Multi-criteria decision-making methods under soft rough fuzzy knowledge

Abstract

Due to the uncertainty of real world problems and the limitation of human’s knowledge to understand the complex problems, it is very difficult for one to apply only a single type of uncertainty method to cope with such problems. One can develop a more powerful new model to solve decision making problems by incorporating the advantages of many other different theories of uncertainty. Fuzzy sets, soft sets and rough sets are very useful mathematical models for dealing with uncertainty. Combinations of these models result into several useful hybrid models. In view of this, in this research paper, the concepts and methods of rough soft sets and fuzzy sets are used to construct a new soft rough fuzzy set model. We employ the concept of soft rough fuzzy sets to graphs and investigate some properties of this model. We apply this new model to describe and resolve some multi-criteria decision-making problems.

Keywords

Soft rough fuzzy relation soft rough fuzzy digraph decision-making 03E72 68R10 68R05

1. Introduction

Vagueness arises in several complex issues of engineering, science and many other fields. These issues cannot be resolved using crisp mathematical methods. Many well-known theories, including probability theory, fuzzy set theory (FST), intuitionistic fuzzy set theory (IFST),vague set theory (VST), interval mathematics theory (IMT) and rough set theory (RST) have been established to explain uncertainty. Molodtsov [22] pointed out the limitations of these theories. To overcome these limitations, he initiated the idea of soft sets, which can be regarded as a useful mathematical model for coping with vagueness. Soft set theory (SST) seems to be free from the limitations affecting the existing methods. SST has very useful applications in many fields, including operational research, Perron integration, probability theory, and measurement theory. Due to practical applications of SST, many researchers worked on it and applied it to different decision-making problems. Maji et al. [19] discussed various operations on soft sets and also gave the idea of hybrid structures, including fuzzy sets and soft sets. They introduced the concept of fuzzy soft sets (FSSs), which can be considered as a fuzzy generalization of soft sets. Majumdar and Samanta [20] modified the definition of FSSs and introduced the notion of generalized fuzzy soft set (GFSS). Yang et al. [31, 32] introduced the notion of interval valued fuzzy soft sets (IVFSSs) by combining interval-valued fuzzy set (IVFS) and soft set models (SSMs) and also discussed multi fuzzy soft sets (MFSSs) with applications in DM.

RST was introduced by Pawlak [26]. The idea of RST is a generalization of crisp set theory (CST) to study the intelligence systems containing incomplete, inexact or uncertain information. It is an effective drive for bestowal with vagueness and incomplete information. RST is an important mathematical approach to imprecise knowledge. RST expresses vagueness in terms of a boundary region of a crisp set. If the boundary region of a set is empty then it is crisp, otherwise it is rough. A subset of a universe in RST is expressed by two approximations which are known as lower approximation (LA) and upper approximation (UA). Equivalence classes are the basic building structures in RST. Classical sets are uniquely described by their elements, it means an element belongs to or not belongs to the set. Therefore, the notion of a set is crisp(precise) one. But in rough sets, we have some additional information related to the elements. Elements which have same information are kept in one equivalence class and with the help of these equivalence classes, we take LA and UA of a set which leads to uncertainty. The main idea of rough sets is truly based on the classification of uncertain information. Thus, rough set theory deals with imprecise data.

Due to attractiveness of RST, many researchers and practitioners have contributed a lot of research to its development and applications. The study of hybrid models combining RSs with other mathematical structures is becoming apparent as an effective research topic of RST (see [6 , 14]). There are many interesting applications of RST in the areas of decision analysis, expert systems, inductive reasoning, knowledge acquisition, machine learning, and pattern recognition. Dubois and Prade [12] investigated rough sets and fuzzy sets and concluded that these two theories are different approaches to handle vagueness. They reported that these are not opposite theories and to obtain beneficial results, both theories can be combined. In result of that investigation, they introduced fuzzy rough sets (FRSs) and rough fuzzy sets (RFSs). Nakamura [24], Nanda [25], Biswas [9], Banerjee and Pal [8] also studied FRSs and RFSs. Rough fuzzy relations (RFRs) were introduced by Pawlak [27]. Zhang et al. [35] discussed the union and intersection operations of rough sets based on various approximation spaces. Furthermore, fuzziness in rough sets was discussed and measured by Chakrabarty et al. [10].

Graph theory is a tremendously helpful tool in solving combinatorial problems in various fields, including computer science, engineering, operations research, optimization, and physical systems. A digraph is a graph having directed edges(arcs). The arrows on the arcs are used to describe the directional information: an arc from vertex w to vertex z shows that one can go from w to z but not from z to w. Directed graphs (or digraphs) are used for modeling distributed and parallel systems. Wu [30] introduced fuzzy digraphs (FDs). Akram et al. [1] presented novel applications of intuitionistic fuzzy digraphs (IFDs) in decision support systems (DSSs). Later, Akram et al. [2] further used bipolar fuzzy digraphs (BFDs) in some DSSs. Rough sets have been used in various DM problems. The existing results of rough sets and other extended rough sets such as RFSs, generalized rough fuzzy sets (GRFSs), soft fuzzy rough sets (SRFSs) and intuitionistic fuzzy soft rough sets (IFSRSs) based DM models have their advantages and limitations(see [28 , 36]). Graphs deal with those DM problems which involve relationships between the given alternatives, so here is convenient to use graphs based DM methods instead of set-based. Recently, Akram et al. [5 , 38] developed some hybrid models of rough sets and fuzzy graphs, including rough fuzzy digraphs and intuitionistic fuzzy rough graphs with their applications in decision-making. Like soft rough set model (SRSM), the hybrid model involving soft rough set (SRS) with fuzzy set, SRFS also can be applied to various applications of real world. Therefore, we apply a new approach to decision making based on soft rough fuzzy digraphs (SRFDs). The paper is organized as follows. In Section 2 we give all basic notions related to our soft rough fuzzy model and introduce some new concepts including soft rough relation (SRR), soft rough digraph (SRD) and soft rough fuzzy relation (SRFR). In Section 3 we introduce SRFDs and discuss the different construction methods of SRFDs with their properties. In Section 4 we develop mainly steps and the algorithm of the DM method based on the soft rough fuzzy model (SRFM). Section 5 concludes the paper with some remarks and future directions of research.

2. Soft rough fuzzy information

Definition 2.1. [7] Let V be a set of universe and W a set of parameters. Let C, D ⊆ W and (F, C) , (G, D) be two soft sets over V, then a relation from (F, C) to (G, D) is a soft subset of (F, C) × (G, D), where (F, C) × (G, D) is Cartesian product of (F, C) and (G, D). That is, (F, C) × (G, D) = (H, C × D), where H : C × D → P (V × V) and H (c, d) = F (c) × G (d), i.e., ∀ (c, d) ∈ C × D $H (c, d) = {(h_{i}, h_{j}); where h_{i} \in F (c) and h_{j} \in G (d)},$

Definition 2.2. [36]Let V be a set of universe and W a set of parameters. Let Q be an arbitrary soft relation over V × W, then a set-valued function (SVF) $Q_{s} : V \to P (W)$ is defined by $Q_{s} (u) = {w \in W | (u, w) \in Q}, \forall u \in V .$

The pair (V, W, Q) is termed as a soft approximation space (SAS). Let L^* ⊆ W, then the lower soft approximation (LSA) and upper soft approximation (USA) of L^* with respect to (V, W, Q), represented by $\underline{Q} L^{*}$ and $\bar{Q} L^{*}$ , respectively, are defined by $\begin{matrix} \underline{Q} L^{*} & = & {u \in V | Q_{s} (u) \subseteq L^{*}}, \\ \bar{Q} L^{*} & = & {u \in V | Q_{s} (u) \cap L^{*} \neg = \emptyset} . \end{matrix}$

The pair $(\underline{Q} L^{*}, \bar{Q} L^{*})$ is termed as a SRS.

Definition 2.3.Let (V, W, Q) and (A, B, R) be two SASs, where A ⊆ V × V, B ⊆ W × W, R is a soft relation over A × B such that $\begin{matrix} (u_{1} u_{2}, w_{1} w_{2}) \in R \Leftrightarrow (u_{1}, w_{1}), (u_{2}, w_{2}) \in Q, \\ \forall u_{1} u_{2} \in A, w_{1} w_{2} \in B . \end{matrix}$

Let $R_{s} : A \to P (B)$ be a SVF, defined by $\begin{matrix} R_{s} (u_{1} u_{2}) = {w_{1} w_{2} \in B | (u_{1} u_{2}, w_{1} w_{2}) \in R}, \\ \forall u_{1} u_{2} \in A . \end{matrix}$

Let N^* ⊆ B, then the LSA and USA of N^* with respect to (A, B, R), represented by $\underline{R} N^{*}$ and $\bar{R} N^{*}$ , respectively, are defined by $\begin{matrix} \underline{R} N^{*} & = & {u_{1} u_{2} \in A | R_{s} (u_{1} u_{2}) \subseteq N^{*}}, \\ \bar{R} N^{*} & = & {u_{1} u_{2} \in A | R_{s} (u_{1} u_{2}) \cap N^{*} \neg = \emptyset} . \end{matrix}$

The pair $(\underline{R} N^{*}, \bar{R} N^{*})$ is termed as a soft rough relation (SRR).

Example 2.1. Let V = {c, d} be a set of universe and W = {p, q, r} a set of parameters. A soft set K over V is defined by $K (p) = {c, d}, K (q) = {d}, K (r) = {c} .$

Then a soft relation Q over V × W can be written as

Q	p	q	r
c	1	0	1
d	1	1	0

Let A = {cc, cd, dc, dd} and B = {pp, pq, pr, qp, qq, qr, rp, rq, rr}. Then a soft relation R over A × B can be written as

R	pp	pq	pr	qp	qq	qr	rp	rq	rr
cc	1	0	1	0	0	0	1	0	1
cd	1	1	0	0	0	0	1	1	0
dc	1	0	1	1	0	1	0	0	0
dd	1	1	0	1	1	0	0	0	0

From Definition 2.3, we have $\begin{matrix} R_{s} (cc) & = & {pp, pr, rp, rr}, \\ R_{s} (cd) & = & {pp, pq, rp, rq}, \\ R_{s} (dc) & = & {pp, pr, qp, qr}, \\ R_{s} (dd) & = & {pp, pq, qp, qq} . \end{matrix}$

Let N^* = {pp, pq, pr, qp, qq, qr}, then $\underline{R} N^{*} = {dc, dd}, \bar{R} N^{*} = {cc, cd, dc, dd} = A .$

Hence, ${RN}^{*} = (\underline{R} N^{*}, \bar{R} N^{*})$ is a SRR.

We now define soft rough digraph (SRD).

Definition 2.4. A SRD on a nonempty set V is an 5-ordered tuple $𝔾 = (W, Q, {QL}^{*}, R, {RN}^{*})$ such that

W is a set of parameters,

Q is an arbitrary soft relation over V × W,

R is an arbitrary soft relation over A × B,

${QL}^{*} = (\underline{Q} L^{*}, \bar{Q} L^{*})$ is a SRS on V,

${RN}^{*} = (\underline{R} N^{*}, \bar{R} N^{*})$ is a SRR on V,

(QL^*, RN^*) is a digraph, where $\underline{𝔾} = (\underline{Q} L^{*}, \underline{R} N^{*})$ and $\bar{𝔾} = (\bar{Q} L^{*}, \bar{R} N^{*})$ are lower and upper approximate subdigraphs of $𝔾$ .

Example 2.2. Let V = {c, d} be a set of universe and W = {p, q, r} a set of parameters. Let Q be an arbitrary soft relation over V × W defined by defined by

Q	p	q	r
c	1	0	1
d	1	1	0

Let L^* = {p, r} ⊆ W, then ${QL}^{*} = (\underline{Q} L^{*}, \bar{Q} L^{*})$ is a SRS, where $\underline{Q} L^{*} = {c, f}, \bar{Q} L^{*} = {c, d, f} .$

Let A = {cc, cd, dc, dd} and B = {pp, pq, pr, qp, qq, qr, rp, rq, rr}. Then a soft relation R over A × B can be written as

R	pp	pq	pr	qp	qq	qr	rp	rq	rr
cc	1	0	1	0	0	0	1	0	1
cd	1	1	0	0	0	0	1	1	0
dc	1	0	1	1	0	1	0	0	0
dd	1	1	0	1	1	0	0	0	0

Let N^* = {pp, pq, pr, qp, qq, qr}, then ${RN}^{*} = (\underline{R} N^{*}, \bar{R} N^{*})$ is a SRR, where $\underline{R} N^{*} = {dc, dd}, \bar{R} N^{*} = {cc, cd, dc, dd} = A .$

Thus, $\underline{𝔾} = (\underline{Q} L^{*}, \underline{R} N^{*})$ and $\bar{𝔾} = (\bar{Q} L^{*}, \bar{R} N^{*})$ are digraphs as shown in Fig. 1.

Fig. 1.

Soft rough digraph $𝔾 = (\underline{𝔾}, \bar{𝔾})$ .

Hence, $𝔾 = (\underline{𝔾}, \bar{𝔾})$ is a SRD.

Definition 2.5. [36] Let V be a set of universe and W a set of parameters. Let Q be an arbitrary soft relation over V × W, then $Q_{s} : V \to P (W)$ is a SVF defined by $Q_{s} (u) = {w \in W | (u, w) \in Q}, \forall u \in V .$

The pair (V, W, Q) is called a SAS. Let L be a fuzzy set on W, then the LSA and USA of L with respect to (V, W, Q), represented by $\underline{Q} L$ and $\bar{Q} L$ , respectively, are defined by $\begin{matrix} (\underline{Q} L) (u) & = & \underset{w \in Q_{s} (u)}{\land} L (w), \\ (\bar{Q} L) (u) & = & \underset{w \in Q_{s} (u)}{\lor} L (w), \forall u \in V . \end{matrix}$

The pair $(\underline{Q} L, \bar{Q} L)$ is termed as a SRFS

Example 2.3. Let V = {c, d, e} be a set of universe and W = {p, q, r} a set of parameters. A soft set K over V is defined by $K (p) = {c, d}, K (q) = φ, K (r) = {d, e} .$

Then a soft relation Q over V × W can be written as

Q	p	q	r
c	1	0	0
d	1	0	1
e	0	0	1

From Definition 2.5, we have $Q_{s} (c) = {p}, Q_{s} (d) = {p, r}, Q_{s} (e) = {r} .$

Let L = {(p, 0.6) , (q, 0.7) , (r, 0.4} be a fuzzy set on W, then $\begin{matrix} \underline{Q} L & = & {(c, 0.6), (d, 0.4), (e, 0.4)}, \\ \bar{Q} L & = & {(c, 0.6), (d, 0.6), (e, 0.4)} . \end{matrix}$

Hence, $QL = (\underline{Q} L, \bar{Q} L)$ is a SRFS.

Definition 2.6. Let (V, W, Q) and (A, B, R) be two SASs, where A ⊆ V × V, B ⊆ W × W, R is a soft relation over A × B such that $\begin{matrix} (u_{1} u_{2}, w_{1} w_{2}) \in R \Leftrightarrow (u_{1}, w_{1}), (u_{2}, w_{2}) \in Q, \\ \forall u_{1} u_{2} \in A, w_{1} w_{2} \in B . \end{matrix}$

Let $R_{s} : A \to P (B)$ be a SVF, defined by $\begin{matrix} R_{s} (u_{1} u_{2}) = {w_{1} w_{2} \in B | (u_{1} u_{2}, w_{1} w_{2}) \in R}, \\ \forall u_{1} u_{2} \in A . \end{matrix}$

Assume that N is a fuzzy set on B defined by $(N) (w_{1} w_{2}) \leq \underset{w \in W}{m i n} {L (w)}, \forall w_{1} w_{2} \in B,$

where L is a fuzzy set on W. Then the LSA and USA of N with respect to (A, B, R), represented by $\underline{R} N$ and $\bar{R} N$ , respectively, are defined by $\begin{array}{l} (\underline{R} N) (u_{1} u_{2}) = \underset{w_{1} w_{2} \in R_{s} (u_{1} u_{2})}{\land} N (w_{1} w_{2}), \\ (\bar{R} N) (u_{1} u_{2}) = \underset{w_{1} w_{2} \in T_{s} (u_{1} u_{2})}{\lor} N (w_{1} w_{2}) . \end{array}$

The pair $(\underline{R} N, \bar{R} N)$ is termed as a SRFR.

Example 2.4. Let V = {c, d, e, f} be a set of universe and W = {p, q, r, s} a set of parameters. A soft set K over V is defined by $\begin{matrix} K (p) = {c, e, f}, K (q) = {c, d, f}, K (r) = {c, d, e}, \\ K (s) = {e} . \end{matrix}$

Then a soft relation Q over V × W can be written as

Q	p	q	r	s
c	1	1	1	0
d	0	1	1	0
e	1	0	1	1
f	1	1	0	0

Let L = {(p, 0.7) , (q, 0.4) , (r, 0.5) , (s, 0.6)} be a fuzzy set on W. Let A = {cc, de, ed, ee, fc, fe} and B = {pq, ps, qq, qr, qs, rs, sq, ss}. Then a soft relation R over A × B can be written as

R	pq	ps	qq	qr	qs	rs	sq	ss
cc	1	0	1	1	0	0	0	0
de	0	0	0	1	1	1	0	0
ed	1	0	0	0	0	0	1	0
ee	0	1	0	0	0	1	0	1
fc	1	0	1	1	0	0	0	0
fe	0	1	0	1	1	0	0	0

From Definition 2.6, we have $\begin{matrix} R_{s} (cc) & = & {pq, qq, qr} = R_{s} (fc), \\ R_{s} (de) & = & {qr, qs, rs}, \\ R_{s} (ed) & = & {pq, sq}, \\ R_{s} (ee) & = & {ps, rs, ss}, \\ R_{s} (fe) & = & {ps, qr, qs} . \end{matrix}$

Let N = {(pq, 0.3) , (ps, 0.2) , (qq, 0.1) , (qr, 0.4) , (qs, 0.2) , (rs, 0.4) , (sq, 0.2) , (ss, 0.3)}, then $\begin{matrix} \underline{R} N & = & {(cc, 0.1), (de, 0.2), (ed, 0.2), (ee, 0.2), \\ (fc, 0.1), (fe, 0.2)}, \\ \bar{R} N & = & {(cc, 0.4), (de, 0.4), (ed, 0.3), (ee, 0.4), \\ (fc, 0.4), (fe, 0.4)} . \end{matrix}$

Hence, $RN = (\underline{R} N, \bar{R} N)$ is a SRFR.

3. Soft rough fuzzy digraphs

Definition 3.1. A SRFD on a nonempty set V is an 5-ordered tuple $\hat{G} = (W, Q, QL, R, RN)$ such that

W is a set of parameters,

Q is an arbitrary soft relation over V × W,

R is an arbitrary soft relation over A × B,

$QL = (\underline{Q} L, \bar{Q} L)$ is a SRFS on V,

$RN = (\underline{R} N, \bar{R} N)$ is a SRFR on V,

(QL, RN) is a fuzzy digraph, where $\underline{\hat{G}} = (\underline{Q} L, \underline{R} N)$ and $\bar{\hat{G}} = (\bar{Q} L, \bar{R} N)$ are lower and upper approximate fuzzy digraphs of $\hat{G}$ .

Example 3.1. Let V = {c, d, e, f, g} be a set of universe and W = {p, q, r, s} a set of parameters. A soft set K over V is defined by $\begin{matrix} K (p) = {c, d}, K (q) = {d, e, f}, K (r) = {f, g}, \\ K (s) = {c, e} . \end{matrix}$

Then a soft relation Q over V × W can be written as

Q	p	q	r	s
c	1	0	0	1
d	1	1	0	0
e	0	1	0	1
f	0	1	1	0
g	0	0	1	0

Assume that L = {(p, 0.9) , (q, 0.6) , (r, 0.8) , (s, 0.5)} is a fuzzy set on W, then $QL = (\underline{Q} L, \bar{Q} L)$ is a SRFS, where $\begin{matrix} \underline{Q} L = {(c, 0.5), (d, 0.6), (e, 0.5), (f, 0.6), (g, 0.8)}, \\ \bar{Q} L = {(c, 0.9), (d, 0.9), (e, 0.6), (f, 0.8), (g, 0.8)} . \end{matrix}$

Let A = {cd, ce, df, ed, eg, fd, ge} and B = {pp, ps, qq, qs, rq, sq}. Then a soft relation R over A × B can be written as

R	pp	ps	qq	qs	rq	sq
cd	1	0	0	0	0	1
ce	0	1	0	0	0	1
df	0	0	1	0	0	0
ed	0	0	1	0	0	1
eg	0	0	0	0	0	0
fd	0	0	1	0	1	0
ge	0	0	0	0	1	0

Let N = {(pp, 0.5) , (ps, 0.2) , (qq, 0.3) , (qs, 0.1) , (rq, 0.5) , (sq, 0.4)}, then $RN = (\underline{R} N, \bar{R} N)$ is a SRFR, where $\begin{matrix} \underline{R} N = {(cd, 0.4), (ce, 0.2), (df, 0.3), (ed, 0.3), (eg, 0), \\ (fd, 0.3), (gd, 0.5)}, \\ \bar{R} N = {(cd, 0.5), (ce, 0.4), (df, 0.3), (ed, 0.4), (eg, 0), \\ (fd, 0.5), (gd, 0.5)} . \end{matrix}$

Thus, $\underline{\hat{G}} = (\underline{Q} L, \underline{R} N)$ and $\bar{\hat{G}} = (\bar{Q} L, \bar{R} N)$ are fuzzy digraphs as shown in Fig. 2.

Fig. 2.

Soft rough fuzzy digraph $\hat{G} = (\underline{\hat{G}}, \bar{\hat{G}})$ .

Hence, $\hat{G} = (\underline{\hat{G}}, \bar{\hat{G}})$ is a SRFD.

Definition 3.2. Let $\hat{G} = (\underline{\hat{G}}, \bar{\hat{G}})$ be a SRFD on a nonempty set V. The order of $\hat{G}$ , denoted by $𝕆 (\hat{G})$ , represented by $𝕆 (\hat{G}) = 𝕆 (\underline{\hat{G}}) + 𝕆 (\bar{\hat{G}}), where$

$\begin{matrix} 𝕆 (\underline{\hat{G}}) = \sum_{u \in V} (\underline{Q} L) (u) and 𝕆 (\bar{\hat{G}}) = \sum_{u \in V} (\bar{Q} L) (u) . \end{matrix}$

The size of $\hat{G}$ , denoted by $𝕊 (\hat{G})$ , represented by $𝕊 (\hat{G}) = 𝕊 (\underline{\hat{G}}) + 𝕊 (\bar{\hat{G}}), where$ $\begin{matrix} 𝕊 (\underline{\hat{G}}) & = & \sum_{u_{1}, u_{2} \in V} (\underline{R} N) (u_{1} u_{2}), \\ 𝕊 (\bar{\hat{G}}) & = & \sum_{u_{1}, u_{2} \in V} (\bar{R} N) (u_{1} u_{2}) . \end{matrix}$

Example 3.2. Let $\hat{G}$ be a SRFD as shown in Fig. 2. Then $\begin{matrix} 𝕆 (\underline{\hat{G}}) & = & 0.5 + 0.6 + 0.5 + 0.6 + 0.8 = 3.0, \\ 𝕆 (\bar{\hat{G}}) & = & 0.9 + 0.9 + 0.6 + 0.8 + 0.8 = 4.0, \\ 𝕆 (\hat{G}) & = & 3.0 + 4.0 = 7.0 . \end{matrix}$ and $\begin{matrix} 𝕊 (\underline{\hat{G}}) = 0.4 + 0.2 + 0.3 + 0.3 + 0.3 + 0.5 = 2.0, \\ 𝕊 (\bar{\hat{G}}) = 0.5 + 0.4 + 0.3 + 0.4 + 0.5 + 0.5 = 2.6, \\ 𝕊 (\hat{G}) = 2.0 + 2.6 = 4.6 . \end{matrix}$

Definition 3.3 The underlying rough digraph of a SRFD $\hat{G} = (\underline{\hat{G}}, \bar{\hat{G}})$ on V denoted by ${\hat{G}}^{*}$ , represented by ${\hat{G}}^{*} = ({\underline{\hat{G}}}^{*}, {\bar{\hat{G}}}^{*})$ , where ${\underline{\hat{G}}}^{*} = ((\underline{Q} L)^{*}, (\underline{R} N)^{*})$ and ${\bar{\hat{G}}}^{*} = ((\bar{Q} L)^{*}, (\bar{R} N)^{*})$ are crisp digraphs such that $\begin{matrix} (\underline{Q} L)^{*} & = & {u \in V | (\underline{Q} L) (u) > 0}, \\ (\bar{Q} L)^{*} & = & {u \in V | (\bar{Q} L) (u) > 0}, \\ (\underline{R} N)^{*} & = & {u_{1} u_{2} \in A | (\underline{R} N) (u_{1} u_{2}) > 0}, \\ (\bar{R} N)^{*} & = & {u_{1} u_{2} \in A | (\bar{R} N) (u_{1} u_{2}) > 0} . \end{matrix}$

Example 3.3. Consider a SRFD $\hat{G} = (\underline{\hat{G}}, \bar{\hat{G}})$ as shown in Fig. 3.

Fig. 3.

Soft rough fuzzy digraph $\hat{G} = (\underline{\hat{G}}, \bar{\hat{G}})$ .

The underlying rough digraph of $\hat{G}$ is ${\hat{G}}^{*} = ({\underline{\hat{G}}}^{*}, {\bar{\hat{G}}}^{*})$ as shown in Fig. 4.

Fig. 4.

Underlying rough digraph ${\hat{G}}^{*}$ of $\hat{G}$ .

We now present methods of construction of SRFDs.

Definition 3.4. Let ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ and ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ be two SRFDs on V. The union of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} ⋓ {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \cup {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \cup {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \cup {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \cup \underline{Q} L_{2}, \underline{R} N_{1} \cup \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \cup {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \cup \bar{Q} L_{2}, \bar{R} N_{1} \cup \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that

{ $\begin{matrix} (\underline{Q} L_{1} \cup \underline{Q} L_{2}) (x) & = {\begin{matrix} (\underline{Q} L_{1}) (x), & if x \in (\underline{Q} L_{1})^{*} but x \notin (\underline{Q} L_{2})^{*}; \\ (\underline{Q} L_{2}) (x), & if x \in (\underline{Q} L_{2})^{*} but x \notin (\underline{Q} L_{1})^{*}; \\ max {(\underline{Q} L_{1}) (x), (\underline{Q} L_{2}) (x)}, & if x \in (\underline{Q} L_{1})^{*} \cap (\underline{Q} L_{2})^{*} . \end{matrix} \\ (\underline{R} N_{1} \cup \underline{R} N_{2}) (xy) & = {\begin{matrix} (\underline{R} N_{1}) (xy), & if xy \in (\underline{R} N_{1})^{*} but xy \notin (\underline{R} N_{2})^{*}; \\ (\underline{R} N_{2}) (xy), & if xy \in \underline{R} N_{2} but xy \notin (\underline{R} N_{1})^{*}; \\ max {(\underline{R} N_{1}) (xy), (\underline{R} N_{2}) (xy)}, & if xy \in (\underline{R} N_{1})^{*} \cap (\underline{R} N_{2})^{*} . \end{matrix} \\ (\bar{Q} L_{1} \cup \bar{Q} L_{2}) (x) & = {\begin{matrix} (\bar{Q} L_{1}) (x), & if x \in (\bar{Q} L_{1})^{*} but x \notin (\bar{Q} L_{2})^{*}; \\ (\bar{Q} L_{2}) (x), & if x \in (\bar{Q} L_{2})^{*} but x \notin (\bar{Q} L_{1})^{*}; \\ max {(\bar{Q} L_{1}) (x), (\bar{Q} L_{2}) (x)}, & if x \in (\bar{Q} L_{1})^{*} \cap (\bar{Q} L_{2})^{*} . \end{matrix} \\ (\bar{R} N_{1} \cup \bar{R} N_{2}) (xy) & = {\begin{matrix} (\bar{R} N_{1}) (xy), & if xy \in (\bar{R} N_{1})^{*} but xy \notin (\bar{R} N_{2})^{*}; \\ (\bar{R} N_{2}) (xy), & if xy \in (\bar{R} N_{2})^{*} but xy \notin (\bar{R} N_{1})^{*}; \\ max {(\bar{R} N_{1}) (xy), (\bar{R} N_{2}) (xy)}, & if xy \in (\bar{R} N_{1})^{*} \cap (\bar{R} N_{2})^{*} . \end{matrix} \end{matrix}$

Example 3.4. Let V = {c, d, e, f} be a set of universe and W = {p, q, r} a set of parameters. Let ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ and ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ be two SRFDs on V, where ${\underline{\hat{G}}}_{1} = (\underline{Q} L_{1}, \underline{R} N_{1})$ and ${\bar{\hat{G}}}_{1} = (\bar{Q} L_{1}, \bar{R} N_{1})$ are fuzzy digraphs as shown in Fig. 5.

Fig. 5.

Soft rough fuzzy digraph ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ .

${\underline{\hat{G}}}_{2} = (\underline{Q} L_{2}, \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{2} = (\bar{Q} L_{2}, \bar{R} N_{2})$ are also fuzzy digraphs as shown in Fig. 6.

Fig. 6.

Soft rough fuzzy digraph ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ .

The union of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} ⋓ {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \cup {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \cup {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \cup {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \cup \underline{Q} L_{2}, \underline{R} N_{1} \cup \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \cup {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \cup \bar{Q} L_{2}, \bar{R} N_{1} \cup \bar{R} N_{2})$ are fuzzy digraphs as shown in Fig. 7.

Fig. 7.

Soft rough fuzzy digraph ${\hat{G}}_{1} ⋓ {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \cup {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \cup {\bar{\hat{G}}}_{2})$ .

Proposition 3.1. Let ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ be two SRFDs. Then their union ${\hat{G}}_{1} ⋓ {\hat{G}}_{2}$ is also a SRFD.

Proof. The proof is obvious from Definition 3.4.

Definition 3.5. Let ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ and ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ be two SRFDs on V. The intersection of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} ⋒ {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \cap {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \cap {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \cap {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \cap \underline{Q} L_{2}, \underline{R} N_{1} \cap \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \cap {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \cap \bar{Q} L_{2}, \bar{R} N_{1} \cap \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that $\begin{matrix} (\underline{Q} L_{1} \cap \underline{Q} L_{2}) (x) & = {\begin{matrix} (\underline{Q} L_{1}) (x), & if x \in (\underline{Q} L_{1})^{*} but x \notin (\underline{Q} L_{2})^{*}; \\ (\underline{Q} L_{2}) (x), & if x \in (\underline{Q} L_{2})^{*} but x \notin (\underline{Q} L_{1})^{*}; \\ min {(\underline{Q} L_{1}) (x), (\underline{Q} L_{2}) (x)}, & if x \in (\underline{Q} L_{1})^{*} \cap (\underline{Q} L_{2})^{*} . \end{matrix} \\ (\underline{R} N_{1} \cap \underline{R} N_{2}) (xy) & = {\begin{matrix} (\underline{R} N_{1}) (xy), & if xy \in (\underline{R} N_{1})^{*} but xy \notin (\underline{R} N_{2})^{*}; \\ (\underline{R} N_{2}) (xy), & if xy \in \underline{R} N_{2} but xy \notin (\underline{R} N_{1})^{*}; \\ min {(\underline{R} N_{1}) (xy), (\underline{R} N_{2}) (xy)}, & if xy \in (\underline{R} N_{1})^{*} \cap (\underline{R} N_{2})^{*} . \end{matrix} \\ (\bar{Q} L_{1} \cap \bar{Q} L_{2}) (x) & = {\begin{matrix} (\bar{Q} L_{1}) (x), & if x \in (\bar{Q} L_{1})^{*} but x \notin (\bar{Q} L_{2})^{*}; \\ (\bar{Q} L_{2}) (x), & if x \in (\bar{Q} L_{2})^{*} but x \notin (\bar{Q} L_{1})^{*}; \\ min {(\bar{Q} L_{1}) (x), (\bar{Q} L_{2}) (x)}, & if x \in (\bar{Q} L_{1})^{*} \cap (\bar{Q} L_{2})^{*} . \end{matrix} \\ (\bar{R} N_{1} \cap \bar{R} N_{2}) (xy) & = {\begin{matrix} (\bar{R} N_{1}) (xy), & if xy \in (\bar{R} N_{1})^{*} but xy \notin (\bar{R} N_{2})^{*}; \\ (\bar{R} N_{2}) (xy), & if xy \in (\bar{R} N_{2})^{*} but xy \notin (\bar{R} N_{1})^{*}; \\ min {(\bar{R} N_{1}) (xy), (\bar{R} N_{2}) (xy)}, & if xy \in (\bar{R} N_{1})^{*} \cap (\bar{R} N_{2})^{*} . \end{matrix} \end{matrix}$

Example 3.5. Consider the SRFDs ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ in Example 3.4.

The intersection of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} ⋒ {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \cap {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \cap {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \cap {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \cap \underline{Q} L_{2}, \underline{R} N_{1} \cap \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \cap {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \cap \bar{Q} L_{2}, \bar{R} N_{1} \cap \bar{R} N_{2})$ are fuzzy digraphs as shown in Fig. 8.

Fig. 8.

Soft rough fuzzy digraph ${\hat{G}}_{1} ⋒ {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \cap {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \cap {\bar{\hat{G}}}_{2})$ .

Proposition 3.2. Let ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ be two SRFDs. Then their intersection ${\hat{G}}_{1} ⋒ {\hat{G}}_{2}$ is a SRFD.

Proof. The proof is obvious from Definition 3.5.

Definition 3.6. Let ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ and ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ be two SRFDs on V. The direct sum of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is ${\hat{G}}_{1} \hat{\oplus} {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \hat{\oplus} {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \hat{\oplus} {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \hat{\oplus} {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \hat{\oplus} \underline{Q} L_{2}, \underline{R} N_{1} \hat{\oplus} \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \hat{\oplus} {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \hat{\oplus} \bar{Q} L_{2},$ $\bar{R} N_{1} \hat{\oplus} \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that $\begin{matrix} (\underline{Q} L_{1} \hat{\oplus} \underline{Q} L_{2}) (x) & = {\begin{matrix} (\underline{Q} L_{1}) (x), & if x \in (\underline{Q} L_{1})^{*} but x \notin (\underline{Q} L_{2})^{*}; \\ (\underline{Q} L_{2}) (x), & if x \in (\underline{Q} L_{2})^{*} but x \notin (\underline{Q} L_{1})^{*}; \\ max {(\underline{Q} L_{1}) (x), (\underline{Q} L_{2}) (x)}, & if x \in (\underline{Q} L_{1})^{*} \cap (\underline{Q} L_{2})^{*} . \end{matrix} \\ (\underline{R} N_{1} \hat{\oplus} \underline{R} N_{2}) (xy) & = {\begin{matrix} (\underline{R} N_{1}) (xy), & if xy \in (\underline{R} N_{1})^{*} but xy \notin (\underline{R} N_{2})^{*}; \\ (\underline{R} N_{2}) (xy), & if xy \in \underline{R} N_{2} but xy \notin (\underline{R} N_{1})^{*}; \\ 0, & if xy \in (\underline{R} N_{1})^{*} \cap (\underline{R} N_{2})^{*} . \end{matrix} \\ (\bar{Q} L_{1} \hat{\oplus} \bar{Q} L_{2}) (x) & = {\begin{matrix} (\bar{Q} L_{1}) (x), & if x \in (\bar{Q} L_{1})^{*} but x \notin (\bar{Q} L_{2})^{*}; \\ (\bar{Q} L_{2}) (x), & if x \in (\bar{Q} L_{2})^{*} but x \notin (\bar{Q} L_{1})^{*}; \\ max {(\bar{Q} L_{1}) (x), (\bar{Q} L_{2}) (x)}, & if x \in (\bar{Q} L_{1})^{*} \cap (\bar{Q} L_{2})^{*} . \end{matrix} \\ (\bar{R} N_{1} \hat{\oplus} \bar{R} N_{2}) (xy) & = {\begin{matrix} (\bar{R} N_{1}) (xy), & if xy \in (\bar{R} N_{1})^{*} but xy \notin (\bar{R} N_{2})^{*}; \\ (\bar{R} N_{2}) (xy), & if xy \in (\bar{R} N_{2})^{*} but xy \notin (\bar{R} N_{1})^{*}; \\ 0, & if xy \in (\bar{R} N_{1})^{*} \cap (\bar{R} N_{2})^{*} . \end{matrix} \end{matrix}$

Remark 3.1. The direct sum of two SRFDs may or may not be a SRFD as it can be seen in the following examples.

Example 3.6. Let V = {c, d, e} be a set of universe and W = {p, q, r, s} a set of parameters. Let ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ and ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ be two SRFDs on V, where ${\underline{\hat{G}}}_{1} = (\underline{Q} L_{1}, \underline{R} N_{1})$ and ${\bar{\hat{G}}}_{1} = (\bar{Q} L_{1}, \bar{R} N_{1})$ are fuzzy digraphs as shown in Fig. 9.

Fig. 9.

Soft rough fuzzy digraph ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ .

${\underline{\hat{G}}}_{2} = (\underline{Q} L_{2}, \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{2} = (\bar{Q} L_{2}, \bar{R} N_{2})$ are also fuzzy digraphs as shown in Fig. 10.

Fig. 10.

Soft rough fuzzy digraph ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ .

The direct sum of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is ${\hat{G}}_{1} \hat{\oplus} {\hat{G}}_{2} =$ $({\underline{\hat{G}}}_{1} \hat{\oplus} {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \hat{\oplus} {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \hat{\oplus} {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \hat{\oplus} \underline{Q} L_{2},$ $\underline{R} N_{1} \hat{\oplus} \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \hat{\oplus} {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \hat{\oplus} \bar{Q} L_{2},$ $\bar{R} N_{1} \hat{\oplus} \bar{R} N_{2})$ are fuzzy digraphs as shown in Fig. 11.

Fig. 11.

${\hat{G}}_{1} \hat{\oplus} {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \hat{\oplus} {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \hat{\oplus} {\bar{\hat{G}}}_{2})$ .

Hence, ${\hat{G}}_{1} \hat{\oplus} {\hat{G}}_{2}$ is a SRFD.

Example 3.7. Let V = {c, d, e, f} be a set of universe and W = {p, q, r, s} a set of parameters. Let ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ and ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ be two SRFDs on V, where ${\underline{\hat{G}}}_{1} = (\underline{Q} L_{1}, \underline{R} N_{1})$ and ${\bar{\hat{G}}}_{1} = (\bar{Q} L_{1}, \bar{R} N_{1})$ are fuzzy digraphs as shown in Fig. 12.

Fig. 12.

Soft rough fuzzy digraph ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ .

${\underline{\hat{G}}}_{2} = (\underline{Q} L_{2}, \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{2} = (\bar{Q} L_{2}, \bar{R} N_{2})$ are also fuzzy digraphs as shown in Fig. 13.

Fig. 13.

Soft rough fuzzy digraph ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ .

The direct sum of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is ${\hat{G}}_{1} \hat{\oplus} {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \hat{\oplus} {\underline{\hat{G}}}_{2},$ ${\bar{\hat{G}}}_{1} \hat{\oplus} {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \hat{\oplus} {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \hat{\oplus} \underline{Q} L_{2}, \underline{R} N_{1} \hat{\oplus} \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \hat{\oplus} {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \hat{\oplus} \bar{Q} L_{2}, \bar{R} N_{1} \hat{\oplus} \bar{R} N_{2})$ are fuzzy digraphs as shown in Fig. 14.

Fig. 14.

${\hat{G}}_{1} \hat{\oplus} {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \hat{\oplus} {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \hat{\oplus} {\bar{\hat{G}}}_{2})$ .

It is clear from Fig. 14, that $\underline{R} N = \underline{R} N_{1} \hat{\oplus} \underline{R} N_{2}$ and $\bar{R} N = \bar{R} N_{1} \hat{\oplus} \bar{R} N_{2}$ are equal fuzzy sets. Thus, $(\underline{R} N, \bar{R} N)$ is not a SRFR on V. Hence, ${\hat{G}}_{1} \hat{\oplus} {\hat{G}}_{2}$ is not a SRFD.

Definition 3.7. The Cartesian product of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} ⋉ {\hat{G}}_{2} =$ $({\underline{\hat{G}}}_{1} ⋉ {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} ⋉ {\bar{\hat{G}}}_{2}),$ where ${\underline{\hat{G}}}_{1} ⋉ {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} ⋉$ $\underline{Q} L_{2}, \underline{R} N_{1} ⋉ \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} ⋉ {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} ⋉ \bar{Q} L_{2}, \bar{R} N_{1} ⋉ \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that

$(\underline{Q} L_{1} ⋉ \underline{Q} L_{2}) (x_{1}, x_{2}) = min {(\underline{Q} L_{1}) (x_{1}), (\underline{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\underline{Q} L_{1})^{*} \times (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} ⋉ \underline{R} N_{2}) ((x, x_{2}) (y, y_{2})) = min {(\underline{Q} L_{1}) (x),$ $(\underline{R} N_{2}) (x_{2} y_{2})}, \forall x \in (\underline{Q} L_{1})^{*}, x_{2} y_{2} \in (\underline{R} N_{2})^{*}$ , $(\underline{R} N_{1} ⋉ \underline{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = min {(\underline{R} N_{1}) (x_{1} y_{1}),$ $(\underline{Q} L_{2}) (z)}, \forall x_{1} y_{1} \in (\underline{R} N_{1})^{*}, z \in (\underline{Q} L_{2})^{*}$ .

$(\bar{Q} L_{1} ⋉ \bar{Q} L_{2}) (x_{1}, x_{2}) = min {(\bar{Q} L_{1}) (x_{1}), (\bar{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\bar{Q} L_{1})^{*} \times (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} ⋉ \bar{R} N_{2}) ((x, x_{2}) (y, y_{2})) = min {(\bar{Q} L_{1}) (x),$ $(\bar{R} N_{2}) (x_{2} y_{2})}, \forall x \in (\bar{Q} L_{1})^{*}, x_{2} y_{2} \in (\bar{R} N_{2})^{*}$ , $(\bar{R} N_{1} ⋉ \bar{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = min {(\bar{R} N_{1}) (x_{1} y_{1}),$ $(\bar{Q} L_{2}) (z)}, \forall x_{1} y_{1} \in (\bar{R} N_{1})^{*}, z \in (\bar{Q} L_{2})^{*}$ .

Example 3.8. Let V = {c, d} be a set of universe and W = {p, q, r} a set of parameters. Consider two SRFDs ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ on V as shown in Fig. 15 and Fig. 16, respectively.

Fig. 15.

Soft rough fuzzy digraph ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ .

Fig. 16.

Soft rough fuzzy digraph ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ .

The Cartesian product of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is $\hat{G} = {\hat{G}}_{1} ⋉ {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} ⋉ {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} ⋉ {\bar{\hat{G}}}_{2})$ , as shown in Fig 17.

Fig. 17.

Soft rough fuzzy digraph ${\hat{G}}_{1} ⋉ {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} ⋉ {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} ⋉ {\bar{\hat{G}}}_{2})$ .

Thus, $\hat{G}$ is a SRFD.

Proposition 3.3. Let ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ be two SRFDs. Then their Cartesian product ${\hat{G}}_{1} ⋉ {\hat{G}}_{2}$ is a SRFD.

Proof. The proof is obvious from Definition 3.7.

Definition 3.8. The maximal product of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} * {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} * {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} * {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} * {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} * \underline{Q} L_{2}, \underline{R} N_{1} * \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} * {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} * \bar{Q} L_{2}, \bar{R} N_{1} * \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that

$(\underline{Q} L_{1} * \underline{Q} L_{2}) (x_{1}, x_{2}) = max {(\underline{Q} L_{1}) (x_{1}), (\underline{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\underline{Q} L_{1})^{*} \times (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} * \underline{R} N_{2}) ((x, x_{2}) (y, y_{2})) = max {(\underline{Q} L_{1}) (x),$ $(\underline{R} N_{2}) (x_{2} y_{2})}, \forall x \in (\underline{Q} L_{1})^{*}, x_{2} y_{2} \in (\underline{R} N_{2})^{*}$ , $(\underline{R} N_{1} * \underline{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = max {(\underline{R} N_{1}) (x_{1} y_{1}),$ $(\underline{Q} L_{2}) (z)}, \forall x_{1} y_{1} \in (\underline{R} N_{1})^{*}, z \in (\underline{Q} L_{2})^{*}$ .

$(\bar{Q} L_{1} * \bar{Q} L_{2}) (x_{1}, x_{2}) = max {(\bar{Q} L_{1}) (x_{1}), (\bar{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\bar{Q} L_{1})^{*} \times (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} * \bar{R} N_{2}) ((x, x_{2}) (y, y_{2})) = max {(\bar{Q} L_{1}) (x),$ $(\bar{R} N_{2}) (x_{2} y_{2})}, \forall x \in (\bar{Q} L_{1})^{*}, x_{2} y_{2} \in (\bar{R} N_{2})^{*}$ , $(\bar{R} N_{1} * \bar{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = max {(\bar{R} N_{1}) (x_{1} y_{1}),$ $(\bar{Q} L_{2}) (z)}, \forall x_{1} y_{1} \in (\bar{R} N_{1})^{*}, z \in (\bar{Q} L_{2})^{*}$ .

Example 3.9. Consider two SRFDs ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ as shown in Fig. 15 and Fig. 16. The maximal product of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is $\hat{G} = {\hat{G}}_{1} * {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} * {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} * {\bar{\hat{G}}}_{2})$ , as shown in Fig 18.

Fig. 18.

Soft rough fuzzy digraph ${\hat{G}}_{1} * {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} * {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} * {\bar{\hat{G}}}_{2})$ .

Thus, $\hat{G}$ is a SRFD.

Proposition 3.4. Let ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ be two SRFDs. Then their maximal product ${\hat{G}}_{1} * {\hat{G}}_{2}$ is a SRFD.

Proof. Let $\hat{G} = {\hat{G}}_{1} * {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} * {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} * {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} * {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} * \underline{Q} L_{2}, \underline{R} N_{1} * \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} * {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} * \bar{Q} L_{2}, \bar{R} N_{1} * \bar{R} N_{2}) .$ We claim that $\hat{G} = {\hat{G}}_{1} * {\hat{G}}_{2}$ is a SRFD. It is enough to show that $\underline{R} N_{1} * \underline{R} N_{2}$ and $\bar{R} N_{1} * \bar{R} N_{2}$ are fuzzy relations on $\underline{Q} L_{1} * \underline{Q} L_{2}$ and $\bar{Q} L_{1} * \bar{Q} L_{2},$ respectively. First, we show that $\underline{R} N_{1} * \underline{R} N_{2}$ is a fuzzy relation on $\underline{Q} L_{1} * \underline{Q} L_{2}$ . If $x \in (\underline{Q} L_{1})^{*}$ , $x_{2} y_{2} \in (\underline{R} N_{2})^{*}$ , then $\begin{matrix} (\underline{R} N_{1} * \underline{R} N_{2}) ((x, x_{2}) (x, y_{2})) & = & (\underline{Q} L_{1}) (x) \lor (\underline{R} N_{2}) (x_{2} y_{2}) \\ \leq & (\underline{Q} L_{1}) (x) \lor ((\underline{Q} L_{2}) (x_{2}) \\ \land (\underline{Q} L_{2}) (y_{2})) \\ = & ((\underline{Q} L_{1}) (x) \lor (\underline{Q} L_{2}) (x_{2})) \\ \land ((\underline{Q} L_{1}) (x) \lor (\underline{Q} L_{2}) (y_{2})) \\ = & (\underline{Q} L_{1} * \underline{Q} L_{2}) (x, x_{2}) \\ \land (\underline{Q} L_{1} * \underline{Q} L_{2}) (x, y_{2}) \\ (\underline{R} N_{1} * \underline{R} N_{2}) ((x, x_{2}) (x, y_{2})) & \leq & (\underline{Q} L_{1} * \underline{Q} L_{2}) (x, x_{2}) \\ \land (\underline{Q} L_{1} * \underline{Q} L_{2}) (x, y_{2}) \end{matrix}$

If $x_{1} y_{1} \in (\underline{R} N_{1})^{*}$ , $z \in (\underline{Q} L_{2})^{*}$ , then $\begin{matrix} (\underline{R} N_{1} * \underline{R} N_{2}) ((x_{1}, z) (y_{1}, z)) & = & (\underline{R} N_{1}) (x_{1} y_{1}) \lor (\underline{Q} L_{2}) (z) \\ \leq & ((\underline{Q} L_{1}) (x_{1}) \land (\underline{Q} L_{1}) (y_{1})) \\ \lor (\underline{Q} L_{2}) (z) \\ = & ((\underline{Q} L_{1}) (x_{1}) \lor (\underline{Q} L_{2}) (z)) \\ \land ((\underline{Q} L_{1}) (y_{1}) \lor (\underline{Q} L_{2}) (z)) \\ = & (\underline{Q} L_{1} * \underline{Q} L_{2}) (x_{1}, z) \\ \land (\underline{Q} L_{1} * \underline{Q} L_{2}) (y_{1}, z) \\ (\underline{R} N_{1} * \underline{R} N_{2}) ((x_{1}, z) (y_{1}, z)) & \leq & (\underline{Q} L_{1} * \underline{Q} L_{2}) (x_{1}, z) \\ \land (\underline{Q} L_{1} * \underline{Q} L_{2}) (y_{1}, z) \end{matrix}$

Thus, $\underline{R} N_{1} * \underline{R} N_{2}$ is a fuzzy relation on $\underline{Q} L_{1} * \underline{Q} L_{2}$ . Similarly, $\bar{R} N_{1} * \bar{R} N_{2}$ is a fuzzy relation on $\bar{Q} L_{1} * \bar{Q} L_{2}$ . Hence, $\hat{G}$ is a SRFD.

Definition 3.9. The residue product of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} • {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} • {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} • {\bar{\hat{G}}}_{2}),$ where ${\underline{\hat{G}}}_{1} • {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} • \underline{Q} L_{2}, \underline{R} N_{1} • \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} • {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} • \bar{Q} L_{2}, \bar{R} N_{1} • \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that

$(\underline{Q} L_{1} • \underline{Q} L_{2}) (x_{1}, x_{2}) = max {(\underline{Q} L_{1}) (x_{1}), (\underline{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\underline{Q} L_{1})^{*} \times (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} • \underline{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (\underline{R} N_{1}) (x_{1} y_{1}), \forall x_{1} y_{1} \in (\underline{R} N_{1})^{*}, x_{2}, y_{2} \in (\underline{Q} L_{2})^{*}$ such that x₂ ≠ y₂.

$(\bar{Q} L_{1} • \bar{Q} L_{2}) (x_{1}, x_{2}) = max {(\bar{Q} L_{1}) (x_{1}), (\bar{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\bar{Q} L_{1})^{*} \times (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} • \bar{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (\bar{R} N_{1}) (x_{1} y_{1}), \forall x_{1} y_{1} \in (\bar{R} N_{1})^{*}, x_{2}, y_{2} \in (\bar{Q} L_{2})^{*}$ such that x₂ ≠ y₂.

Example 3.10. Consider two SRFDs ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ as shown in Fig. 15 and Fig. 16. The residue product of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is $\hat{G} = {\hat{G}}_{1} • {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} • {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} • {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} • {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} • \underline{Q} L_{2}, \underline{R} N_{1} • \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} • {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} • \bar{Q} L_{2}, \bar{R} N_{1} • \bar{R} N_{2})$ are fuzzy digraphs as shown in Fig. 19.

Fig. 19.

Soft rough fuzzy digraph ${\hat{G}}_{1} • {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} • {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} • {\bar{\hat{G}}}_{2})$ .

Thus, $\hat{G}$ is a SRFD.

Proposition 3.5. Let ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ be two SRFDs. Then their residue product ${\hat{G}}_{1} • {\hat{G}}_{2}$ is a SRFD.

Proof. Let $\hat{G} = {\hat{G}}_{1} • {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} • {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} • {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} • {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} • \underline{Q} L_{2}, \underline{R} N_{1} • \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} • {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} • \bar{Q} L_{2}, \bar{R} N_{1} • \bar{R} N_{2}) .$ We claim that $\hat{G} = {\hat{G}}_{1} • {\hat{G}}_{2}$ is a SRFD. It is enough to show that $\underline{R} N_{1} • \underline{R} N_{2}$ and $\bar{R} N_{1} • \bar{R} N_{2}$ are fuzzy relations on $\underline{Q} L_{1} • \underline{Q} L_{2}$ and $\bar{Q} L_{1} • \bar{Q} L_{2},$ respectively. First, we show that $\underline{R} N_{1} • \underline{R} N_{2}$ is a fuzzy relation on $\underline{Q} L_{1} • \underline{Q} L_{2}$ . If $x_{1} y_{1} \in (\underline{R} N_{1})^{*}$ , $x_{2}, y_{2} \in (\underline{Q} L_{2})^{*}$ such that x₂ ≠ y₂, then $\begin{matrix} (\underline{R} N_{1} • \underline{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = (\underline{R} N_{1}) (x_{1} y_{1}) \\ \leq ((\underline{Q} L_{1}) (x_{1}) \land (\underline{Q} L_{1}) (y_{1})) \\ = ((\underline{Q} L_{1}) (x_{1}) \lor (\underline{Q} L_{2}) (x_{2})) \\ \land ((\underline{Q} L_{1}) (y_{1}) \lor (\underline{Q} L_{2}) (y_{2})) \\ = (\underline{Q} L_{1} • \underline{Q} L_{2}) (x_{1}, x_{2}) \\ \land (\underline{Q} L_{1} • \underline{Q} L_{2}) (y_{1}, y_{2}) \\ (\underline{R} N_{1} • \underline{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) \leq (\underline{Q} L_{1} • \underline{Q} L_{2}) (x_{1}, x_{2}) \\ \land (\underline{Q} L_{1} • \underline{Q} L_{2}) (y_{1}, y_{2}) \end{matrix}$

Thus, $\underline{R} N_{1} • \underline{R} N_{2}$ is a fuzzy relation on $\underline{Q} L_{1} • \underline{Q} L_{2}$ . Similarly, we can show that $\bar{R} N_{1} • \bar{R} N_{2}$ is a fuzzy relation on $\bar{Q} L_{1} • \bar{Q} L_{2}$ . Hence, $\hat{G}$ is a SRFD.

Definition 3.10. The composition of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} [{\hat{G}}_{2}] = ({\underline{\hat{G}}}_{1} \times {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \times {\bar{\hat{G}}}_{2}),$ where ${\underline{\hat{G}}}_{1} \times {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \times \underline{Q} L_{2}, \underline{R} N_{1} \times \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \times {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \times \bar{Q} L_{2}, \bar{R} N_{1} \times \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that

$(\underline{Q} L_{1} \times \underline{Q} L_{2}) (x_{1}, x_{2}) = min {(\underline{Q} L_{1}) (x_{1}), (\underline{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\underline{Q} L_{1})^{*} \times (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} \times \underline{R} N_{2}) ((x, x_{2}) (y, y_{2})) = min {(\underline{Q} L_{1}) (x),$ $(\underline{R} N_{2}) (x_{2} y_{2})}, \forall x \in (\underline{Q} L_{1})^{*}, x_{2} y_{2} \in (\underline{R} N_{2})^{*},$ $(\underline{R} N_{1} \times \underline{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = min {(\underline{R} N_{1}) (x_{1} y_{1}),$ $(\underline{Q} L_{2}) (z)}, \forall x_{1} y_{1} \in (\underline{R} N_{1})^{*}, z \in (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} \times \underline{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = min {(\underline{R} N_{1}) (x_{1} y_{1}),$ $(\underline{Q} L_{2}) (x_{2}), (\underline{Q} L_{2}) (y_{2})}, \forall x_{1} y_{1} \in (\underline{R} N_{1})^{*}, x_{2}, y_{2} \in (\underline{Q} L_{2})^{*}$ such that x₂ ≠ y₂.

$(\bar{Q} L_{1} \times \bar{Q} L_{2}) (x_{1}, x_{2}) = min {(\bar{Q} L_{1}) (x_{1}), (\bar{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\bar{Q} L_{1})^{*} \times (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} \times \bar{R} N_{2}) ((x, x_{2}) (y, y_{2})) = min {(\bar{Q} L_{1}) (x),$ $(\bar{R} N_{2}) (x_{2} y_{2})}, \forall x \in (\bar{Q} L_{1})^{*}, x_{2} y_{2} \in (\bar{R} N_{2})^{*},$ $(\bar{R} N_{1} \times \bar{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = min {(\bar{R} N_{1}) (x_{1} y_{1}),$ $(\bar{Q} L_{2}) (z)}, \forall x_{1} y_{1} \in (\bar{R} N_{1})^{*}, z \in (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} \times \bar{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = min {(\bar{R} N_{1}) (x_{1} y_{1}),$ $(\bar{Q} L_{2}) (x_{2}), (\bar{Q} L_{2}) (y_{2})}, \forall x_{1} y_{1} \in (\bar{R} N_{1})^{*}, x_{2}, y_{2} \in (\bar{Q} L_{2})^{*}$ such that x₂ ≠ y₂.

Example 3.11. Consider two SRFDs ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ as shown in Fig. 15 and Fig. 16. The composition of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is $\hat{G} = {\hat{G}}_{1} \times {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \times {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \times {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \times {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \times \underline{Q} L_{2}, \underline{R} N_{1} \times \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \times {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \times \bar{Q} L_{2}, \bar{R} N_{1} \times \bar{R} N_{2})$ are fuzzy digraphs as shown in Fig. 20.

Fig. 20.

Soft rough fuzzy digraph ${\hat{G}}_{1} \times {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \times {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \times {\bar{\hat{G}}}_{2})$ .

Thus, $\hat{G}$ is a SRFD.

Proposition 3.6. Let ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ be two SRFDs. Then their composition ${\hat{G}}_{1} \times {\hat{G}}_{2}$ is also a SRFD.

Proof. The proof is obvious from Definition 3.10.

Definition 3.11 The rejection of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is ${\hat{G}}_{1} | {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} | {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} | {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} | {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} | \underline{Q} L_{2}, \underline{R} N_{1} | \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} | {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} | \bar{Q} L_{2}, \bar{R} N_{1} | \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that

$(\underline{Q} L_{1} | \underline{Q} L_{2}) (x_{1}, x_{2}) = min {(\underline{Q} L_{1}) (x_{1}), (\underline{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\underline{Q} L_{1})^{*} \times (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} | \underline{R} N_{2}) ((x, x_{2}) (y, y_{2})) = min {(\underline{Q} L_{1}) (x),$ $(\underline{Q} L_{2}) (x_{2}), (\underline{Q} L_{2}) (y_{2})}, \forall x \in (\underline{Q} L_{1})^{*}, x_{2} y_{2} \notin (\underline{R} N_{2})^{*},$ $(\underline{R} N_{1} | \underline{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = min {(\underline{Q} L_{1}) (x_{1}),$ $(\underline{Q} L_{1}) (y_{1}), (\underline{Q} L_{2}) (z)}, \forall x_{1} y_{1} \notin (\underline{R} N_{1})^{*}, z \in (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} | \underline{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = min {(\underline{Q} L_{1}) (x_{1}),$ $(\underline{Q} L_{1}) (y_{1}), (\underline{Q} L_{2}) (x_{2}), (\underline{Q} L_{2}) (y_{2})}, \forall x_{1} y_{1} \notin (\underline{R} N_{1})^{*}, x_{2} y_{2} \notin (\underline{R} N_{2})^{*} .$

$(\bar{Q} L_{1} | \bar{Q} L_{2}) (x_{1}, x_{2}) = min {(\bar{Q} L_{1}) (x_{1}), (\bar{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\bar{Q} L_{1})^{*} \times (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} | \bar{R} N_{2}) ((x, x_{2}) (y, y_{2})) = min {(\bar{Q} L_{1}) (x),$ $(\bar{Q} L_{2}) (x_{2}), (\bar{Q} L_{2}) (y_{2})}, \forall x \in (\bar{Q} L_{1})^{*}, x_{2} y_{2} \notin (\bar{R} N_{2})^{*},$ $(\bar{R} N_{1} | \bar{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = min {(\bar{Q} L_{1}) (x_{1}),$ $(\bar{Q} L_{1}) (y_{1}), (\bar{Q} L_{2}) (z)}, \forall x_{1} y_{1} \notin (\bar{R} N_{1})^{*}, z \in (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} | \bar{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = min {(\bar{Q} L_{1}) (x_{1}),$ $(\bar{Q} L_{1}) (y_{1}), (\bar{Q} L_{2}) (x_{2}), (\bar{Q} L_{2}) (y_{2})}, \forall x_{1} y_{1} \notin (\bar{R} N_{1})^{*}, x_{2} y_{2} \notin (\bar{R} N_{2})^{*} .$

Remark 3.2. The rejection of two SRFDs is not a SRFD in general, as it can be seen in the following example.

Example 3.12. Let V = {c, d} be a set of universe and W = {p, q, r} a set of parameters. Consider the two SRFDs ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ on V as shown in Fig. 21 and Fig. 22, respectively.

Fig. 21.

Soft rough fuzzy digraph ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ .

Fig. 22.

Soft rough fuzzy digraph ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ .

The rejection of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is ${\hat{G}}_{1} | {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} | {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} | {\bar{\hat{G}}}_{2})$ as shown in Fig. 23.

Fig. 23.

${\hat{G}}_{1} | {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} | {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} | {\bar{\hat{G}}}_{2})$ .

It is clear from Fig. 23, that $\underline{R} N = \underline{R} N_{1} | \underline{R} N_{2}$ and $\bar{R} N = \bar{R} N_{1} | \bar{R} N_{2}$ are equal fuzzy sets. Thus, $(\underline{R} N, \bar{R} N)$ is not a SRFR on V. Hence, ${\hat{G}}_{1} | {\hat{G}}_{2}$ is not a SRFD.

Definition 3.12. The symmetric difference of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} \oplus {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \oplus {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \oplus {\bar{\hat{G}}}_{2})$ , where ${\underline{\hat{G}}}_{1} \oplus {\underline{\hat{G}}}_{2} = (\underline{Q} L_{1} \oplus \underline{Q} L_{2}, \underline{R} N_{1} \oplus \underline{R} N_{2})$ and ${\bar{\hat{G}}}_{1} \oplus {\bar{\hat{G}}}_{2} = (\bar{Q} L_{1} \oplus \bar{Q} L_{2}, \bar{R} N_{1} \oplus \bar{R} N_{2})$ are fuzzy digraphs, respectively, such that

$(\underline{Q} L_{1} \oplus \underline{Q} L_{2}) (x_{1}, x_{2}) = min {(\underline{Q} L_{1}) (x_{1}), (\underline{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\underline{Q} L_{1})^{*} \times (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} \oplus \underline{R} N_{2}) ((x, x_{2}) (y, y_{2})) = min {(\underline{Q} L_{1}) (x),$ $(\underline{R} N_{2}) (x_{2} y_{2})}, \forall x \in (\underline{Q} L_{1})^{*}, x_{2} y_{2} \in (\underline{R} N_{2})^{*},$ $(\underline{R} N_{1} \oplus \underline{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = min {(\underline{R} N_{1}) (x_{1} y_{1}),$ $(\underline{Q} L_{2}) (z)}, \forall x_{1} y_{1} \in (\underline{R} N_{1})^{*}, z \in (\underline{Q} L_{2})^{*},$ $(\underline{R} N_{1} \oplus \underline{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = {\begin{matrix} min {(\underline{Q} L_{1}) (x_{1}), (\underline{Q} L_{1}) (y_{1}), (\underline{R} N_{2}) (x_{2} y_{2})}, \\ \forall x_{1} y_{1} \notin (\underline{R} N_{1})^{*}, x_{2} y_{2} \in (\underline{R} N_{2})^{*}, \\ or \\ min {(\underline{R} N_{1}) (x_{1} y_{1}), (\underline{Q} L_{2}) (x_{2}), (\underline{Q} L_{2}) (y_{2})}, \\ \forall x_{1} y_{1} \in (\underline{R} N_{1})^{*}, x_{2} y_{2} \notin (\underline{R} N_{2})^{*} . \end{matrix}$

$(\bar{Q} L_{1} \oplus \bar{Q} L_{2}) (x_{1}, x_{2}) = min {(\bar{Q} L_{1}) (x_{1}), (\bar{Q} L_{2}) (x_{2})}, \forall (x_{1}, x_{2}) \in (\bar{Q} L_{1})^{*} \times (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} \oplus \bar{R} N_{2}) ((x, x_{2}) (y, y_{2})) = min {(\bar{Q} L_{1}) (x),$ $(\bar{R} N_{2}) (x_{2} y_{2})}, \forall x \in (\bar{Q} L_{1})^{*}, x_{2} y_{2} \in (\bar{R} N_{2})^{*},$ $(\bar{R} N_{1} \oplus \bar{R} N_{2}) ((x_{1}, z) (y_{1}, z)) = min {(\bar{R} N_{1}) (x_{1} y_{1}),$ $(\bar{Q} L_{2}) (z)}, \forall x_{1} y_{1} \in (\bar{R} N_{1})^{*}, z \in (\bar{Q} L_{2})^{*},$ $(\bar{R} N_{1} \oplus \bar{R} N_{2}) ((x_{1}, x_{2}) (y_{1}, y_{2})) = {\begin{matrix} min {(\bar{Q} L_{1}) (x_{1}), (\bar{Q} L_{1}) (y_{1}), (\bar{R} N_{2}) (x_{2} y_{2})}, \\ \forall x_{1} y_{1} \notin (\bar{R} N_{1})^{*}, x_{2} y_{2} \in (\bar{R} N_{2})^{*}, \\ or \\ min {(\bar{R} N_{1}) (x_{1} y_{1}), (\bar{Q} L_{2}) (x_{2}), (\bar{Q} L_{2}) (y_{2})}, \\ \forall x_{1} y_{1} \in (\bar{R} N_{1})^{*}, x_{2} y_{2} \notin (\bar{R} N_{2})^{*} . \end{matrix}$

Example 3.13. Let V = {c, d} be a set of universe and W = {p, q, r} a set of parameters. Consider two SRFDs ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ on V as shown in Fig. 24 and Fig. 26, respectively.

The symmetric difference of ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ is a SRFD $\hat{G} = {\hat{G}}_{1} \oplus {\hat{G}}_{2} = ({\underline{\hat{G}}}_{1} \oplus {\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{1} \oplus {\bar{\hat{G}}}_{2})$ as shown in Fig. 26.

Fig. 24.

Soft rough fuzzy digraph ${\hat{G}}_{1} = ({\underline{\hat{G}}}_{1}, {\bar{\hat{G}}}_{1})$ .

Fig. 25.

Soft rough fuzzy digraph ${\hat{G}}_{2} = ({\underline{\hat{G}}}_{2}, {\bar{\hat{G}}}_{2})$ .

Fig. 26.

Soft rough fuzzy digraph $\hat{G} = {\hat{G}}_{1} \oplus {\hat{G}}_{2}$ .

Thus, $\hat{G}$ is a SRFD.

Proposition 3.7. Let ${\hat{G}}_{1}$ and ${\hat{G}}_{2}$ be two SRFDs. Then their symmetric difference ${\hat{G}}_{1} \oplus {\hat{G}}_{2}$ is also a SRFD. Proof. The proof is obvious from Definition 3.12.

4. Applications

A thought process of choosing a logical choice from the given objects is called DM. Before making a good decision, a decision-maker must consider the advantages and limitations of each object and for this purpose, he must analyze the characteristics of each object. Considering all these characteristics, he should find the best option for that particular situation. Every DM process yields a final choice. DM is an analyzing and selecting process of alternatives based on the beliefs, preferences and values of the decision-maker. The given DM method can be used to evaluate upper and lower approximations to develop deep considerations of the problem. The presented algorithms can be applied to avoid lengthy calculations when dealing with large number of objects. This method can be applied in various domains for multi-criteria selection of objects. (1) Analysis of different techniques used in a pharmaceutical industry: To understand the physical and chemical process of pharmaceutical materials, one needs to analyze the techniques which are used for characterization of products. There are some well-known techniques for characterization of products:

Ultra-violet (UV) spectrophotometry

Fourier transform Infra-Red (FTIR) spectroscopy

Nuclear magnetic resonance (NMR) spectroscopy

Gas chromatography mass spectrometry (GC-MS)

Scanning electron microscopy (SEM)

Thermogravimetric analysis (TGA)

Let V = {t₁ = UVspectrophotometry, t₂ = FTIR spectroscopy, t₃ = NMRspectroscopy, t₄ = GC - MS, t₅ = SEM, t₆ = TGA} be a set of analytic techniques under consideration and W = {c₁ = sample preparation, c₂ = costly, c₃ = sensitive, c₄ = destructive, c₅ = complicatedinstrumentation} a set of parameters. Assume that a pharmaceutical industry wants to select the best technique for products analysis with respect to the parameter set. A soft set K over V is defined by $\begin{matrix} K (a) = {t_{1}, t_{2}, t_{3}, t_{5}}, K (c_{2}) = {t_{3}, t_{4}}, \\ K (c_{3}) = {t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}}, K (c_{4}) = {t_{4}, t_{6}}, \\ K (c_{5}) = {t_{2}, t_{3}, t_{4}} . \end{matrix}$

A soft relation Q over V × W can be written as

Q	c ₁	c ₂	c ₃	c ₄	c ₅
t ₁	1	0	1	0	0
t ₂	1	0	1	0	1
t ₃	1	1	1	0	1
t ₄	0	1	1	1	1
t ₅	1	0	1	0	0
t ₆	0	0	1	1	0

Let L be the optimum normal decision object given by the pharmaceutical industry, defined by L = {(c₁, 0.4) , (c₂, 0.9) , (c₃, 0.5) , (c₄, 0.7) , (c₅, 0.6)}, then $QL = (\underline{Q} L, \bar{Q} L)$ is a SRFS, where $\begin{matrix} \underline{Q} L = {(t_{1}, 0.4), (t_{2}, 0.4), (t_{3}, 0.4), (t_{4}, 0.5), (t_{5}, 0.4), \\ (t_{6}, 0.5)}, \\ \bar{Q} L = {(t_{1}, 0.5), (t_{2}, 0.6), (t_{3}, 0.9), (t_{4}, 0.9), (t_{5}, 0.5), \\ (t_{6}, 0.7)} . \end{matrix}$ Let A = {t₁t₂, t₁t₄, t₂t₆, t₃t₁, t₃t₆, t₄t₂, t₄t₃, t₄t₅, t₅t₃, t₅t₆, t₆t₃} and B = {c₁c₂, c₁c₄, c₂c₃, c₃c₄, c₃c₅, c₄c₅, c₅c₁, c₅c₂}. Then a soft relation R over A × B can be written as

c ₁	c ₁	c ₂	c ₃	c ₃	c ₄	c ₅	c ₅
R	c ₂	c ₄	c ₃	c ₄	c ₅	c ₅	c ₁	c ₂
t ₁ t ₂	0	0	0	0	1	0	0	0
t ₁ t ₄	1	1	0	1	1	0	0	0
t ₂ t ₆	0	1	0	1	0	0	0	0
t ₃ t ₁	0	0	1	0	0	0	1	0
t ₃ t ₆	0	1	1	1	0	0	0	0
t ₄ t ₂	0	0	1	0	1	1	1	0
t ₄ t ₃	0	0	1	0	1	1	1	1
t ₄ t ₅	0	0	1	0	0	0	1	0
t ₅ t ₃	1	0	0	0	1	0	0	0
t ₅ t ₆	0	1	0	1	0	0	0	0
t ₆ t ₃	0	0	0	0	1	1	0	0

Let N = {(c₁c₂, 0.40) , (c₁c₄, 0.30) , (c₂c₃, 0.35) , (c₃c₄, 0.32) , (c₃c₅, 0.25) , (c₄c₅, 0.36) , (c₅c₁, 0.28) , (c₅c₂, 0.15)} be a fuzzy set on W, which describes some relationship between the parameters under consideration. Then $RN = (\underline{R} N, \bar{R} N)$ is a SRFR, where $\begin{matrix} \underline{R} N & = & {(t_{1} t_{2}, 0.25), (t_{1} t_{4}, 0.25), (t_{2} t_{6}, 0.30), \\ (t_{3} t_{1}, 0.28), (t_{3} t_{6}, 0.30), (t_{4} t_{2}, 0.25), \\ (t_{4} t_{3}, 0.15), (t_{4} t_{5}, 0.28), (t_{5} t_{3}, 0.25), \\ (t_{5} t_{6}, 0.30), (t_{6} t_{3}, 0.25)}, \\ \bar{R} N & = & {(t_{1} t_{2}, 0.25), (t_{1} t_{4}, 0.40), (t_{2} t_{6}, 0.32), \\ (t_{3} t_{1}, 0.35), (t_{3} t_{6}, 0.35), (t_{4} t_{2}, 0.36), \\ (t_{4} t_{3}, 0.36), (t_{4} t_{5}, 0.35), (t_{5} t_{3}, 0.40), \\ (t_{5} t_{6}, 0.32), (t_{6} t_{3}, 0.36)} . \end{matrix}$

Thus, $\underline{\hat{G}} = (\underline{Q} L, \underline{R} N)$ and $\bar{\hat{G}} = (\bar{Q} L, \bar{R} N)$ are fuzzy digraphs as shown in Fig. 27 and Fig. 28, respectively.

Fig. 27.

Fuzzy digraph $\underline{\hat{G}} = (\underline{Q} L, \underline{R} N)$ .

Fig. 28.

Fuzzy digraph $\bar{\hat{G}} = (\bar{Q} L, \bar{R} N)$ .

The pharmaceutical industry defines a score function S (t_i) for each t_i ∈ V, $S (t_{i}) = \sum_{t_{i} t_{j} \in A} {\frac{(\underline{Q} L) (t_{j}) + (\bar{Q} L) (t_{j})}{1 - (\underline{R} N) (t_{i} t_{j}) . (\bar{R} N) (t_{i} t_{j})}},$ and the industry’s decision is t_k if $t_{k} = max_{i} S (t_{i})$ . By calculations, we have $\begin{matrix} S (t_{1}) = 2.623, S (t_{2}) = 1.327, S (t_{3}) = 2.339, \\ S (t_{4}) = 3.471, S (t_{5}) = 2.771, S (t_{6}) = 1.428 . \end{matrix}$

Here, t₄ is the optimal(maximum) decision, therefore the industry selects GC-MS for the analysis of products (see Table 1).

Table 1

Algorithm for analysis of techniques used in a pharmaceutical industry

Algorithm
1. Begin
2. Input the set V of analytical techniques t₁, t₂, …, t_n
and set W of parameters c₁, c₂, …, c_m.
3. Input a soft relation Q over V × W.
4. Input the set A = [a_ik] _n×n of relations where, a_ik = t_it_k.
5. Input the set B of relations b₁, b₂, …, b_s where, b_i = c_jc_k,
for some j, k ∈ {1, 2, …, m}.
6. Input a soft relation R over A × B.
7. doi from 1 to m
8. read*, L (c_i)
9. end do
10. doi from 1 to n
11. $(\underline{Q} L) (t_{i}) = 1$
12. $(\bar{Q} L) (t_{i}) = 0$
13. doj from 1 to m
14. if (Q (t_i, c_j) = =1) then
15. $(\underline{Q} L) (t_{i}) = min {(\underline{Q} L) (t_{i}), L (c_{j})}$
16. $(\bar{Q} L) (t_{i}) = max {(\bar{Q} L) (t_{i}), L (c_{j})}$
17. end if
18. end do
19. end do
20. doi from 1 to s
21. read*, N (b_i)
22. end do
23. doi from 1 to n
24. dok from 1 to n
25. if (a_ik = =1) then
26. $(\underline{R} N) (a_{ik}) = 1$
27. $(\bar{R} N) (a_{ik}) = 0$
28. doj from 1 to s
29. if (R (a_ik, b_j) = =1) then
30. $(\underline{R} N) (a_{ik}) = min {(\underline{R} N) (a_{ik}), N (b_{j})}$
31. $(\bar{R} N) (a_{ik}) = max {(\bar{R} N) (a_{ik}), N (b_{j})}$
32. end if
33. end do
34. end if
35. end do
36. end do
37. doi from 1 to n
38. S (t_i) =0
39. dok from 1 to n
40. if (a_ik ≠ 0) then
41. $S (t_{i}) = S (t_{i}) + \frac{(\underline{Q} L) (t_{k}) + (\bar{Q} L) (t_{k})}{1 - (\underline{R} N) (a_{ik}) \times (\bar{R} N) (a_{ik})}$
42. end if
43. end do
44. end do
45. M = 0
46. doi from 1 to n
47. M = max {M, S (t_i)}
48. end do
49. doi from 1 to n
50. if (S (t_i) = = M) then
51. print*,t_i is the decision
52. end if
53. end do
54. End

(2) Selection of plumbing pipes for a house: Plumbing pipes are present everywhere whether these are under a house or in the walls of a house. We must know the different types of plumbing pipes, it will help us to select the right type of pipe for our homes. Pipes are available in a market with different shapes, sizes, and materials. Some of them carry gas and others carry water. Each type of pipe has its advantages and disadvantages also with its special usage. These are some residential plumbing pipes available in market:

Cast iron pipes

PVC (polyvinyl-chloride) plastic pipes

CPVC (chlorinated polyvinyl-chloride) pipes

PEX (cross-linked polyethylene) pipes

Copper pipes

Let V = {p₁ = Castiron, p₂ = PVC, p₃ = CPVC, p₄ = PEX, p₅ = Copper} be a set of different types of plumbing pipes under consideration and W = {c₁ = durable, c₂ = expensive, c₃ = corrosionresistant, c₄ = heattolerant, c₅ = pressureflow} a set of parameters. Assume that a person Mr. Ali wants the best type of plumbing pipe for his house. A soft set K over V is defined by $\begin{matrix} K (c_{1}) = {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}}, K (c_{2}) = {p_{5}}, \\ K (c_{3}) = {p_{2}, p_{3}, p_{4}}, K (c_{4}) = {p_{1}, p_{3}, p_{4}, p_{5}}, \\ K (c_{5}) = {p_{2}, p_{3}, p_{4}, p_{5}} . \end{matrix}$

A soft relation Q over V × W can be written as

Q	c ₁	c ₂	c ₃	c ₄	c ₅
p ₁	1	0	0	1	0
p ₂	1	0	1	0	1
p ₃	1	0	1	1	1
p ₄	1	0	1	1	1
p ₅	1	1	0	1	1

Let L be the optimum normal decision object given by Mr. Ali, defined by L = {(c₁, 0.8) , (c₂, 0.4) , (c₃, 1) , (c₄, 0.6) , (c₅, 0.9)}, then $QL = (\underline{Q} L, \bar{Q} L)$ is a SRFS, where $\begin{matrix} \underline{Q} L = {(p_{1}, 0.6), (p_{2}, 0.8), (p_{3}, 0.6), (p_{4}, 0.6), \\ (p_{5}, 0.4)}, \\ \bar{Q} L = {(p_{1}, 0.8), (p_{2}, 1), (p_{3}, 1), (p_{4}, 1), (p_{5}, 0.9)} . \end{matrix}$

Let A = {p₁p₅, p₂p₅, p₃p₁, p₃p₄, p₃p₅, p₄p₂, p₅p₄} and B = {c₁c₃, c₂c₃, c₃c₅, c₄c₂, c₅c₄}. Then a soft relation R over A × B can be written as

R	c ₁ c ₃	c ₂ c ₃	c ₃ c ₅	c ₄ c ₂	c ₅ c ₄
p ₁ p ₅	0	0	0	1	0
p ₂ p ₅	0	0	1	0	1
p ₃ p ₁	0	0	0	0	1
p ₃ p ₄	1	0	1	0	1
p ₃ p ₅	0	0	1	1	1
p ₄ p ₂	1	0	1	0	0
p ₅ p ₄	1	1	0	0	1

Let N = {(c₁c₃, 0.30) , (c₂c₃, 0.20) , (c₃c₅, 0.35) , (c₄c₂, 0.40) , (c₅c₄, 0.25)} be a fuzzy set on W, which describes some relationships between the parameters under consideration. Then $RN = (\underline{R} N, \bar{R} N)$ is a SRFR, where $\begin{matrix} \underline{R} N = {(p_{1} p_{5}, 0.40), (p_{2} p_{5}, 0.25), (p_{3} p_{1}, 0.25), \\ (p_{3} p_{4}, 0.25), (p_{3} p_{5}, 0.25), (p_{4} p_{2}, 0.30), \\ (p_{5} p_{4}, 0.20)}, \\ \bar{R} N = {(p_{1} p_{5}, 0.40), (p_{2} p_{5}, 0.35), (p_{3} p_{1}, 0.25), \\ (p_{3} p_{4}, 0.35), (p_{3} p_{5}, 0.40), (p_{4} p_{2}, 0.35), \\ (p_{5} p_{4}, 0.30)} . \end{matrix}$

Thus, $\underline{\hat{G}} = (\underline{Q} L, \underline{R} N)$ and $\bar{\hat{G}} = (\bar{Q} L, \bar{R} N)$ are fuzzy digraphs as shown in Fig. 29.

Fig. 29.

Soft rough fuzzy digraph $\hat{G} = (\underline{\hat{G}}, \bar{\hat{G}})$ .

Mr. Ali defines a score function S (p_i) for each p_i ∈ V, $S (p_{i}) = \sqrt{\sum_{p_{i} p_{j} \in A} {\frac{(\underline{R} N) (p_{i} p_{j}) + (\bar{R} N) (p_{i} p_{j})}{1 - (\underline{Q} L) (p_{j}) . (\bar{Q} L) (p_{j})}}},$ and his decision is p_k if $p_{k} = max_{i} S (p_{i})$ . By calculations, we have $\begin{matrix} S (p_{1}) = 1.118 = S (p_{5}), S (p_{2}) = 0.968, \\ S (p_{3}) = 1.865, S (p_{4}) = 1.803 . \end{matrix}$

Here, p₃ is the optimal(maximum) decision, therefore Mr. Ali selects CPVC plasic pipes for the plumbing system in his house(see Table 2).

Table 2

Algorithm for finding the best plumbing pipe

Algorithm 2
1. Begin
2. Input the set V of pipes p₁, p₂, …, p_n
and set W of parameters c₁, c₂, …, c_m.
3. Follow steps 3 to 36 of Algorithm 1.
4. doi from 1 to n
5. S (p_i) =0
6. dok from 1 to n
7. if (a_ik ≠ 0) then
8. $S (p_{i}) = \sqrt{S (p_{i}) + \frac{(\underline{R} N) (a_{ik}) + (\bar{R} N) (a_{ik})}{1 - (\underline{Q} L) (p_{k}) \times (\bar{Q} L) (p_{k})}}$
9. end if
10. end do
11. end do
12. M = 0
13. doi from 1 to n
14. M = max {M, S (p_i)}
15. end do
16. doi from 1 to n
17. if (S (p_i) = = M) then
18. print*,p_i is the decision
19. end if
20. end do
21. End

5. Conclusions and future directions

RST can be considered as an extension of CST in one way. This is a very useful mathematical model to handle vagueness. FST, RST and SST are three useful distinguished approaches to deal with vagueness. We have applied the new theory formed by combining these three theories to digraphs. In the existing literature, generalized RST is based on an arbitrary relation but we have developed a method considering Pawlak RST which is based upon equivalence relation (ER). In comparison to the existing DM models based on generalized rough sets (GRSs), the proposed method deals with the objects which are indiscernible. A new component is involved in the method to cope with uncertainty and vagueness in data which refers to degree of dependence of an element to the other. Both fuzzy and rough models are vague concepts and handle uncertainty, combination of these two models with another remarkable model of soft sets, give more precise results of real world problems. SST involves parametrization of objects which leads to multi-criteria DM. We have first presented the concept of SRFDs in this paper. Furthermore, we have investigated some properties of SRFDs in detail. In addition, a new DM method based on SRFDs is proposed. Also some real world multi-criteria decision problems have been discussed to check the validity of proposed method. Soft rough fuzzy digraphs can also be applied to feature selection, attribute analysis and network problem. This research work may be extended to (1) Intuitionistic fuzzy rough hypergraphs, (2) Decision support systems based on intuitionistic fuzzy soft rough information.

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R	pp	pq	pr	qp	qq	qr	rp	rq	rr
cc	1	0	1	0	0	0	1	0	1
cd	1	1	0	0	0	0	1	1	0
dc	1	0	1	1	0	1	0	0	0
dd	1	1	0	1	1	0	0	0	0

R	pp	pq	pr	qp	qq	qr	rp	rq	rr
cc	1	0	1	0	0	0	1	0	1
cd	1	1	0	0	0	0	1	1	0
dc	1	0	1	1	0	1	0	0	0
dd	1	1	0	1	1	0	0	0	0

R	pq	ps	qq	qr	qs	rs	sq	ss
cc	1	0	1	1	0	0	0	0
de	0	0	0	1	1	1	0	0
ed	1	0	0	0	0	0	1	0
ee	0	1	0	0	0	1	0	1
fc	1	0	1	1	0	0	0	0
fe	0	1	0	1	1	0	0	0

R	pp	ps	qq	qs	rq	sq
cd	1	0	0	0	0	1
ce	0	1	0	0	0	1
df	0	0	1	0	0	0
ed	0	0	1	0	0	1
eg	0	0	0	0	0	0
fd	0	0	1	0	1	0
ge	0	0	0	0	1	0

R	pp	pq	pr	qp	qq	qr	rp	rq	rr
cc	1	0	1	0	0	0	1	0	1
cd	1	1	0	0	0	0	1	1	0
dc	1	0	1	1	0	1	0	0	0
dd	1	1	0	1	1	0	0	0	0

R	pp	pq	pr	qp	qq	qr	rp	rq	rr
cc	1	0	1	0	0	0	1	0	1
cd	1	1	0	0	0	0	1	1	0
dc	1	0	1	1	0	1	0	0	0
dd	1	1	0	1	1	0	0	0	0

R	pq	ps	qq	qr	qs	rs	sq	ss
cc	1	0	1	1	0	0	0	0
de	0	0	0	1	1	1	0	0
ed	1	0	0	0	0	0	1	0
ee	0	1	0	0	0	1	0	1
fc	1	0	1	1	0	0	0	0
fe	0	1	0	1	1	0	0	0

R	pp	ps	qq	qs	rq	sq
cd	1	0	0	0	0	1
ce	0	1	0	0	0	1
df	0	0	1	0	0	0
ed	0	0	1	0	0	1
eg	0	0	0	0	0	0
fd	0	0	1	0	1	0
ge	0	0	0	0	1	0

R	pp	pq	pr	qp	qq	qr	rp	rq	rr
cc	1	0	1	0	0	0	1	0	1
cd	1	1	0	0	0	0	1	1	0
dc	1	0	1	1	0	1	0	0	0
dd	1	1	0	1	1	0	0	0	0

R	pp	pq	pr	qp	qq	qr	rp	rq	rr
cc	1	0	1	0	0	0	1	0	1
cd	1	1	0	0	0	0	1	1	0
dc	1	0	1	1	0	1	0	0	0
dd	1	1	0	1	1	0	0	0	0

R	pq	ps	qq	qr	qs	rs	sq	ss
cc	1	0	1	1	0	0	0	0
de	0	0	0	1	1	1	0	0
ed	1	0	0	0	0	0	1	0
ee	0	1	0	0	0	1	0	1
fc	1	0	1	1	0	0	0	0
fe	0	1	0	1	1	0	0	0

R	pp	ps	qq	qs	rq	sq
cd	1	0	0	0	0	1
ce	0	1	0	0	0	1
df	0	0	1	0	0	0
ed	0	0	1	0	0	1
eg	0	0	0	0	0	0
fd	0	0	1	0	1	0
ge	0	0	0	0	1	0