degree of object $x \in {\tilde{a}}^{I}$ ) is 0.7 and non membership grade $ν_{{\tilde{a}}^{I}} (x)$ (the degree of object $x \notin {\tilde{a}}^{I}$ ) is 0.2 then degree of hesitancy $π_{{\tilde{a}}^{I}} (x)$ is 0.1. Therefore $\partial_{{\tilde{a}}^{I}} (x)$ is 0.77 and $η_{{\tilde{a}}^{I}} (x)$ is 0.22.

Definition 3. The membership function and non-membership function of a generalized triangular intuitionistic fuzzy number (GTIFN) ${\tilde{a}}^{I}$ with parameters $a_{1}^{'} \leq a_{1} \leq a_{2}^{'} \leq a_{2} \leq a_{3} \leq a_{3}^{'}$ is denoted by ${\tilde{a}}^{I} = {(a_{1}, a_{2}, a_{3}), (a_{1}^{'}, a_{2}^{'}, a_{3}^{'}); w_{\tilde{a}}, ν_{\tilde{a}}},$ $0 \leq w_{\tilde{a}}, ν_{\tilde{a}} \leq 1$ and $0 \leq w_{\tilde{a}} + ν_{\tilde{a}} \leq 1$ , and defined as follows

$μ_{{\tilde{a}}^{I}} (x) = {\begin{matrix} \begin{matrix} 0 & for x < a_{1} \\ \frac{w_{\tilde{a}} (x - a_{1})}{a_{2} - a_{1}} & for a_{1} \leq x \leq a_{2} \\ \frac{w_{\tilde{a}} (a_{3} - x)}{a_{3} - a_{2}} & for a_{2} \leq x \leq a_{3} \\ 0 & for x > a_{3} \end{matrix} \end{matrix}$ and

$γ_{{\tilde{a}}^{I}} (x) = {\begin{matrix} \begin{matrix} 1 & for x < a_{1}^{'} \\ 1 + \frac{(ν_{\tilde{a}} - 1) (x - a_{1}^{'})}{a_{2}^{'} - a_{1}^{'}} & for a_{1}^{'} \leq x \leq a_{2}^{'} \\ v_{\tilde{a}} + \frac{(1 - v_{\tilde{a}}) (x - a_{1}^{'})}{a_{1}^{'} - a_{2}^{'}} & for a_{2}^{'} \leq x \leq a_{3}^{'} \\ 1 & for x > a_{3}^{'} \end{matrix} \end{matrix}$

We use G(R) to denote the set of all GTIFNs. The left spread and right spread of membership function is denoted as α₁ = (a₂ - a₁) , β₁ = (a₃ - a₂) respectively. In the same way the left spread and right spread of non- membership function is denoted as respectively. Also $m = a_{2}, m' = a_{2}^{'}$ represents the modal value (or) midpoint of membership and non membership functions respectively.

Definition 4. GTIFN ${\tilde{a}}^{I} \in F (R)$ can also be represented as a pair ${\tilde{a}}^{I} = (\underline{a}, \bar{a}; \underline{a}', \bar{a}')$ of functions $\underline{a} (r), \bar{a} (r), \underline{a}' (r^{'})$ and $\bar{a}' (r^{'})$ which satisfies the following requirements:

$\underline{a} (r)$ is a bounded monotonic increasing left continuous function for membership function.

$\bar{a} (r)$ is a bounded monotonic decreasing left continuous function for membership function.

$\underline{a} (r) \leq \bar{a} (r), 0 \leq r \leq 1$ .

$\underline{a}' (r^{*})$ is a bounded monotonic decreasing left continuous function for non- membership function.

$\bar{a}' (r^{*})$ is a bounded monotonic increasing left continuous function for non-membership function.

$\underline{a}' (r^{*}) \leq \bar{a}' (r^{*}), 0 \leq r^{*} \leq 1$ .

Definition 5. For an arbitrary GTIFN ${\tilde{a}}^{I} = (\underline{a}, \bar{a}; \underline{a}', \bar{a}')$ the location index number of membership and non membership function is represented as $a_{0} = (\frac{\underline{a} (1) + \bar{a} (1)}{2}), a_{0}^{'} = (\frac{\underline{a}' (1) + \bar{a}' (1)}{2})$ respectively. The non-decreasing left continuous functions $a_{*} = (a_{0} - \underline{a})$ and the non-increasing left continuous function $a^{*} = (\bar{a} - a_{0})$ are called the left fuzziness index function and the right fuzziness index function for the membership function. And in the same way the non-increasing left continuous functions $a_{*}^{'} = (a_{0}^{'} - \underline{a'})$ and the non-decreasing left continuous function $a^{' *} = ({\bar{a}}^{'} - a_{0}^{'})$ are called the left fuzziness index function and the right fuzziness index function for non-membership function respectively. Hence every GTIFN ${\tilde{a}}^{I}$ can also be represented by ${\tilde{a}}^{I} = (〈 a_{0}, a_{*}, a^{*} 〉, 〈 a_{0}^{'}, a_{*}^{'}, a^{' *} 〉; w_{\tilde{a}}, ν_{\tilde{a}})$ .

2.1 Arithmetic operations on GTIFN

For any two GTIFNs

${\tilde{a}}^{I} = (〈 a_{0}, a_{*}, a^{*} 〉, 〈 a_{0}^{'}, a_{*}^{'}, a^{' *} 〉; w_{\tilde{a}}, ν_{\tilde{a}})$ and ${\tilde{b}}^{I} = (〈 b_{0}, b_{*}, b^{*} 〉, 〈 b_{0}^{'}, b_{*}^{'}, b^{' *} 〉; w_{\tilde{b}}, ν_{\tilde{b}})$ we define [19].

Addition: $\begin{matrix} {\tilde{a}}^{I} + {\tilde{b}}^{I} = & (〈 a_{0}, a_{*}, a^{*} 〉, 〈 a_{0}^{'}, a_{*}^{'}, a^{' *} 〉; w_{\tilde{a}}, ν_{\tilde{a}}) \\ + (〈 b_{0}, b_{*}, b^{*} 〉, 〈 b_{0}^{'}, b_{*}^{'}, b^{' *} 〉; w_{\tilde{b}}, ν_{\tilde{b}}) \\ = & (〈 a_{0} + b_{0}, max {a_{*}, b_{*}}, max {a^{*}, b^{*}} 〉, \\ 〈 a_{0}^{'} + b_{0}^{'}, max {a_{*}^{'}, b_{*}^{'}}, max {a^{' *}, b^{' *}} 〉; m, n) \end{matrix}$ where $m = min {w_{\tilde{a}}, w_{\tilde{b}}}, n = max {ν_{\tilde{a}}, ν_{\tilde{b}}}$ .

Subtraction: $\begin{matrix} {\tilde{a}}^{I} - {\tilde{b}}^{I} = & (〈 a_{0}, a_{*}, a^{*} 〉, 〈 a_{0}^{'}, a_{*}^{'}, a^{' *} 〉; w_{\tilde{a}}, ν_{\tilde{a})} \\ - (〈 b_{0}, b_{*}, b^{*} 〉, 〈 b_{0}^{'}, b_{*}^{'}, b^{' *} 〉; w_{\tilde{b}}, ν_{\tilde{b}}) \\ = & (〈 a_{0} - b_{0}, max {a_{*}, b_{*}}, max {a^{*}, b^{*}} 〉, \\ 〈 a_{0}^{'} - b_{0}^{'}, max {a_{*}^{'}, b_{*}^{'}}, max {a^{' *}, b^{' *}} 〉; m, n) \end{matrix}$ where $m = min {w_{\tilde{a}}, w_{\tilde{b}}}, n = max {ν_{\tilde{a}}, ν_{\tilde{b}}}$ .

Scalar multiplication:

$λ {\tilde{a}}^{I}$ = ${\begin{matrix} (〈 λ a_{0}, λ a_{*}, λ a^{*} 〉, 〈 λ a_{0}^{'}, λ a_{*}^{'}, λ a^{' *} 〉; w_{\tilde{a}}, v_{\tilde{a}}) \\ for λ > 0 \\ (〈 λ a_{0}, - λ a_{*}, - λ a^{*} 〉, 〈 λ a_{0}^{'}, - λ a_{*}^{'}, - λ a^{' *} 〉; w_{\tilde{a}}, v_{\tilde{a}}) \\ for λ > 0 \end{matrix}$

3 Ranking of generalized triangular intuitionistic fuzzy numbers

In this section we define the centroid point or geometric center of membership and non membership functions of GTIFNs in terms of parametric form. The geometric center of GTIFNs corresponds to $x ({\tilde{a}}^{I})$ value on the horizontal axis and $y ({\tilde{a}}^{I})$ value on the vertical axis. The centroid point of GTIFN is denoted by $(x ({\tilde{a}}_{μ}^{I}), y ({\tilde{a}}_{μ}^{I})); (x ({\tilde{a}}_{γ}^{I}), y ({\tilde{a}}_{γ}^{I}))$ and is defined as

$\begin{matrix} x ({\tilde{a}}_{μ}^{I}) = & \frac{1}{3} [3 b + \frac{w_{\tilde{a}} (a^{*} - a_{*})}{w_{\tilde{a}} - r}] \\ y ({\tilde{a}}_{μ}^{I}) = & \frac{w_{\tilde{a}}}{3} \\ x ({\tilde{a}}_{γ}^{I}) = & [\frac{(2 - ν_{\tilde{a}} (1 + ν_{\tilde{a}})) (a^{' *} - a_{*}^{'})}{3 (r^{*} - ν_{\tilde{a}}) (1 + ν_{\tilde{a}})}] \\ + & [\frac{3 (1 + ν_{\tilde{a}}) (r^{*} - ν_{\tilde{a}}) a_{0}^{'}}{3 (r^{*} - ν_{\tilde{a}}) (1 + ν_{\tilde{a}})}] \\ y ({\tilde{a}}_{γ}^{I}) = & \frac{ν_{\tilde{a}}}{3} \end{matrix}$ Definition 6.The ranking function of the GTIFN ${\tilde{a}}^{I}$ is defined by

$R ({\tilde{a}}^{I}) = \sqrt{\frac{1}{2} ({[x ({\tilde{a}}_{μ}^{I}) - y ({\tilde{a}}_{μ}^{I})]}^{2} + {[x ({\tilde{a}}_{γ}^{I}) - y ({\tilde{a}}_{γ}^{I})]}^{2})}$ .

The proposed ranking function for $R ({\tilde{a}}^{I})$ satisfies the properties of wang and kerre [36].

For any two GTIFNs ${\tilde{a}}^{I} = (〈 a_{0}, a_{*}, a^{*} 〉, 〈 a_{0}^{'}, a_{*}^{'}, a^{' *} 〉; w_{\tilde{a}}, ν_{\tilde{a}})$ and ${\tilde{b}}^{I} = (〈 b_{0}, b_{*}, b^{*} 〉, 〈 b_{0}^{'}, b_{*}^{'}, b^{' *} 〉; w_{\tilde{b}}, ν_{\tilde{b}})$ in G(R), we have

If ${\tilde{a}}^{I} ≺ {\tilde{b}}^{I}$ then $R ({\tilde{a}}^{I}) < R ({\tilde{b}}^{I})$ .

If ${\tilde{a}}^{I} ≻ {\tilde{b}}^{I}$ then $R ({\tilde{a}}^{I}) > R ({\tilde{b}}^{I})$ .

If ${\tilde{a}}^{I} \approx {\tilde{b}}^{I}$ then $R ({\tilde{a}}^{I}) = R ({\tilde{b}}^{I})$ .

3.1 Examples

We consider some examples from [13] and comparison for the proposed ranking method with existing methods are given in Table 1.

Table 1
Comparison table for the proposed method

Ranking Process Example 3.1 Example 3.2 Example 3.3

Wei’s Process [3] ${\tilde{a}}_{1}^{I} \approx {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} \approx {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} \approx {\tilde{b}}_{3}^{I}$

Wang &Zhang’s ${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} > {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} > {\tilde{b}}_{3}^{I}$

Process [12]

Rezvani’s Process [31] ${\tilde{a}}_{1}^{I} < {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} < {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$

Li’s Process [5] ${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} \approx {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} \approx {\tilde{b}}_{3}^{I}$

Debey &Mehra’s ${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} \approx {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} \approx {\tilde{b}}_{3}^{I}$

Process [4]

Satyajit’s Process [27] ${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} < {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$

Sagaya’s Process [32] ${\tilde{a}}_{1}^{I} \approx {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} < {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$

Arun prakash ${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} > {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$

Process [13]

Proposed method ${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$ ${\tilde{a}}_{2}^{I} < {\tilde{b}}_{2}^{I}$ ${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$

Ranking Process	Example 3.1	Example 3.2	Example 3.3
Wei’s Process [3]	${\tilde{a}}_{1}^{I} \approx {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} \approx {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} \approx {\tilde{b}}_{3}^{I}$
Wang &Zhang’s	${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} > {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} > {\tilde{b}}_{3}^{I}$
Process [12]
Rezvani’s Process [31]	${\tilde{a}}_{1}^{I} < {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} < {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$
Li’s Process [5]	${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} \approx {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} \approx {\tilde{b}}_{3}^{I}$
Debey &Mehra’s	${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} \approx {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} \approx {\tilde{b}}_{3}^{I}$
Process [4]
Satyajit’s Process [27]	${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} < {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$
Sagaya’s Process [32]	${\tilde{a}}_{1}^{I} \approx {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} < {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$
Arun prakash	${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} > {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$
Process [13]
Proposed method	${\tilde{a}}_{1}^{I} > {\tilde{b}}_{1}^{I}$	${\tilde{a}}_{2}^{I} < {\tilde{b}}_{2}^{I}$	${\tilde{a}}_{3}^{I} < {\tilde{b}}_{3}^{I}$

Example 3.1: Let ${\tilde{a}}_{1}^{I} = (〈 0.57, 0.73, 0.83 〉, 〈 0.57, 0.73, 0.83 〉; 0.73, 0.2)$ and ${\tilde{b}}_{1}^{I} = (〈 0.58, 0.74, 0.819 〉, 〈 0.58, 0.74, 0.819 〉; 0.72$ ,0.2) be two GTIFNs. Then, $R ({\tilde{a}}_{1}^{I}) = 0.587$ , $R ({\tilde{b}}_{1}^{I}) = 0.455$ . Hence $R ({\tilde{a}}_{1}^{I}) > R ({\tilde{b}}_{1}^{I})$ .

Example 3.2: Let ${\tilde{a}}_{2}^{I} = (〈 3, 4, 5 〉, 〈 3, 4, 5 〉; 0.8, 0.2)$ and ${\tilde{b}}_{2}^{I} = (〈 6, 8, 10 〉, 〈 6, 8, 10 〉; 0.4, 0.6)$ be two GTIFNs. Then, $R ({\tilde{a}}_{2}^{I}) = 3.834$ , $R ({\tilde{b}}_{2}^{I}) = 7.834$ . Hence $R ({\tilde{a}}_{2}^{I}) < R ({\tilde{b}}_{2}^{I})$ .

Example 3.3: Let ${\tilde{a}}_{3}^{I} = (〈 1, 2, 3 〉, 〈 1, 2, 3 〉; 0.6, 0.4)$ and ${\tilde{b}}_{3}^{I} = (〈 2, 4, 6 〉, 〈 2, 4, 6 〉; 0.3, 0.7)$ be two GTIFNs. Then, $R ({\tilde{a}}_{3}^{I}) = 1.834$ , $R ({\tilde{b}}_{3}^{I}) = 3.834$ . Hence $R ({\tilde{a}}^{I}) < R ({\tilde{b}}^{I})$ .

4 Generalized intuitionistic fuzzy flow shop scheduling problem

Given n jobs to be processed on three machines M_j (j = 1, 2, 3) in the same technological order. Let ${\tilde{p}}_{ij}^{I}$ denote the processing time of job i on machine j represented by GTIFNs. Let ${\tilde{t}}_{i, j \to k}^{I}$ be the transportation time of the job i from machine j to machine k. Aim of the proposed method is to find the sequence S_q of the given jobs which minimize the rental cost of all the three machines while minimizing the total elapsed time. The mathematical model of the problem is represented in Table 3 and Fig. 1 shows the structure of solving the n-jobs 3-machine flow shop scheduling problem. We use the following notation for the formulation and solution of GIFN flow shop scheduling problem.

Fig.1

Structure of industrialized system solving the n-job 3-machine flow shop scheduling problem.

Sequence of jobs 1, 2, 3, ⋯ n .

S_q:

Sequence obtained by applying Johnson’s rule, q = 1, 2, 3, ⋯

M_j:

Machine j, j=1, 2, 3.

{\tilde{p}}_{ij}^{I}

Intuitionistic fuzzy Processing time of job i on machine M_j.

{\tilde{L}}_{j}^{I} (S_{q})

The latest starting time of machine M_j for the sequence S_q when it is taken on rent.

{\tilde{T}}_{ij}^{I} (S_{q})

Completion time of job i on machine M_j for the sequence S_q.

{\tilde{T}}_{ij}^{I^{'}} (S_{q})

Completion time of job i on machine M_j for the sequence S_q when machine M_j starts processing jobs at time ${\tilde{L}}_{j}^{I} (S_{q})$ .

{\tilde{I}}_{ij}^{I} (S_{q})

Idle time of machine M_j for job i in the sequence S_q.

{\tilde{t}}_{i, j \to k}^{I}

Transportation time of job i from machine j to machine k.

{\tilde{U}}_{j}^{I} (S_{q})

Usage time for machine M_j for the sequence S_q when M_j starts processing jobs at the time ${\tilde{L}}_{j}^{I} (S_{q})$ .

{\tilde{R}}^{I} (S_{q})

Total rental cost for the sequence S_q of all machines.

Definition 7. Completion time of job i on machine M_j is denoted by ${\tilde{T}}_{ij}^{I}$ and is defined as ${\tilde{T}}_{ij}^{I} = max ({\tilde{T}}_{(i - 1) j}^{I}, {\tilde{T}}_{i (j - 1)}^{I}) + {\tilde{t}}_{i (j - 1) \to j}^{I} + {\tilde{p}}_{ij}^{I}$ for j ≥ 2.

Definition 8. Completion time of job i on machine M_j when M_j starts processing jobs at time ${\tilde{L}}_{j}^{I}$ is denoted by ${\tilde{T}}_{ij}^{I^{'}}$ and is defined as ${\tilde{T}}_{ij}^{I^{'}} = {\tilde{L}}_{j}^{I} + \sum_{k = 1}^{i} {\tilde{p}}_{kj}^{I} = \sum_{k = 1}^{i} {\tilde{I}}_{kj}^{I} + \sum_{k = 1}^{i} {\tilde{p}}_{kj}^{I}$ also ${\tilde{T}}_{ij}^{I^{'}} = max ({\tilde{T}}_{i (j - 1)}^{I}, {\tilde{T}}_{(i - 1) j}^{I^{'}}) + {\tilde{p}}_{ij}^{I}$ .

4.1 Assumptions

A single job cannot be processed simultaneously by more than one machine.

The process time and transportation time of each job is known.

Jobs are processed as soon as possible.

n jobs should be processed through three machines M₁, M₂ and M₃ in the same order M₁M₂M₃ i.e. no passing is allowed.

There is only one machine of each type in the shop.

A machine is not kept idle if it can take up a job.

The storage space is available and the cost of holding inventory for each job is either same or negligible.

Time intervals for processing are independent of the order in which operations are performed.

4.2 Rental policy

The machines will be taken on rent as and when they are required and are returned as and when they are no longer required i.e. the first machine will be taken on rent in the starting of the processing the jobs, 2nd machine will be taken on rent at time when 1st job is completed on 1st machine and transported to 2nd machine, 3rd machine will be take on rent at time when 1st job is completed on the 2nd machine and transported.

Theorem [1]: [26]

The processing of jobs on M₃ at time ${\tilde{L}}_{3}^{I} = \sum_{i = 1}^{n} {\tilde{I}}_{i 3}^{I}$ keeps ${\tilde{T}}_{n 3}^{I}$ unaltered.

Theorem [2]: [26]

The processing of jobs on M₂ at time ${\tilde{L}}_{2}^{I} = min_{i \leq k \leq n} {{\tilde{X}}_{k}^{I}}$ keeps total elapsed time unaltered where ${\tilde{X}}_{1}^{I} = {\tilde{L}}_{3}^{I} - {\tilde{p}}_{12}^{I} - {\tilde{t}}_{1, 2 \to 3}^{I}$ and ${\tilde{X}}_{k}^{I} = {\tilde{T}}_{(k - 1) 3}^{I^{'}} - \sum_{i = 1}^{k} {\tilde{p}}_{i 2}^{I} - \sum_{i = 1}^{k} {\tilde{t}}_{i, 2 \to 3}^{I}; k > 1$ .

Remark: If M₃ starts processing jobs at time ${\tilde{L}}_{3}^{I} = {\tilde{T}}_{n 3}^{I} - \sum_{i = 1}^{k} {\tilde{p}}_{i 3}^{I}$ then the total elapsed time ${\tilde{T}}_{n 3}^{I}$ is not altered and M₃ is engaged for minimum time equal to sum of processing times of all the jobs on M₃, i.e. reducing the idle time of M₃ to zero. Moreover total elapsed time/rental cost of M₁ is always least as idle time of M₁ is always zero. Therefore the objective remains to minimize the elapsed time and hence the rental cost of M₂.

5 Algorithm

The following algorithm is proposed to obtain optimal sequence having minimum total elapsed time and reduced rental cost.

Step 1: Transform the given GTIFNs into ${\tilde{a}}^{I} = (〈 a_{0}, a_{*}, a^{*} 〉, 〈 a_{0}^{'}, a_{*}^{'}, a'^{*} 〉; w_{\tilde{a}}, ν_{\tilde{a}})$

Step 2: Check the condition. Either $min {{\tilde{p}}_{i 1}^{I} + {\tilde{t}}_{i, 1 \to 2}^{I}}$ $\geq max {{\tilde{p}}_{i 2}^{I} + {\tilde{t}}_{i, 1 \to 2}^{I}}$ or $min {{\tilde{p}}_{i 3}^{I} + {\tilde{t}}_{i, 2 \to 3}^{I}} \geq max {{\tilde{p}}_{i 2}^{I} + {\tilde{t}}_{i, 2 \to 3}^{I}}$ or both for all i. If the conditions are satisfied then go to step 3, else the given data is not in the standard form.

Step 3: Introduce the two fictitious machines G and H with processing times ${\tilde{G}}_{i}^{I}$ and ${\tilde{H}}_{i}^{I}$ as ${\tilde{G}}_{i}^{I} = {\tilde{p}}_{i 1}^{I} + {\tilde{t}}_{i, 1 \to 2}^{I} + {\tilde{p}}_{i 2}^{I} + {\tilde{t}}_{i, 2 \to 3}^{I}$ and ${\tilde{H}}_{i}^{I} = {\tilde{t}}_{i, 1 \to 2}^{I} + {\tilde{p}}_{i 2}^{I} + {\tilde{t}}_{i, 2 \to 3}^{I} + {\tilde{p}}_{i 3}^{I}$ for all i.

Step 4: Using Johnson’s procedure, obtain the sequence S_q having minimum elapsed time.

Step 5: Prepare In-Out tables for S_q and compute total elapsed time ${\tilde{T}}_{ij}^{I} (S_{q})$ .

Step 6: Compute latest starting time for machine M₃ for the sequence S_q as ${\tilde{L}}_{3}^{I} (S_{q}) = {\tilde{T}}_{n 3}^{I} (S_{q}) - \sum_{i = 1}^{n} {\tilde{p}}_{i 3}^{I}$ .

Step 7: For the sequence S_q, compute

${\tilde{T}}_{n 2}^{I} (S_{q})$

${\tilde{X}}_{1}^{I} (S_{q}) = {\tilde{L}}_{3}^{I} (S_{q}) - {\tilde{p}}_{12}^{I} (S_{q}) - {\tilde{t}}_{1, 2 \to 3}^{I}$

${\tilde{X}}_{k}^{I} (S_{q}) = {\tilde{L}}_{3}^{I} (S_{q}) - \sum_{i = 1}^{k} {\tilde{p}}_{i 2}^{I} (S_{q}) - \sum_{i = 1}^{k} {\tilde{t}}_{i, 2 \to 3}^{I} + \sum_{i = 1}^{k - 1} {\tilde{p}}_{i 3}^{I} (S_{q}) + \sum_{i = 1}^{k - 1} {\tilde{t}}_{i, 1 \to 2}^{I}; k = 2, 3, \dots, n$

${\tilde{L}}_{2}^{I} (S_{q}) = min_{1 \leq k \leq n} {{\tilde{X}}_{k}^{I} (S_{q})}$

${\tilde{U}}_{2}^{I} (S_{q}) = {\tilde{T}}_{n 2}^{I} (S_{q}) - {\tilde{L}}_{2}^{I} (S_{q})$

Step 8: Compute total rental cost of all the three machines for sequence S_q as

{\tilde{R}}^{I} (S_{q}) = \sum_{i = 1}^{n} {\tilde{p}}_{i 1}^{I} (S_{q}) \times C_{1} + {\tilde{U}}_{2}^{I} (S_{q}) \times C_{2} + {\tilde{U}}_{3}^{I} (S_{q}) \times C_{3}

5.1 Example

The given 5 jobs, 3 machine flow shop problem is considered with processing duration and transportation duration as a GTIFNs. The rental cost per unit time for machines M1 is 4 units, M2 is 2 units and M3 is 3 units, subject to the rental policy P.

Our objective is to obtain an optimal sequence to minimize the total rental cost of machines by applying the proposed arithmetic operations and ranking grade for GTIFNs. The processing time and transportation time with GTIFNs is given in Tables 2 and 4 represents the parametric form of processing time and transportation time. Table 5 represents the fictitious machines.

Table 2
Processing durations with GTIFNs [26]

Jobs Machine M₁ ${\tilde{t}}_{i, 1 \to 2}^{I}$ Machine M₂ ${\tilde{t}}_{i, 2 \to 3}^{I}$ Machine M₃

${\tilde{p}}_{i 1}^{I}$ ${\tilde{p}}_{i 2}^{I}$ ${\tilde{p}}_{i 3}^{I}$

1 (〈7, 8, 9〉, (〈2, 3, 4〉, (〈6, 7, 8〉, (〈1, 2, 3〉, (〈3, 4, 5〉,

〈6, 7.5, 10〉 ; 〈1, 3, 5〉 ; 〈5, 7, 9〉 ; 〈0.5, 2.5, 4〉 ; 〈2, 4, 6〉 ;

0.8, 0.2) 0.8, 0.2) 0.8, 0.2) 0.8, 0.2)

2 (〈12, 13, 14〉, (〈4, 5, 6〉, (〈5, 6, 7〉, (〈2, 3, 4〉, (〈4, 5, 6〉,

〈11, 13, 15〉 ; 〈3, 6, 7〉 ; 〈4, 5, 9〉 ; 〈1, 3.5, 6〉 ; 〈3, 6, 7〉 ;

0.7, 0.2) 0.7, 0.2) 0.7, 0.2) 0.7, 0.2) 0.7, 0.2)

3 (〈8, 10, 12〉, (〈5, 6, 7〉, (〈4, 5, 6〉, (〈3, 4, 5〉, (〈6, 7, 8〉,

〈7, 9, 13〉 ; 〈4, 5, 9〉 ; 〈3, 6, 7〉 ; 〈2, 4, 6〉 ; 〈5, 7.5, 9〉 ;

0.5, 0.3) 0.5, 0.3) 0.5, 0.3) 0.5, 0.3) 0.5, 0.3)

4 (〈10, 11, 12〉, (〈2, 3, 4〉, (〈5, 6, 7〉, (〈1, 2, 3〉, (〈11, 12, 13〉,

〈9, 10.5, 13〉 ; 〈1, 3, 5〉 ; 〈4, 5, 9〉 ; 〈0.5, 2, 4〉 ; 〈10, 12, 14〉 ;

0.6, 0.3) 0.6, 0.3) 0.6, 0.3) 0.6, 0.3) 0.6, 0.3)

5 (〈9, 10, 11〉, (〈5, 6, 7〉, (〈6, 7, 8〉, (〈3, 4, 5〉, (〈8, 9, 10〉,

〈8, 9, 12〉 ; 〈4, 5, 9〉 ; 〈5, 7, 10〉 ; 〈2, 4, 6〉 ; 〈7, 8, 11〉 ;

0.7, 0.2) 0.7, 0.2) 0.7, 0.2) 0.7, 0.2) 0.7, 0.2)

Jobs	Machine M₁	${\tilde{t}}_{i, 1 \to 2}^{I}$	Machine M₂	${\tilde{t}}_{i, 2 \to 3}^{I}$	Machine M₃
1	(〈7, 8, 9〉,	(〈2, 3, 4〉,	(〈6, 7, 8〉,	(〈1, 2, 3〉,	(〈3, 4, 5〉,
	〈6, 7.5, 10〉 ;	〈1, 3, 5〉 ;	〈5, 7, 9〉 ;	〈0.5, 2.5, 4〉 ;	〈2, 4, 6〉 ;
	0.8, 0.2)	0.8, 0.2)	0.8, 0.2)	0.8, 0.2)
2	(〈12, 13, 14〉,	(〈4, 5, 6〉,	(〈5, 6, 7〉,	(〈2, 3, 4〉,	(〈4, 5, 6〉,
	〈11, 13, 15〉 ;	〈3, 6, 7〉 ;	〈4, 5, 9〉 ;	〈1, 3.5, 6〉 ;	〈3, 6, 7〉 ;
	0.7, 0.2)	0.7, 0.2)	0.7, 0.2)	0.7, 0.2)	0.7, 0.2)
3	(〈8, 10, 12〉,	(〈5, 6, 7〉,	(〈4, 5, 6〉,	(〈3, 4, 5〉,	(〈6, 7, 8〉,
	〈7, 9, 13〉 ;	〈4, 5, 9〉 ;	〈3, 6, 7〉 ;	〈2, 4, 6〉 ;	〈5, 7.5, 9〉 ;
	0.5, 0.3)	0.5, 0.3)	0.5, 0.3)	0.5, 0.3)	0.5, 0.3)
4	(〈10, 11, 12〉,	(〈2, 3, 4〉,	(〈5, 6, 7〉,	(〈1, 2, 3〉,	(〈11, 12, 13〉,
	〈9, 10.5, 13〉 ;	〈1, 3, 5〉 ;	〈4, 5, 9〉 ;	〈0.5, 2, 4〉 ;	〈10, 12, 14〉 ;
	0.6, 0.3)	0.6, 0.3)	0.6, 0.3)	0.6, 0.3)	0.6, 0.3)
5	(〈9, 10, 11〉,	(〈5, 6, 7〉,	(〈6, 7, 8〉,	(〈3, 4, 5〉,	(〈8, 9, 10〉,
	〈8, 9, 12〉 ;	〈4, 5, 9〉 ;	〈5, 7, 10〉 ;	〈2, 4, 6〉 ;	〈7, 8, 11〉 ;
	0.7, 0.2)	0.7, 0.2)	0.7, 0.2)	0.7, 0.2)	0.7, 0.2)

Table 3

Representation of the mathematical model of the problem in matrix form

Jobs	Machine M₁	${\tilde{t}}_{i, 1 \to 2}^{I}$	Machine M₂	${\tilde{t}}_{i, 2 \to 3}^{I}$	Machine M₃
i	${\tilde{p}}_{i 1}^{I}$		${\tilde{p}}_{i 2}^{I}$		${\tilde{p}}_{i 3}^{I}$
1	${\tilde{p}}_{11}^{I}$	${\tilde{t}}_{1, 1 \to 2}^{I}$	${\tilde{p}}_{12}^{I}$	${\tilde{t}}_{1, 2 \to 3}^{I}$	${\tilde{p}}_{13}^{I}$
2	${\tilde{p}}_{21}^{I}$	${\tilde{t}}_{2, 1 \to 2}^{I}$	${\tilde{p}}_{22}^{I}$	${\tilde{t}}_{2, 2 \to 3}^{I}$	${\tilde{p}}_{23}^{I}$
3	${\tilde{p}}_{31}^{I}$	${\tilde{t}}_{3, 1 \to 2}^{I}$	${\tilde{p}}_{32}^{I}$	${\tilde{t}}_{3, 2 \to 3}^{I}$	${\tilde{p}}_{33}^{I}$
4	${\tilde{p}}_{41}^{I}$	${\tilde{t}}_{4, 1 \to 2}^{I}$	${\tilde{p}}_{42}^{I}$	${\tilde{t}}_{4, 2 \to 3}^{I}$	${\tilde{p}}_{43}^{I}$
–	–	–	–	–	–
–	–	–	–	–	–
n	${\tilde{p}}_{n 1}^{I}$	${\tilde{t}}_{n, 1 \to 2}^{I}$	${\tilde{p}}_{n 2}^{I}$	${\tilde{t}}_{n, 2 \to 3}^{I}$	${\tilde{p}}_{n 3}^{I}$

Table 4

Parametric form of processing durations

Jobs	Machine M₁	${\tilde{t}}_{i, 1 \to 2}^{I}$	Machine M₂	${\tilde{t}}_{i, 2 \to 3}^{I}$	Machine M₃
	${\tilde{p}}_{i 1}^{I}$		${\tilde{p}}_{i 2}^{I}$		${\tilde{p}}_{i 3}^{I}$
1	(〈8, 1 - 1.25r, 1 - 1.25r〉,	(〈3, 1 - 1.25r, 1 - 1.25r〉,	(〈7, 1 - 1.25r, 1 - 1.25r〉,	(〈2, 1 - 1.25r, 1 - 1.25r〉,	(〈4, 1 - 1.25r, 1 - 1.25r〉
	〈8.01, 0.13 + 1.88r * ,	〈3, - 0.5 + 2.5r * ,	〈7, - 0.5 + 2.5r * ,	〈2.26, - 0.74 + 2.5r * ,	〈4, - 0.5 + 2.5r * ,
	-1.13 + 3.13r * 〉;0.8, 0.2)	-0.5 + 2.5r * 〉;0.8, 0.2)	-0.5 + 2.5r * 〉;0.8, 0.2)	-0.13 + 1.88r * 〉;0.8, 0.2)	-0.5 + 2.5r * 〉;0.8, 0.2)
2	(〈13, 1 - 1.43r, 1 - 1.43r〉,	(〈5, 1 - 1.43r, 1 - 1.43r〉,	(〈6, 1 - 1.43r, 1 - 1.43r〉,	(〈3, 1 - 1.43r, 1 - 1.43r〉,	(〈5, 1 - 1.43r, 1 - 1.43r〉,
	〈13, - 0.5 + 2.5r * ,	〈5, - 1.75 + 3.75r * ,	〈6.5, 1.25 + 1.25r * ,	〈3.51, - 0.62 + 3.13r * ,	〈5, - 1.75 + 3.75r * ,
	-0.5 + 2.5r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.7, 0.2)	-2.5 + 5r * 〉;0.7, 0.2)	-0.63 + 3.13r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.7, 0.2)
3	(〈10, 2 - 4r, 2 - 4r〉,	(〈6, 1 - 2r, 1 - 2r〉,	(〈5, 1 - 2r, 1 - 2r〉,	(〈4, 1 - 2r, 1 - 2r〉,	(〈7, 1 - 2r, 1 - 2r〉,
	〈10, 0.14 + 2.86r * ,	〈6.5, 1.07 + 1.43r * ,	〈5, - 2.29 + 4.29r * ,	〈4, - 0.86 + 2.86r * ,	〈7, - 1.57 + 3.57r * ,
	-2.71 + 5.71r * 〉;0.5, 0.3)	-3.21 + 5.71r * 〉;0.5, 0.3)	-0.57 + 1.43r * 〉;0.5, 0.3)	-0.86 + 2.86r * 〉;0.5, 0.3)	-0.14 + 2.14r * 〉;0.5, 0.3)
4	(〈11, 1 - 1.66r, 1 - 1.66r〉,	(〈3, 1 - 1.66r, 1 - 1.66r〉,	(〈6, 1 - 1.66r, 1 - 1.66r〉,	(〈2, 1 - 1.66r, 1 - 1.66r〉,	(〈12, 1 - 1.66r, 1 - 1.66r〉,
	〈11, - 0.14 + 2.14r * ,	〈3, - 0.86 + 2.86r * ,	〈6.49, 1.06 + 1.43r * ,	〈2.25, - 0.39 + 2.14r * ,	〈12, - 0.86 + 2.86r * ,
	-1.57 + 3.57r * 〉;0.6, 0.3)	-0.86 + 2.86r * 〉;0.6, 0.3)	-3.21 + 5.71r * 〉;0.6, 0.3)	-1.11 + 2.86r * 〉;0.6, 0.3)	-0.86 + 2.86r * 〉;0.6, 0.3)
5	(〈10, 1 - 1.43r, 1 - 1.43r〉,	(〈6, 1 - 1.43r, 1 - 1.43r〉,	(〈7, 1 - 1.43r, 1 - 1.43r〉,	(〈4, 1 - 1.43r, 1 - 1.43r〉,	(〈9, 1 - 1.43r, 1 - 1.43r〉,
	〈11.43, - 0.75 + 1.25r * ,	〈6.5, 1.25 + 1.25r * ,	〈7.5, 2.5r * ,	〈4, - 0.5 + 2.5r * ,	〈9, 0.75 + 1.25r * ,
	-1.75 + 3.75r * 〉;0.7, 0.2)	-2.5 + 5r * 〉;0.7, 0.2)	-1.25 + 3.75r * 〉;0.7, 0.2)	-0.5 + 2.5r * 〉;0.7, 0.2)	-1.75 + 3.75r * 〉;0.7, 0.2)

Table 5

Representation of two fictitious machines

Jobs	${\tilde{G}}_{i}^{I} = {\tilde{p}}_{i 1}^{I} + {\tilde{t}}_{i, 1 \to 2} + {\tilde{p}}_{i 2}^{I} + {\tilde{t}}_{i, 2 \to 3}$	${\tilde{H}}_{i}^{I} = {\tilde{t}}_{i, 1 \to 2} + {\tilde{p}}_{i 2}^{I} + {\tilde{t}}_{i, 2 \to 3} + {\tilde{p}}_{i 3}^{I}$
1	(〈20, 1 - 1.25r, 1 - 1.25r〉,	(〈16, 1 - 1.25r, 1 - 1.25r〉,
	〈20.27, 0.13 + 1.88r * ,	〈16.26, - 0.5 + 2.5r * ,
	-0.13 + 1.88r * 〉;0.8, 0.2)	-0.13 + 1.88r * 〉;0.8, 0.2)
2	(〈27, 1 - 1.43r, 1 - 1.43r〉,	(〈19, 1 - 1.43r, 1 - 1.43r〉,
	〈28.01, 1.25 + 1.25r * ,	〈20.01, 1.25 + 1.25r * ,
	0.75 + 1.25r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.7, 0.2)
3	(〈25, 2 - 4r, 2 - 4r〉,	(〈22, 1 - 2r, 1 - 2r〉,
	〈25.5, 1.07 + 1.43r * ,	〈22.5, 1.07 + 1.43r * ,
	0.57 + 1.43r * 〉;0.5, 0.3)	0.57 + 1.43r * 〉;0.5, 0.3)
4	(〈22, 1 - 1.66r, 1 - 1.66r〉,	(〈23, 1 - 1.66r, 1 - 1.66r〉,
	〈22.74, 1.06 + 1.43r * ,	〈23.74, 1.06 + 1.43r * ,
	-0.86 + 2.86r * 〉;0.6, 0.3)	-0.86 + 2.86r * 〉;0.6, 0.3)
5	(〈27, 1 - 1.43r, 1 - 1.43r〉,	(〈26, 1 - 1.43r, 1 - 1.43r〉,
	〈29.43, 1.25 + 1.25r * ,	〈27, 1.25 + 1.25r * ,
	-0.5 + 2.5r * 〉;0.7, 0.2)	-0.5 + 2.5r * 〉;0.7, 0.2)

From the Tables 6 and 7 we have obtained total elapsed time ${\tilde{T}}_{n 3}^{I} (S_{q}) = (〈 68, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈70.19, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{L}}_{3}^{I} (S_{q}) = {\tilde{T}}_{n 3}^{I} (S_{q}) - \sum_{i = 1}^{n} {\tilde{p}}_{i 3}^{I}$ ${\tilde{L}}_{3}^{I} (S_{q}) = (〈 31, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈33.19, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) For the sequence S_q, we have ${\tilde{T}}_{n 2}^{I} (S_{q}) = (〈 62, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈63.93, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{X}}_{1}^{I} (S_{q}) = (〈 23, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈24.45, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{X}}_{2}^{I} (S_{q}) = (〈 27, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈27.95, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{X}}_{3}^{I} (S_{q}) = (〈 33, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈34.45, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{X}}_{4}^{I} (S_{q}) = (〈 37, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈37.94, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{X}}_{5}^{I} (S_{q}) = (〈 38, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈38.68, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{L}}_{2}^{I} (S_{q}) = min {{\tilde{X}}_{k}^{I}}$ , where $min {{\tilde{X}}_{k}^{I}}$ is calculated by the proposed ranking function. ${\tilde{L}}_{2}^{I} (S_{q}) = (〈 23, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈24.45, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{U}}_{2}^{I} (S_{q}) = {\tilde{T}}_{n 2}^{I} (S_{q}) - {\tilde{L}}_{2}^{I} (S_{q})$ ${\tilde{U}}_{2}^{I} (S_{q}) = (〈 39, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈39.48, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) ${\tilde{U}}_{3}^{I} (S_{q}) = {\tilde{T}}_{n 3}^{I} (S_{q}) - {\tilde{L}}_{3}^{I} (S_{q})$ ${\tilde{U}}_{3}^{I} (S_{q}) = (〈 37, 1 - 1.25 r, 1 - 1.25 r 〉,$ 〈37, 1.25 + 1.25r * , 0.75 + 1.25r * 〉;0.5, 0.3) . From the reduced In-Out Tables 8 and 9. The latest possible time at which machine M₂ should be taken on rent is ${\tilde{L}}_{2}^{I} (S_{q}) = (〈 23, 1 - 1.25 r, 1 - 1.25 r 〉, 〈 24.45, 1.25 + 1.25 r *, 0.75 + 1.25 r * 〉; 0.5, 0.3)$ The utilization time of machine M₂ is $M_{2} = {\tilde{U}}_{2}^{I} (S_{q}) = {\tilde{T}}_{n 2}^{I} (S_{q}) - {\tilde{L}}_{2}^{I} (S_{q})$ ${\tilde{U}}_{2}^{I} (S_{q}) = (〈 39, 1 - 1.25 r, 1 - 1.25 r 〉, 〈 39.48, 1.25 + 1.25 r *, 0.75 + 1.25 r * 〉; 0.5, 0.3)$ Total minimum rental cost is ${\tilde{R}}^{I} (S_{q}) = \sum_{i = 1}^{n} {\tilde{p}}_{i 1}^{I} (S_{q}) \times C_{1} + {\tilde{U}}_{2}^{I} (S_{q}) \times C_{2} + {\tilde{U}}_{3}^{I} (S_{q}) \times C_{3}$ . ${\tilde{R}}^{I} (S_{q}) = (〈 397, 4 - 5 r, 4 - 5 r 〉,$ 〈403.72, 0.56 + 11.44r * , 2.25 + 3.75r * 〉;0.5, 0.3)

Table 6

In-Out table

Jobs	Machine ${\tilde{p}}_{i 1}^{I}$		${\tilde{t}}_{i, 1 \to 2}^{I}$	Machine ${\tilde{p}}_{i 2}^{I}$
	In	Out		In	Out
4	(〈0, 0, 0〉,	(〈11, 1 - 1.66r, 1 - 1.66r〉,	(〈3, 1 - 1.66r, 1 - 1.66r〉,	(〈14, 1 - 1.66r, 1 - 1.66r〉,	(〈20, 1 - 1.66r, 1 - 1.66r〉,
	〈0, 0, 0〉;0.5, 0.3)	〈11, - 0.14 + 2.14r * ,	〈3, - 0.86 + 2.86r * ,	〈14, - 0.14 + 2.14r * ,	〈20.49, 1.06 + 1.43r * ,
	-1.57 + 3.57r * 〉;0.5, 0.3)	-0.86 + 2.86r * 〉;0.6, 0.3)	-0.86 + 2.86r * 〉;0.5, 0.3	-0.86 + 2.86r * 〉;0.5, 0.3)
5	(〈11, 1 - 1.66r, 1 - 1.66r〉,	(〈21, 1 - 1.43r, 1 - 1.43r〉,	(〈6, 1 - 1.43r, 1 - 1.43r〉,	(〈27, 1 - 1.43r, 1 - 1.43r〉,	(〈34, 1 - 1.43r, 1 - 1.43r〉,
	〈11, - 0.14 + 2.14r * ,	〈22.43, 0.75 + 1.25r * ,	〈6.5, 1.25 + 1.25r * ,	〈28.93, 1.25 + 1.25r * ,	〈36.43, 1.25 + 1.25r * ,
	-1.57 + 3.57r * 〉;0.6, 0.3)	-1.57 + 3.57r * 〉;0.5, 0.3)	-2.5 + 5r * 〉;0.7, 0.2)	-1.57 + 3.57r * 〉;0.5, 0.3)	-1.25 + 3.75r * 〉;0.5, 0.3)
3	(〈21, 1 - 1.43r, 1 - 1.43r〉,	(〈31, 1 - 1.43r, 1 - 1.43r〉,	(〈6, 1 - 2r, 1 - 2r〉,	(〈37, 1 - 1.43r, 1 - 1.43r〉,	(〈42, 1 - 1.43r, 1 - 1.43r〉,
	〈22.43, 0.75 + 1.25r * ,	〈32.43, 0.14 + 2.86r * ,	〈6.5, 1.07 + 1.43r * ,	〈38.93, 1.07 + 1.43r * ,	〈43.93, 1.07 + 1.43r * ,
	-1.57 + 3.57r * 〉;0.5, 0.3)	-1.57 + 3.57r * 〉;0.5, 0.3)	-3.21 + 5.71r * 〉;0.5, 0.3)	-1.25 + 3.57r * 〉;0.5, 0.3)	0.57 + 1.43r * 〉;0.5, 0.3)
2	(〈31, 1 - 1.43r, 1 - 1.43r〉,	(〈44, 1 - 1.43r, 1 - 1.43r〉,	(〈5, 1 - 1.43r, 1 - 1.43r〉,	(〈49, 1 - 1.43r, 1 - 1.43r〉,	(〈55, 1 - 1.43r, 1 - 1.43r〉,
	〈32.43, 0.14 + 2.86r * ,	〈45.43, 0.14 + 2.86r * ,	〈5, - 1.75 + 3.75r * ,	〈50.43, 0.14 + 2.86r * ,	〈56.93, 1.25 + 1.25r * ,
	-1.57 + 3.57r * 〉;0.5, 0.3)	-0.5 + 2.5r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
1	(〈44, 1 - 1.43r, 1 - 1.43r〉,	(〈52, 1 - 1.25r, 1 - 1.25r〉,	(〈3, 1 - 1.25r, 1 - 1.25r〉,	(〈55, 1 - 1.43r, 1 - 1.43r〉,	(〈62, 1 - 1.25r, 1 - 1.25r〉,
	〈45.43, 0.14 + 2.86r * ,	〈53.44, 0.14 + 2.86r * ,	〈3, - 0.5 + 2.5r * ,	〈56.93, 1.25 + 1.25r * ,	〈63.93, 1.25 + 1.25r * ,
	-0.5 + 2.5r * 〉;0.5, 0.3)	-0.5 + 2.5r * 〉;0.5, 0.3)	-0.5 + 2.5r * 〉;0.8, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)

Table 7

In-Out table

Jobs	${\tilde{t}}_{i, 2 \to 3}^{I}$	Machine ${\tilde{p}}_{i 3}^{I}$
		In	Out
4	(〈2, 1 - 1.66r, 1 - 1.66r〉,	(〈22, 1 - 1.66r, 1 - 1.66r〉,	(〈34, 1 - 1.66r, 1 - 1.66r〉,
	〈2.25, - 0.39 + 2.14r * ,	〈22.74, 1.06 + 1.43r * ,	〈34.74, 1.06 + 1.43r * ,
	-1.11 + 2.86r * 〉;0.6, 0.3)	-0.86 + 2.86r * 〉;0.5, 0.3)	-0.86 + 2.86r * 〉;0.5, 0.3)
5	(〈4, 1 - 1.43r, 1 - 1.43r〉,	(〈38, 1 - 1.43r, 1 - 1.43r〉,	(〈47, 1 - 1.43r, 1 - 1.43r〉,
	〈4, - 0.5 + 2.5r * ,	〈40.43, 1.25 + 1.25r * ,	〈49.43, 1.25 + 1.25r * ,
	-0.5 + 2.5r * 〉;0.7, 0.2)	-0.5 + 2.5r * 〉;0.5, 0.3)	-0.5 + 2.5r * 〉;0.5, 0.3)
3	(〈4, 1 - 2r, 1 - 2r〉,	(〈47, 1 - 1.43r, 1 - 1.43r〉,	(〈54, 1 - 1.43r, 1 - 1.43r〉,
	〈4, - 0.86 + 2.86r * ,	〈49.43, 1.25 + 1.25r * ,	〈56.43, 1.25 + 1.25r * ,
	-0.86 + 2.86r * 〉;0.5, 0.3)	-0.5 + 2.5r * 〉;0.5, 0.3)	-0.14 + 2.14r * 〉;0.5, 0.3)
2	(〈3, 1 - 1.43r, 1 - 1.43r〉,	(〈58, 1 - 1.43r, 1 - 1.43r〉,	(〈63, 1 - 1.43r, 1 - 1.43r〉,
	〈3.51, - 0.62 + 3.13r * ,	〈60.44, 1.25 + 1.25r * ,	〈65.44, 1.25 + 1.25r * ,
	-0.63 + 3.13r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
1	(〈2, 1 - 1.25r, 1 - 1.25r〉,	(〈64, 1 - 1.25r, 1 - 1.25r〉,	(〈68, 1 - 1.25r, 1 - 1.25r〉,
	〈2.26, - 0.74 + 2.5r * ,	〈66.19, 1.25 + 1.25r * ,	〈70.19, 1.25 + 1.25r * ,
	-0.13 + 1.88r * 〉;0.8, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)

Table 8

The reduced In-Out table

Jobs	Machine ${\tilde{p}}_{i 1}^{I}$		${\tilde{t}}_{i, 1 \to 2}^{I}$	Machine ${\tilde{p}}_{i 2}^{I}$
	In	Out		In	Out
4	(〈0, 0, 0〉,	(〈11, 1 - 1.66r, 1 - 1.66r〉,	(〈3, 1 - 1.66r, 1 - 1.66r〉, **	(〈23, 1 - 1.25r, 1 - 1.25r〉,	(〈29, 1 - 1.25r, 1 - 1.25r〉,
	〈0, 0, 0〉;0.5, 0.3)	〈11, - 0.14 + 2.14r * ,	〈3, - 0.86 + 2.86r * ,	〈24.45, 1.25 + 1.25r * ,	〈30.94, 1.25 + 1.25r * ,
	-1.57 + 3.57r * 〉;0.5, 0.3)	-0.86 + 2.86r * 〉;0.6, 0.3)	-0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
5	(〈11, 1 - 1.66r, 1 - 1.66r〉,	(〈21, 1 - 1.43r, 1 - 1.43r〉,	(〈6, 1 - 1.43r, 1 - 1.43r〉,	(〈29, 1 - 1.25r, 1 - 1.25r〉,	(〈36, 1 - 1.25r, 1 - 1.25r〉,
	〈11, - 0.14 + 2.14r * ,	〈22.43, 0.75 + 1.25r * ,	〈6.5, 1.25 + 1.25r * ,	〈30.94, 1.25 + 1.25r * ,	〈38.44, 1.25 + 1.25r * ,
	-1.57 + 3.57r * 〉;0.6, 0.3)	-1.57 + 3.57r * 〉;0.5, 0.3)	-2.5 + 5r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
3	(〈21, 1 - 1.43r, 1 - 1.43r〉,	(〈31, 1 - 1.43r, 1 - 1.43r〉,	(〈6, 1 - 2r, 1 - 2r〉,	(〈37, 1 - 1.43r, 1 - 1.43r〉,	(〈42, 1 - 1.43r, 1 - 1.43r〉,
	〈22.43, 0.75 + 1.25r * ,	〈32.43, 0.14 + 2.86r * ,	〈6.5, 1.07 + 1.43r * ,	〈38.93, 1.07 + 1.43r * ,	〈43.93, 1.07 + 1.43r * ,
	-1.57 + 3.57r * 〉;0.5, 0.3)	-1.57 + 3.57r * 〉;0.5, 0.3)	-3.21 + 5.71r * 〉;0.5, 0.3)	-1.57 + 3.57r * 〉;0.5, 0.3)	0.57 + 1.43r * 〉;0.5, 0.3)
2	(〈31, 1 - 1.43r, 1 - 1.43r〉,	(〈44, 1 - 1.43r, 1 - 1.43r〉,	(〈5, 1 - 1.43r, 1 - 1.43r〉,	(〈49, 1 - 1.43r, 1 - 1.43r〉,	(〈55, 1 - 1.43r, 1 - 1.43r〉,
	〈32.43, 0.14 + 2.86r * , *	〈45.43, 0.14 + 2.86r * ,	〈5, - 1.75 + 3.75r * ,	〈50.43, 0.14 + 2.86r * ,	〈56.93, 1.25 + 1.25r * ,
	-1.57 + 3.57r * 〉;0.5, 0.3)*	-0.5 + 2.5r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
1	(〈44, 1 - 1.43r, 1 - 1.43r〉,	(〈52, 1 - 1.25r, 1 - 1.25r〉,	(〈3, 1 - 1.25r, 1 - 1.25r〉,	(〈55, 1 - 1.43r, 1 - 1.43r〉,	(〈62, 1 - 1.25r, 1 - 1.25r〉,
	〈45.43, 0.14 + 2.86r * ,	〈53.44, 0.14 + 2.86r * ,	〈3, - 0.5 + 2.5r * ,	〈56.93, 1.25 + 1.25r * ,	〈63.93, 1.25 + 1.25r * ,
	-0.5 + 2.5r * 〉;0.5, 0.3)	-0.5 + 2.5r * 〉;0.5, 0.3)	-0.5 + 2.5r * 〉;0.8, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)

Table 9

The reduced In-Out table

Jobs	${\tilde{t}}_{i, 2 \to 3}^{I}$	Machine ${\tilde{p}}_{i 3}^{I}$
		In	Out
4	(〈2, 1 - 1.66r, 1 - 1.66r〉,	(〈31, 1 - 1.25r, 1 - 1.25r〉,	(〈43, 1 - 1.25r, 1 - 1.25r〉,
	〈2.25, - 0.39 + 2.14r * ,	〈33.19, 1.25 + 1.25r * ,	〈45.19, 1.25 + 1.25r * ,
	-1.11 + 2.86r * 〉;0.6, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
5	(〈4, 1 - 1.43r, 1 - 1.43r〉,	(〈43, 1 - 1.25r, 1 - 1.25r〉,	(〈52, 1 - 1.25r, 1 - 1.25r〉,
	〈4, - 0.5 + 2.5r * ,	〈45.19, 1.25 + 1.25r * ,	〈54.19, 1.25 + 1.25r * ,
	-0.5 + 2.5r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
3	(〈4, 1 - 2r, 1 - 2r〉,	(〈52, 1 - 1.25r, 1 - 1.25r〉,	(〈59, 1 - 1.25r, 1 - 1.25r〉,
	〈4, - 0.86 + 2.86r * ,	〈54.19, 1.25 + 1.25r * ,	〈61.19, 1.25 + 1.25r * ,
	-0.86 + 2.86r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
2	(〈3, 1 - 1.43r, 1 - 1.43r〉,	(〈59, 1 - 1.25r, 1 - 1.25r〉,	(〈64, 1 - 1.25r, 1 - 1.25r〉,
	〈3.51, - 0.62 + 3.13r * ,	〈61.19, 1.25 + 1.25r * ,	〈66.19, 1.25 + 1.25r * ,
	-0.63 + 3.13r * 〉;0.7, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)
1	(〈2, 1 - 1.25r, 1 - 1.25r〉,	(〈64, 1 - 1.25r, 1 - 1.25r〉,	(〈68, 1 - 1.25r, 1 - 1.25r〉,
	〈2.26, - 0.74 + 2.5r * ,	〈66.19, 1.25 + 1.25r * ,	〈70.19, 1.25 + 1.25r * ,
	-0.13 + 1.88r * 〉;0.8, 0.2)	0.75 + 1.25r * 〉;0.5, 0.3)	0.75 + 1.25r * 〉;0.5, 0.3)

6 Discussion

A flow shop scheduling problems under GIFN is discussed by few authors. Uthra et al., obtained a solution for intuitionistic fuzzy flow shop scheduling problem by converting it into an equivalent crisp problem to avoid the computational complexity. In such case they obtained mean completion time and minimized rental time which is a crisp number and may not represent the original problem. The crisp equivalent model of the problem is not exactly interpreting the intuitionistic fuzzy nature of the intuitionistic fuzzy flow shop scheduling problem. To overcome this issue we have attempt to solve intuitionistic fuzzy flow shop scheduling problem, in which any GIFNs can be expressed to its parametric form and the arithmetic operations proposed by Ming ma [19] has been extended for the set of GTIFNs which is computationally easy. The proposed centroid based ranking method is used to find the optimal sequence having minimum total elapsed time. The In-Out tables are computed using proposed arithmetic operations. The obtained total elapsed time and reduced rental cost are {(67, 68, 69) , (67.7, 70.19, 72.19) ;0.5, 0.3} and {(393, 397, 401) , (391.73, 403.72, 409.73) ;0.5, 0.3}. This explains the intuitionistic fuzzy nature of the intuitionistic fuzzy flow shop scheduling problem effectively.

7 Conclusion

In this paper a new centroid based ranking grade for GIFN is proposed. The centroid points ofmembership and non membership functions of GIFNs in terms of its parametric form is used for grading. To expose the performance of the proposed ranking grade, we have attempted to solve flow shop scheduling problems involving generalized triangular intuitionistic fuzzy processing time and transportation time without converting to crisp equivalent form. The proposed centroid based ranking method is used to find the optimal sequence S_q having minimum total elapsed time. The main advantage of this method is easy and simple for computational purpose.

8 Future work

We planned to continue our work on Picture fuzzy set, which is a generalization of the fuzzy set and intuitionistic fuzzy set. Picture fuzzy set [18, 29] based decision making models are also available in literature which is adequate only for situations of human opinions.

Footnotes

Acknowledgement

The authors would like to thank the anonymous referees for their valuable reviews and constructive suggestions for the improvement of this article.

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