Decision making using algebraic operations on soft effect matrix as new category of similarity measures and study their application in medical diagnosis problems

Abstract

A new category of similarity measures is investigated in this work, we first introduce the concept of effect matrix $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple and study some of their properties. Moreover, from the soft set we find the pictures of type regular and irregular using effect matrix $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple. An effect matrix $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple is better than effect matrix $\overset{\land}{E} (R, p_{n}, q_{m})$ of type 1-tuple, because we can deal with two different sets $R_{1}$ (a family of objectives) and $R_{2}$ (a family of parameters) in the same problem. This burden can be alleviated by application of type 2-tuple. Some applications of soft effect matrix of type 2-tuple $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ in decision making problems are studied and explained. In this work we deal with pictures. Moreover, the similarity measure between two different soft sets under the same universal soft set $(R_{1} \times R_{2})$ can be studied and explained its applications in medical diagnosis problems.

Keywords

Decision making algebraic operations fuzzy soft sets theory model equivalence relation similarity measures 54A10 54C10 54A05

1 Introduction

The problem of decision making with similarity measurement between fuzzy numbers, fuzzy sets, soft sets and vague sets have been studied by many mathematicians in different ways see ([4 , 34]). In 2013, [11] the concept of effect matrix $\overset{\land}{E} (R, p_{n}, q_{m})$ is studied and then the pictures of type regular and irregular which are found by effect matrix $\overset{\land}{E} (R, p_{n}, q_{m})$ from the sound. There are many problems in economics, medical science, environments, engineering, and so forth, have various uncertainties. Molodtsov [28] initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties. After Molodtsov’s work, some operations and applications of soft sets in algebras and topologies were studied by many researchers see ([1 , 26]). To address decision making problems based on fuzzy soft sets, Feng et al. introduced the concept of soft level sets of fuzzy soft sets and initiated an adjustable decision making scheme using fuzzy soft sets [6] and then operations and applications of fuzzy soft sets in algebras and topologies are introduced see ([10 , 31]).

In 2014, the concept of induced fuzzy soft set $\underset{H (A)}{ξ}$ is introduced [9]. The colours and soft sets are used in this paper. An induced fuzzy soft set $\underset{H (A)}{ξ}$ is the key to soft set and colour. Some researchers studied the colours and their applications. Further, there are many an important different applications are discussed (see 2, 3, 29).

In this work for any soft set (H, A) over a universe U and the parameter set A we can generate a picture using effect matrix $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple.. This method is useful and it will help us to study many of the soft sets from the different problems in economics, medical science, environments, engineering, and so forth and then make comparison between them over the same universal soft set $R_{1} \times R_{2}$ to help us in our everyday lives in many ways. Some applications of effect matrix $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple in in medical diagnosis problems are illustrated.

2 Preliminaries and basic results

We will show some past results and basic definitions in this section.

2.1 Definition ([28, 32])

Assume that E is a set of parameters, U is an initial universe set, P (U) is the power set of U and K is a subset of E. We say (F, K) is a soft set over U if F is a multi-valued function of K into P (X). The family of all soft sets over a universe U and the parameter set E is denoted by SS (U_E) and denoted by SS (U_A) when the parameter set is a subset A of E.

2.2 Definition ([25])

Assume that U is an initial universe set and E is a set of parameters. Let I^U denotes the collection of all fuzzy subsets of U. We say F_A ; A → I^U is a fuzzy soft set over (U, E), where A ⊆ E and $μ_{F_{A}}^{e} = \bar{0}$ if e ∈ E ∖ A and $μ_{F_{A}}^{e} \neq \bar{0}$ if e ∈ A. Also, it is defined by $F_{A} (e) = μ_{F_{A}}^{e}$ (a fuzzy subset of U) and we refer to family of all fuzzy soft sets over (U, E) byFS (U, E).

2.3 Definition ([9])

Assume that (H, A) is a soft set over U. Then $\underset{H (A)}{ξ}$ is called an induced fuzzy soft set over (U, E), whereξ : SS (U_A)→ FS (U, E) is a mapping which is given as:

for (H, A) ∈ SS (U_A) the image of (H, A) under ξ denoted by $\underset{H (A)}{ξ}$ , is defended as following:

$\underset{H (A)}{ξ} (e) = \underset{H (A)}{μ^{e}}$ (a fuzzy subset of U), where $\underset{H (A)}{μ^{e}} (x) = \bar{0}$ if e ∈ E ∖ A and $\begin{matrix} \underset{H (A)}{μ^{e}} (x) = \\ {\begin{matrix} 1, & if x \neq H (e) & x \in H (e) \\ 1 / n (SS (U_{A})), & if x = H (e) \\ n (SS (U_{A})) / n (SS (U_{E})), & if x \neq H (e) & \\ x \notin H (e) \neq φ \\ 0, & if H (e) = φ \end{matrix} \end{matrix}$

For all x ∈ U, if e ∈ A.

2.4 Effect graph ([11])

Assume {p₁, p₂, p₃, . . . , p_m} is a family has m members of mathematical operations on $R = {y_{i} / i = 1, 2, 3, . . ., I}$ , where I ∈ N and each member of the family {p₁, p₂, p₃, . . . , p_m} can be applied on $R$ in the form p_j (y_i), where j = 1, 2, 3, . . . , m ≤ I. Assume {q₁, q₂, q₃, . . . , q_n} is a family has n members of mathematical operations on $R$ which can be also applied on $R$ in the form of q_k (y_i), where k = 1, 2, 3, . . . , n ≤ I. Let p_j (j = 1, 2, 3, . . . , m) and q_k (k = 1, 2, 3, . . . , n) be arranged in rectangular matrix, where the operations p_j and q_k represent the rows and columns respectively of matrix as the following arrangement.

$\begin{matrix} y_{11} & y_{12} & y_{13} & . & . & . & y_{1 n} \\ y_{21} & y_{22} & y_{23} & . & . & . & y_{2 n} \\ y_{31} & y_{32} & y_{33} & . & . & . & y_{3 n} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ y_{m 1} & y_{m 2} & y_{m 3} & . & . & . & y_{mn} \end{matrix}$ (1)

Where y_jk = (y_j, y_k) ;1 ≤ j ≤ m, 1 ≤ k ≤ n}. Then next step is to imagine that there exists some graph formed from that two sets of operations p_j and q_k such operator p_j is connected with each q_k. This oriented connection between p_j to q_k [(p_j, q_k)]. This effect is oriented and the corresponding rectangular graph of effects is called the “effect graph” it’s convenient to symbolize effect between p_j and q_k. The notation $(p_{j} \overset{\land}{,} q_{k})$ refers to the presence of the effect. Therefore the, above arrangement can be represented as follows:

$\begin{matrix} (p_{1} \overset{\land}{,} q_{1}) & (p_{1} \overset{\land}{,} q_{2}) & (p_{1} \overset{\land}{,} q_{3}) & . & . & . & (p_{1} \overset{\land}{,} q_{n}) \\ (p_{2} \overset{\land}{,} q_{1}) & (p_{2} \overset{\land}{,} q_{2}) & (p_{2} \overset{\land}{,} q_{3}) & . & . & . & (p_{2} \overset{\land}{,} q_{n}) \\ (p_{3} \overset{\land}{,} q_{1}) & (p_{3} \overset{\land}{,} q_{2}) & (p_{3} \overset{\land}{,} q_{3}) & . & . & . & (p_{3} \overset{\land}{,} q_{n}) \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ (p_{m} \overset{\land}{,} q_{1}) & (p_{m} \overset{\land}{,} q_{2}) & (p_{m} \overset{\land}{,} q_{3}) & . & . & . & (p_{m} \overset{\land}{,} q_{n}) \end{matrix}$ (2)

Assume that there exists a rectangular matrix of coefficients in the, following form:

$E = [\begin{matrix} y_{11} & y_{12} & y_{13} & . & . & . & y_{1 n} \\ y_{21} & y_{22} & y_{23} & . & . & . & y_{2 n} \\ y_{31} & y_{32} & y_{33} & . & . & . & y_{3 n} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ y_{m 1} & y_{m 2} & y_{m 3} & . & . & . & y_{mn} \end{matrix}]$

Let us symbolize the effect graph by the symbol $\overset{\land}{E}$ i.e, to

$\overset{\land}{E} = [\begin{matrix} (p_{1} \overset{\land}{,} q_{1}) & (p_{1} \overset{\land}{,} q_{2}) & (p_{1} \overset{\land}{,} q_{3}) & . & . & . & (p_{1} \overset{\land}{,} q_{n}) \\ (p_{2} \overset{\land}{,} q_{1}) & (p_{2} \overset{\land}{,} q_{2}) & (p_{2} \overset{\land}{,} q_{3}) & . & . & . & (p_{2} \overset{\land}{,} q_{n}) \\ (p_{3} \overset{\land}{,} q_{1}) & (p_{3} \overset{\land}{,} q_{2}) & (p_{3} \overset{\land}{,} q_{3}) & . & . & . & (p_{3} \overset{\land}{,} q_{n}) \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ (p_{m} \overset{\land}{,} q_{1}) & (p_{m} \overset{\land}{,} q_{2}) & (p_{m} \overset{\land}{,} q_{3}) & . & . & . & (p_{m} \overset{\land}{,} q_{n}) \end{matrix}]$ (3)

Where $\overset{\land}{E}$ is an effect graph. If the oriented effect (p_j, q_k) could be interpreted mathematically as some oriented algebraic operation ∧ originating at p_j and ending at q_k (i.e.il from p_j toil q_k) $(p_{j} \overset{\land}{,} q_{k}) \to [p_{j}] \land [q_{k}]$ (4)

This algebraic operator ∧ may refer to addition multiplication inner product subtraction ... etc.). The quantity [p_j] ∧ [q_k] may be scaled (or normalized) simply by multiplication with the corresponding scaled (normalization) coefficient,y_jk: that is

$(p_{j} \overset{\land}{,} q_{k}) \to y_{jk} ([p_{j}] \land [q_{k}]) = [p_{j} (y_{j})] \land [q_{k} (y_{k})]$ (5)

Forming the matrix of (5) yields

$y_{jk} ([p_{j}] \land [q_{k}]); (\begin{matrix} j = 1, 2, 3, . . ., m \\ k = 1, 2, 3, . . ., n \end{matrix}) \to E • \overset{\land}{E}$ (6)

2.5 Definition ([11])

The symbol (•) refers to a special type of matrix multiplication in which each element of the matrix E is multiplied with the corresponding element in the matrix $\overset{\land}{E}$ and this special type of the multiplication matrix is called “Effect product”.

2.6 Definition ([11])

The special type of the quantity $E • \overset{\land}{E}$ is a matrix and its called “Effect matrix of $R$ ” and denoted by $\overset{\land}{E} (R, p_{m}, q_{n})$ .

3 Algebraic operations on effect matrix of type 2-tuple

Assume that R₁ = {x_i|i = 1, 2, 3, . . . , I} and R₂ = {y_j|j = 1, 2, 3, . . . , L} are two collections of distinct objects where I, L ∈ N. Assume that there exists some finite positive number m of algebraic operations; p₁, p₂, p₃, . . . , p_m, which can be applied on R₁ in the formp_t (x_i), where t = 1, 2, 3, . . . , m ≤ I . Assume that there exist another finite positive number n of other algebraic operations; q₁, q₂, q₃, . . . , q_n, which can be also applied on R₂ in the form of q_k (y_j), where k = 1, 2, 3, . . . , n ≤ L . Assume that there exists a rectangular matrix of coefficients in the following form: $T = [\begin{matrix} a_{11} a_{12} a_{13} . . . a_{1 n} \\ a_{21} a_{22} a_{23} . . . a_{2 n} \\ a_{31} a_{32} a_{33} . . . a_{3 n} \\ . . . . . . . \\ . . . . . . . \\ . . . . . . . \\ a_{m 1} a_{m 2} a_{m 3} . . . a_{mn} \end{matrix}]$

Where a_tk = (x_t, y_k) for all t = 1, 2, 3, . . . , m ≤ I and k = 1, 2, 3, . . . , n ≤ L. Let us symbolize the effect graph by the symbol $\overset{\land}{T}$ in the following form: $\overset{\land}{T} = [\begin{matrix} (p_{1} \overset{\land}{,} q_{1}) & (p_{1} \overset{\land}{,} q_{2}) & (p_{1} \overset{\land}{,} q_{3}) & . & . & . & (p_{1} \overset{\land}{,} q_{n}) \\ (p_{2} \overset{\land}{,} q_{1}) & (p_{2} \overset{\land}{,} q_{2}) & (p_{2} \overset{\land}{,} q_{3}) & . & . & . & (p_{2} \overset{\land}{,} q_{n}) \\ (p_{3} \overset{\land}{,} q_{1}) & (p_{3} \overset{\land}{,} q_{2}) & (p_{3} \overset{\land}{,} q_{3}) & . & . & . & (p_{3} \overset{\land}{,} q_{n}) \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ (p_{m} \overset{\land}{,} q_{1}) & (p_{m} \overset{\land}{,} q_{2}) & (p_{m} \overset{\land}{,} q_{3}) & . & . & . & (p_{m} \overset{\land}{,} q_{n}) \end{matrix}]$

Then the special type of the quantity $T • \overset{\land}{T}$ is a matrix and it’s called “Effect matrix of typei2-tuple” and denoted by $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ .

3.1 Definition

Let (F, A) be a soft set over a universe U = {u₁, u₂, …, u_I} and the parameter set A ⊆ E = {e₁, e₂, …, e_L}. Also, let p₁, p₂, p₃, . . . , p_I and q₁, q₂, q₃, . . . , q_L be algebraic operations on U and on E, respectively. Define ${p_{t}}_{t = 1}^{I}$ as follows:

$p_{t} = {\begin{matrix} l \underset{F (A)}{ξ} (e_{t}) if t \leq L \\ \underset{F (A)}{ξ} (e_{h}) if t > L \end{matrix}$ , where h ≤ I with t ≡ h ( I). And define ${q_{k}}_{k = 1}^{L}$ byq_k = F. Then $\overset{\land}{T} (U \times E, p_{m}, q_{n})$ is called Soft effect matrix of type 2- $tuple, where (p_{t} \overset{\land}{,} q_{k}) \to a_{tk} ([p_{t}] \land [q_{k}]) = [p_{t} (u_{t})] \land [q_{k} (e_{k})]$ $= \underset{F (A)}{ξ} (e_{t}) (u_{t}) +$ $\frac{| F (e_{k}) |}{SS (U_{A})}$ , and a_tk = (u_t, e_k) ∈ U × E

3.2 Lemma

If e_k ∈ E - A, then a_tk ([p_t] ∧ [q_k]) $= \underset{F (A)}{ξ} (e_{t}) (u_{t})$ , for all u_t ∈ U.

Proof: Since e_k ∈ E - A, thus F (e_k) = φ and hence |F (e_k) |=0. However, a_tk ([p_t] ∧ [q_k]) = [p_t (u_t)] ∧ $[q_{k} (e_{k})] = \underset{F (A)}{ξ} (e_{t}) (u_{t}) +$ $\frac{| F (e_{k}) |}{SS (U_{A})}$ , for all a_tk = (u_t, e_k)∈ U × E. Therefore we consider thata_tk ([p_t] ∧ [q_k]) $= \underset{F (A)}{ξ} (e_{t}) (u_{t})$ , for all u_t ∈ U.

3.3 Lemma

If $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ is a soft effect matrix of type 2-tuple, then the cardinality of {p_t} _t≤m in $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ satisfies | {p_t} _t≤m| = n.

Proof:

Since $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ is a soft effect matrix of type 2-tuple, then there exists soft set over a universe U = {u₁, u₂, …, u_m} and the parameter set A ⊆ E = {e₁, e₂, …, e_n} with $p_{i} = \underset{F (A)}{ξ} (e_{i})$ and q_j = F. However, $p_{i} = \underset{F (A)}{ξ} (e_{i})$ then the cardinality of {p_i} _i≤m in $\overset{\land}{T} (R_{1} \times R_{2}, p_{m}, q_{n})$ dependent on cardinality of E. That means | {p_t} _t≤m| = |E|. But |E| = | {e₁, e₂, …, e_n} | = n. Hence | {p_t} _t≤m| = n.

3.4 Remark

we will consider new definitions more general by dealing with (m × n) cells instead of (n²) cells.

3.5 Definition

Let ${A_{i}}_{i = 1}^{μ}$ be a collection of colours and S be a quadrilateral shape which is divided into (m × n) of equal sub-square shapes (cells), where each cell has colour A_i for some (1 ≤ i ≤ μ). Then the line which is passing through at least two cells of colour A_i is called path and denoted by P (A_i).

3.6 Example

Let S be a quadrilateral shape consists (30) cells, where each cell has colour A_i for (1 ≤ i ≤ 7) as follows:

Then the following are hold in above example:

There are 9 paths which are passing through 26.

There are two paths of types P (A₁),P (A₂) and P (A₆).

There is a single path for each type of {P (A₃) , P (A₄) , P (A₅)}.

There exists no path of type P (A₇).

There is no path of type P (A₄) between cell 9 and cell 18.

3.7 Definition

Let $\overset{\land}{T} (U \times E, p_{m}, q_{n})$ be soft effect matrix of type 2-tuple generated a picture consists of (m × n) cells, where the area of all the cells are equal and each cell has a special colour. Then for each pair of cells c_ij and c_kl whose colour A are called A- connected, if there is a path P (A) between them. In every other case are called A- disconnected.

3.8 Definition

Let $\overset{\land}{T} (U \times E, p_{m}, q_{n})$ be soft effect matrix of type 2-tuple generated a picture consists of (m × n) cells, where the area of all the cells are equal and each cell has a special colour, this picture is called regular if all pairs of cells whose colour A are A- connected.

3.9 Definition

Let $\overset{\land}{T} (U \times E, p_{m}, q_{n})$ be soft effect matrix of type 2-tuple generated a picture consists of (m × n) cells, where the area of all the cells are equal and each cell has a special colour, this picture is called irregular if there are at least two cells whose colour A are A- disconnected. Moreover, if any picture has more one path of the same type. Then it is an irregular picture.

3.10 Steps of the work

In the present work, for any soft set (F, A) over a universe U = {u₁, u₂, …, u_m} and the parameter set A ⊆ E = {e₁, e₂, …, e_n}, we can find picture consists of (n × m) square shapes, where the area of all square shapes are equal and each square shape is called cell and has a special colour. Let μ be the number of colours in a given problem and c_ij, for all (1 ≤ i ≤ m ; 1 ≤ j ≤ n) be elements in soft effect matrix of type 2-tuple $\overset{\land}{T} (U \times E, p_{m}, q_{n})$ which is introduced in this work. Then for each colour given, there is a semi open set $[\frac{(2 μ - 3) λ}{2}, \infty) or [\frac{(2 r - 3) λ}{2}, \frac{(2 r - 1) λ}{2})$ or $[0, \frac{λ}{2})$ in real numbers R, where (2 ≤ r ≤ μ - 1) and $λ = \frac{Max (c_{ij}) - Min (c_{ij})}{μ}$ , it is represent the level of this colour. Moreover, this generated picture may be regular or irregular. The procedure of this work can be compressed as following steps:

Find the soft effect matrix of type 2-tuple $\overset{\land}{T} (U \times E, p_{m}, q_{n})$ .

Assume Max [c_ij] = L , Min [c_ij] = K and μ is a number of the colours.

$\begin{matrix} 3) - Let c_{ij} = \\ {\begin{matrix} A_{1} & if 0 \leq c_{ij} < \frac{λ}{2} \\ A_{r} & if \frac{(2 r - 3) λ}{2} \leq c_{ij} < \frac{(2 r - 1) λ}{2}, \\ (2 \leq r \leq μ - 1) \\ A_{μ} & if \frac{(2 μ - 3) λ}{2} \leq c_{ij} \end{matrix} \end{matrix}$ Where $λ = \frac{L - K}{μ}$ and each of A_i (i = 1, 2, . . . . . , μ) represent a special colour.

3.11 Remarks

If more of the colour are given, then a good picture will be seen.

3.12 Remark

If we choose (μ) colours it is not necessary all (μ) colours are appear in a picture. [see Example (3.13)].

3.13 Example

Suppose that U is the set of men whose looking for job under consideration, say U = {x₁, x₂, x₃}. Let E be the set of some attributes of such men, say E = {e₁, e₂, e₃, e₄}, where e₁, e₂, e₃, e₄ stand for the attribute “healthy body”, “hearing activity”, “smell activity”, “sight activity”, respectively. Suppose that the soft set (F, A) where A = {e₁, e₃, e₄} , describing the Dr. X opinion to detect the right man for the job was defined by F (e₁) = {x₁} , F (e₃) = {x₁, x₂} , F (e₄) = φ. Find and determine the type of the picture (regular or irregular) that is generated by the soft effect matrix of type 2-tuple $\overset{\land}{T} (U \times E, p_{m}, q_{n})$ .

Solution.

Assume A₁ = white, A₂ = yellow, A₃ = orange, A₄ = rose, A₅ = red, A₆ = green, A₇ = blue, A₈ = violet, A₉ = brown, A₁₀ = black, thus μ = 10. Also, we consider that $T = [\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \end{matrix}]$ where a_tk = (x_t, e_k) ∈ U × E and $\overset{\land}{T} [\begin{matrix} (p_{1} \overset{\land}{,} q_{1}) & (p_{1} \overset{\land}{,} q_{2}) & (p_{1} \overset{\land}{,} q_{3}) & (p_{1} \overset{\land}{,} q_{4}) \\ (p_{2} \overset{\land}{,} q_{1}) & (p_{2} \overset{\land}{,} q_{2}) & (p_{2} \overset{\land}{,} q_{3}) & (p_{2} \overset{\land}{,} q_{4}) \\ (p_{3} \overset{\land}{,} q_{1}) & (p_{3} \overset{\land}{,} q_{2}) & (p_{3} \overset{\land}{,} q_{3}) & (p_{3} \overset{\land}{,} q_{4}) \end{matrix}]$

where ${p_{t}}_{t = 1}^{3}$ and ${q_{k}}_{k = 1}^{4}$ are defined as follows; $p_{t} = {\begin{matrix} \underset{F (A)}{ξ} (e_{t}) if t \leq 4 \\ \underset{F (A)}{ξ} (e_{h}) if t > 4 \end{matrix}$ where h ≤ 3 with t ≡ h ( 3), and q_k = F. Now, we need to find [p_t (x_t)] and [q_k (e_k)] for all (1 ≤ t ≤ 3) and (1 ≤ k ≤ 4) as the following:

$\underset{F (A)}{ξ} (e_{1}) (x_{1}) = (1 / n (SS (U_{A})) = 0 . 001953125,$ $\underset{F (A)}{ξ} (e_{2}) (x_{2}) = 0$ ,

$\underset{F (A)}{ξ} (e_{3}) (x_{3}) = n (SS (U_{A})) / n (SS (U_{E})) = 0 . 125 .$

Where n (SS (U_A)) =2⁹ = 512 and n (SS (U_E)) = 2¹² = 4096.

Also, |F (e₁) |=1, |F (e₂) |=0, |F (e₃) |=2, |F (e₄) |=0. However, a_tk ([p_t] ∧ [q_k]) = [p_t (x_t)] ∧ [q_k (e_k)] = $\underset{F (A)}{ξ} (e_{t}) (x_{t}) + \frac{| F (e_{k}) |}{SS (U_{A})}$ . Then, $\begin{matrix} a_{11} ([p_{1}] \land [q_{1}]) = [p_{1} (x_{1})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 0 . 00390625 \end{matrix}$ $\begin{matrix} a_{12} ([p_{1}] \land [q_{2}]) = [p_{1} (x_{1})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 0 . 001953125 \end{matrix}$ $\begin{matrix} a_{13} ([p_{1}] \land [q_{3}]) = [p_{1} (x_{1})] \land [q_{3} (e_{3})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 0 . 005859375 \end{matrix}$ $\begin{matrix} a_{14} ([p_{1}] \land [q_{4}]) = [p_{1} (x_{1})] \land [q_{4} (e_{4})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{4}) |}{SS (U_{A})} = 0 . 001953125 \end{matrix}$ $\begin{matrix} a_{21} ([p_{2}] \land [q_{1}]) = [p_{2} (x_{2})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 0 . 001953125 \end{matrix}$ $\begin{matrix} a_{22} ([p_{2}] \land [q_{2}]) = [p_{2} (x_{2})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 0 \end{matrix}$ $\begin{matrix} a_{23} ([p_{2}] \land [q_{3}]) = [p_{2} (x_{2})] \land [q_{3} (e_{3})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 0 . 00390625 \end{matrix}$ $\begin{matrix} a_{24} ([p_{2}] \land [q_{4}]) = [p_{2} (x_{2})] \land [q_{4} (e_{4})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{4}) |}{SS (U_{A})} = 0 \end{matrix}$

$\begin{matrix} a_{31} ([p_{3}] \land [q_{1}]) = [p_{3} (x_{3})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 0 . 126953125 \end{matrix}$ $\begin{matrix} a_{32} ([p_{3}] \land [q_{2}]) = [p_{3} (x_{3})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 0 . 125 \end{matrix}$ $\begin{matrix} a_{33} ([p_{3}] \land [q_{3}]) = [p_{3} (x_{3})] \land [q_{3} (e_{3})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 0 . 12890625 \end{matrix}$ $\begin{matrix} a_{34} ([p_{3}] \land [q_{4}]) = [p_{3} (x_{3})] \land [q_{4} (e_{4})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{4}) |}{SS (U_{A})} = 0 . 125 . \end{matrix}$

Hence we have $\overset{\land}{T} (U \times E, p_{3}, q_{4}) = T • \overset{\land}{T} = [\begin{matrix} c_{11} & c_{12} & c_{13} & c_{14} \\ c_{21} & c_{22} & c_{23} & c_{24} \\ c_{31} & c_{32} & c_{33} & c_{34} \end{matrix}]$ $(\begin{matrix} 0 . 00390625 & 0 . 001953125 & 0 . 005859375 & 0 . 001953125 \\ 0 . 001953125 & 0 & 0 . 00390625 & 0 \\ 0.126953125 & 0.125 & 0.12890625 & 0.125 \end{matrix})$

Table 1

λ 0.12890625

λ/2 0.064453125

3λ/2 0.193359375

5λ/2 0.322265625

7λ/2 0.4511718

9λ/2 0.580078125

11λ/2 0.708984375

13λ/2 0.837890625

15λ/2 0.966796875

17λ/2 1.095703125

Let $\begin{matrix} c_{ij} = \\ {\begin{matrix} A_{1} & if 0 \leq c_{ij} < \frac{λ}{2} \\ A_{r} & if \frac{(2 r - 3) λ}{2} \leq c_{ij} < \frac{(2 r - 1) λ}{2}, (2 \leq r \leq μ - 1) \\ A_{μ} & if \frac{(2 μ - 3) λ}{2} \leq c_{ij} \end{matrix} \end{matrix}$

Where $λ = \frac{L - K}{μ}$ and each of A_i (i = 1, 2, . . . . . , μ) represent a special colour

Then there are eight cells c_ij∈ {c₁₁, c₁₂, c₁₃, c₁₄, c₂₁, c₂₂, c₂₃, c₂₄} whose colourA₁ and all of them are A₁- connected of type p (A₁) and four cells c_kl∈ {c₃₁, c₃₂, c₃₃, c₃₄} whose colourA₂ and all of them are A₂- connected of typeP (A₂). So the picture which is generated by soft effect matrix of type 2-tuple $\overset{\land}{T} (U \times E, p_{3}, q_{4})$ is a regular picture (see Fig. 3).

Fig.1

Shape consists 30 cells.

Fig.2

Shape has two paths.

Fig.3

Regular picture with two paths.

3.14 Remark

If we replace the soft set (F, A) over the same an initial universe set U and the same set of parameters E in example (3.13). Then,

We can turn Fig. (3), see Figs. (4-a,b,c). For example, let (G, B) be a soft set where B = {e₁, e₂, e₃} , G (e₁) = {x₂, x₃} , G (e₂) = φ, G (e₃) = {x₃}. Then we have:

$\begin{matrix} \overset{\land}{T} (U \times E, p_{3}, q_{4}) = T • \overset{\land}{T} = [\begin{matrix} c_{11} & c_{12} & c_{13} & c_{14} \\ c_{21} & c_{22} & c_{23} & c_{24} \\ c_{31} & c_{32} & c_{33} & c_{34} \end{matrix}] = \end{matrix}$ $(\begin{matrix} 0.126953125 & 0.125 & 0.12890625 & 0.125 \\ 0 . 00390625 & 0 & 0 . 001953125 & 0 \\ 0 . 005859375 & 0 . 001953125 & 0 . 00390625 & 0 . 001953125 \end{matrix})$

That means we turn Fig. (3) two times and hence we consider Fig. (4-c)

Fig.4

New pictures induced by turning one picture.

we can consider new picture see Fig. (6) in the following example.

3.15 Example

See example 3.13, If we replace redefine soft set (F, A) as follows: F (e₁) = {x₁, x₃} , F (e₃) = {x₁, x₂} , F (e₄) = φ. Then we consider the following, $\underset{F (A)}{ξ} (e_{1}) (x_{1}) = 1$ , $\underset{F (A)}{ξ} (e_{2}) (x_{2}) = 0$ , $\underset{F (A)}{ξ} (e_{3}) (x_{3}) =$ n (SS (U_A))/n (SS (U_E)) = 0 . 125 . Also, |F (e₁) |=2, |F (e₂) |=0, |F (e₃) |=2, |F (e₄) |=0. Hence we consider that; $\begin{matrix} a_{11} ([p_{1}] \land [q_{1}]) = [p_{1} (x_{1})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 1 . 00390625 \end{matrix}$ $\begin{matrix} a_{12} ([p_{1}] \land [q_{2}]) = [p_{1} (x_{1})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 1 \end{matrix}$ $\begin{matrix} a_{13} ([p_{1}] \land [q_{3}]) = [p_{1} (x_{1})] \land [q_{3} (e_{3})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 1 . 00390625 \end{matrix}$ $\begin{matrix} a_{14} ([p_{1}] \land [q_{4}]) = [p_{1} (x_{1})] \land [q_{4} (e_{4})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{4}) |}{SS (U_{A})} = 1 \end{matrix}$ $\begin{matrix} a_{21} ([p_{2}] \land [q_{1}]) = [p_{2} (x_{2})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 0 . 00390625 \end{matrix}$ $\begin{matrix} a_{22} ([p_{2}] \land [q_{2}]) = [p_{2} (x_{2})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 0 \end{matrix}$ $\begin{matrix} a_{23} ([p_{2}] \land [q_{3}]) = [p_{2} (x_{2})] \land [q_{3} (e_{3})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 0 . 00390625 \end{matrix}$ $\begin{matrix} a_{24} ([p_{2}] \land [q_{4}]) = [p_{2} (x_{2})] \land [q_{4} (e_{4})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{4}) |}{SS (U_{A})} = 0 \end{matrix}$ $\begin{matrix} a_{31} ([p_{3}] \land [q_{1}]) = [p_{3} (x_{3})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 0 . 126953125 \end{matrix}$ $\begin{matrix} a_{32} ([p_{3}] \land [q_{2}]) = [p_{3} (x_{3})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 0 . 125 \end{matrix}$ $\begin{matrix} a_{33} ([p_{3}] \land [q_{3}]) = [p_{3} (x_{3})] \land [q_{3} (e_{3})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 0 . 12890625 \end{matrix}$ $\begin{matrix} a_{34} ([p_{3}] \land [q_{4}]) = [p_{3} (x_{3})] \land [q_{4} (e_{4})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{4}) |}{SS (U_{A})} = 0 . 125 . \end{matrix}$

Then, $\begin{matrix} \overset{\land}{T} (U \times E, p_{3}, q_{4}) = T • \overset{\land}{T} = [\begin{matrix} c_{11} & c_{12} & c_{13} & c_{14} \\ c_{21} & c_{22} & c_{23} & c_{24} \\ c_{31} & c_{32} & c_{33} & c_{34} \end{matrix}] \\ = (\begin{matrix} 1 . 00390625 & 1 & 1 . 00390625 & 1 \\ 0 . 00390625 & 0 & 0 . 00390625 & 0 \\ 0.126953125 & 0.125 & 0.12890625 & 0.125 \end{matrix}) \end{matrix}$

Table 2

λ 0.100390625

λ/2 0.0501953125

3λ/2 0.1505859375

5λ/2 0.2509765625

7λ/2 0.3513671875

9λ/2 0.4517578125

11λ/2 0.5521484375

13λ/2 0.6525390625

15λ/2 0.752929875

17λ/2 0.8533203125

Hence we consider Fig. (5)

Fig.5

Shape consists 16 cells.

Then there are three cells c_ij∈ {c₁₁, c₁₂, c₁₃, c₁₄} whose colour A₁₀ and all of them are A₁₀- connected of type P (A₁₀), three cells c_kl∈ {c₂₁, c₂₂, c₂₃, c₂₄} whose colourA₁ and all of them are A₁- connected of type p (A₁) and three cells c_kl∈ {c₃₁, c₃₂, c₃₃, c₃₄} whose colourA₂ and all of them are A₂- connected of type P (A₂). Therefore we generated new regular picture see Fig. (6).

Fig.6

Regular picture with three paths.

3.16 Definition

Let (F, A) and (G, B) be two different soft sets over the common universal soft set (U, E). Then (F, A) and (G, B) are called equivalence and they are denoted by (F, A) ≈ (G, B) if and only if they have the same soft effect matrix of type 2-tuple $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ .

3.17 Remark

It is clearly, the relation ≈ is an equivalence relation on SS (U_E). Verifications of reflexivity, symmetry and transitivity. Then we can divide the set SS (U_E) into equivalence classes. Denote the equivalence class of (F, A) by [F, A] . Where [F, A] = {(G, B) ∈ SS (U_E) | (G, B) ≈ (F, A)} . Further, if any soft set (G, B) ∈ SS (U_E) has regular (irregular) picture, and (F, A) has irregular (regular) picture then we say (G, B) and (F, A) are not equivalent. Moreover, if(G, B) , (F, A) ∈ [F, A]. Then [G, B] = [F, A] and either both of the soft sets (F, A) and (G, B) have regular pictures or irregular pictures.

3.18 Example

Suppose that U = {x₁, x₂, x₃, x₄} is a set of four patients. Let E be a set of some diseases, say E = {e₁, e₂, e₃, e₄}, where e₁, e₂, e₃, e₄ stand for the attributes “body temperature”, “cough with chest congestion”, “ cough with no chest congestion”, respectively. Suppose that the soft set (F, A) describing the Dr.X opinion to detect the emergency state was defined by A = {e₁, e₃, e₄} , F (e₁) = {x₁} , F (e₃) = {x₁, x₂} , F (e₄) = φ. In addition, we assume that the “emergency state” in the opinion of another Doctor, say Dr.Y, is described by the soft set (G, B), where B = {e₁, e₂, e₃}, G (e₁) = {x₁} , G (e₂) = φ, G (e₃) = {x₁, x₄}. Now, we need to find [p_t (x_t)] and [q_k (e_k)] for all (1 ≤ t ≤ 4) and (1 ≤ k ≤ 4) as the following: $\underset{F (A)}{ξ} (e_{1}) (x_{1}) = (1 / n (SS (U_{A})) = 0 . 000244140625,$ $\underset{F (A)}{ξ} (e_{2}) (x_{2}) = 0$ , $\underset{F (A)}{ξ} (e_{3}) (x_{3}) = n (SS (U_{A})) / n (SS (U_{E})) = 0 . 0625$ , $\underset{F (A)}{ξ} (e_{4}) (x_{4}) = 0$ .

Where n (SS (U_A)) =2¹² = 4096 and n (SS (U_E)) =2¹⁶ = 65536.

Also, |F (e₁) |=1, |F (e₂) |=0, | F (e₃) |=2, |F (e₄) |=0. Then, $\begin{matrix} a_{11} ([p_{1}] \land [q_{1}]) = [p_{1} (x_{1})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 0 . 00048828125 \end{matrix}$ $\begin{matrix} a_{12} ([p_{1}] \land [q_{2}]) = [p_{1} (x_{1})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 0 . 000244140625 \end{matrix}$ $\begin{matrix} a_{13} ([p_{1}] \land [q_{3}]) = [p_{1} (x_{1})] \land [q_{3} (e_{3})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 0 . 000732421875 \end{matrix}$ $\begin{matrix} a_{14} ([p_{1}] \land [q_{4}]) = [p_{1} (x_{1})] \land [q_{4} (e_{4})] = \\ \underset{F (A)}{ξ} (e_{1}) (x_{1}) + \frac{| F (e_{4}) |}{SS (U_{A})} = 0 . 000244140625 \end{matrix}$ $\begin{matrix} a_{21} ([p_{2}] \land [q_{1}]) = [p_{2} (x_{2})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 0 . 000244140625 \end{matrix}$ $\begin{matrix} a_{22} ([p_{2}] \land [q_{2}]) = [p_{2} (x_{2})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 0 \end{matrix}$ $\begin{matrix} a_{23} ([p_{2}] \land [q_{3}]) = [p_{2} (x_{2})] \land [q_{3} (e_{3})] = \end{matrix}$

$\begin{matrix} \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 0 . 00048828125 \end{matrix}$ $\begin{matrix} a_{24} ([p_{2}] \land [q_{4}]) = [p_{2} (x_{2})] \land [q_{4} (e_{4})] = \\ \underset{F (A)}{ξ} (e_{2}) (x_{2}) + \frac{| F (e_{4}) |}{SS (U_{A})} = 0 \end{matrix}$ $\begin{matrix} a_{31} ([p_{3}] \land [q_{1}]) = [p_{3} (x_{3})] \land [q_{1} (e_{1})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{1}) |}{SS (U_{A})} = 0 . 062744140625 \end{matrix}$ $\begin{matrix} a_{32} ([p_{3}] \land [q_{2}]) = [p_{3} (x_{3})] \land [q_{2} (e_{2})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{2}) |}{SS (U_{A})} = 0 . 0625 \end{matrix}$ $\begin{matrix} a_{33} ([p_{3}] \land [q_{3}]) = [p_{3} (x_{3})] \land [q_{3} (e_{3})] = \\ \underset{F (A)}{ξ} (e_{3}) (x_{3}) + \frac{| F (e_{3}) |}{SS (U_{A})} = 0 . 06298828125 \end{matrix}$ $\begin{matrix} a_{34} ([p_{3}] \land [q_{4}]) = [p_{3} (x_{3})] \land [q_{4} (e_{4})] = \\ 0 . 0625 . \end{matrix}$

Then,

a₄₁ ([p₄] ∧ [q₁]) = [p₄ (x₄)] ∧ [q₁ (e₁)] =0 . 000244140625

a₄₂ ([p₄] ∧ [q₂]) = [p₄ (x₄)] ∧ [q₂ (e₂)] =0

a₄₃ ([p₄] ∧ [q₃]) = [p₄ (x₄)] ∧ [q₃ (e₃)] =0 . 00048828125

a₄₄ ([p₄] ∧ [q₄]) = [p₄ (x₄)] ∧ [q₄ (e₄)] =0. Then, the soft set (F, A) generated soft effect matrix of type 2 tuple as follows: $(\begin{array}{l} \begin{matrix} 0 .00048828125 & 0 .000244140625 & 0 .000732421875 & 0 .000244140625 \\ 0 .000244140625 & 0 & 0 .00048828125 & 0 \\ 0 .062744140625 & 0.0625 & 0 .06298828125 & 0.0625 \end{matrix} \\ \begin{matrix} 0 .000244140625 & 0 & 0 .00048828125 & 0 \end{matrix} \end{array})$

Similarity, we can find the soft effect matrix of type 2-tuple for (G, B) which is defined as follows: $(\begin{array}{l} \begin{matrix} 0 .00048828125 & 0 .000244140625 & 0 .000732421875 & 0 .000244140625 \\ 0 .000244140625 & 0 & 0 .00048828125 & 0 \\ 0 .062744140625 & 0.0625 & 0 .06298828125 & 0.0625 \end{matrix} \\ \begin{matrix} 0 .000244140625 & 0 & 0 .00048828125 & 0 \end{matrix} \end{array})$

Hence [G, B] = [F, A]. That means both of the soft sets (F, A) and (G, B) have the same picture, because they have the same soft effect matrix of type 2-tuple. Then Dr. X and Dr. Y their opinions are equivalence.

Table 3

λ 0.006298828125

λ/2 0.0031494140625

3λ/2 0.0094482421875

5λ/2 0.0157470703125

7λ/2 0.0220458984375

9λ/2 0.0283447265625

11λ/2 0.0346435546875

13λ/2 0.0409423828125

15λ/2 0.0472412109375

17λ/2 0.0535400390625

Hence we consider Fig. (7)

Fig.7

Shape has three paths.

Here we consider picture has more one path of the same type. Hence it is irregular picture see Fig. (8).

Fig.8

An irregular picture.

3.19 Similarity Measurement and Their Applications in Medical Diagnosis Problems

Several researchers have studied the problem of similarity measurement between fuzzy sets, fuzzy numbers, vague sets and soft sets in different ways. In this work a measure of similarity between two soft sets has been given using effect matrix $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple. In another side, sometimes there are two soft sets (F, A) and (G, B) in SS (U_E) nearly they are similar. That means F (e) = G (e) for most of e ∈ E. So its not necessary they have different pictures. That means if there are two Doctors say Dr. X (his opinion is(F, A)) and Dr. Y (his opinion is (G, B)), then we say Dr. X and Dr. Y their opinions are similarity or equivalence if and only if they have the same soft effect matrix of type 2-tuple $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ and hence we consider that their pictures are similar. This idea can be extension in many fields to solve some decision making problems, if we assume (F, A) is our model and {(F_i, A_i) |1 ≤ i ≤ k} is the family of soft sets over ( $R_{1} \times R_{2}$ ). Then we recommend to choose (F_i, A_i) for some 1 ≤ i ≤ k) if (F_i, A_i) is equivalence with our model (F, A).

3.20 Example

Let U=c₁,c₂, c₃,c₄,c₅,c₆ which is the set of six patients. Let E be a set of some diseases, say E = {e₁, e₂, e₃, e₄}, where e₁, e₂, e₃, e₄ stand for the attributes “body ache”, “headache”, “loose motion”, “breathing trouble”, respectively. Suppose Γ = {(F_i, A_i) |1 ≤ i ≤ 5} is a family of soft sets describing the emergency state of the patients with respect to the given diseases. Moreover, Mr. Z is a employee in the hospital and he need to know which one of the patients need to enter to the intensive care unit. Suppose that the soft set (F, A) describing hospital’s model to choose the emergency state of the patients that need to enter to the intensive care unit with respect to the given diseases. For any soft set (F, A) let us refer to its generated picture by P^F. Let P^F,P^{F
_i} be defined respectively, where 1 ≤ i ≤ 5 as follows:

Fig.9

The collection of pictures.

That means hospital’s model is P^F. Further, Mr. Z need to choose which one of the patients is equivalence with hospital’s model and then he will decide to enter him to the intensive care unit or no. Now, to help Mr. Z we need to answer the questions:

Is there any member (F_i, A_i) in Γsuch that P^{F
_i} =P^F?

Is there any member (F_i, A_i) in Γsuch that P^{F
_i}≠P^F?

We consider the following two pictures {P^{F
₂}, P^{F
₅}}are equivalence with hospital’s model. Then we recommend Mr. Z to choose the second and fifth of the patients. Moreover, we consider the following three pictures {P^{F
₁}, P^{F
₃}, P^{F
₄}} are not equivalence with hospital’s model. Then we do not recommend Mr. Z to choose the first, third and fourth of the patients.

3.21 Remark

It is clearly, If hospital’s model has regular (irregular) picture, then we do not recommend Mr. Z to choose any one of the patients that is described as irregular (regular) picture.

4 Conclusion

In this paper, we have introduced the notions of the effect matrix $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple and their picture. Using this, idea of picture consists of (m × n) cells, we have studied A- connected, A- disconnected, regular pictures and irregular pictures. Further, for any pair of soft sets (F, A) and (G, B) in SS (U_E) we defined relation ≈ on SS (U_E) by (F, A) ≈(G, B) if they have the same effect matrix $\overset{\land}{T} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple. That means if there are two persons say Mr. X (his opinion is (F, A)) and Mr.Y (his opinion is (G, B)), then we say Mr. X and Mr.Y their opinions are equivalence if and only if (F, A) ≈ (G, B). Similarity measure of two soft sets has been given using effect matrix $\overset{\land}{E} (R_{1} \times R_{2}, p_{n}, q_{m})$ of type 2-tuple and an application of this to solve a decision making problem in medical diagnosis has been shown. The author is hopeful that this modified concept will yield more natural results and will be more useful in dealing with some problems related to uncertainty.

References

Aktaş

and Çağman

, “Soft sets and soft groups,”, Information Sciences 177(13) (2007), 2726–2735.

Ballerini

, Hogberg

, Lundstrom

and Borgefors

, Colour image analysis technique for measuring of fat in meat: An application for the meat industry, Electronic Imaging 4301 (2001), 113–142.

Ballerini

, Genetic snakes for medical images segmentation, Lectures Notes in Computer Science 1596 (1999), 59–73.

Cavone

, Dotoli

, Epicoco

and Seatzu

, A decision making procedure for robust train rescheduling based on mixed integer linear programming and Data Envelopment Analysis, Applied Mathematical Modelling 52 (2017), 255–273.

Chen

, Tsang

E.C.

, Yeung

D.S.

and Wang

, The parameterization reduction of soft sets and its applications, Computers & Mathematics with Applications 49(5-6) (2005), 757–763.

Feng

, Jun

Y.B.

, Liu

X.Y.

and Li

, “An adjustable approach to fuzzy soft set based decision making,”, Journal of Computational and Applied Mathematics 234(1) (2010), 10–20.

Herrera

, Alonso

, Chiclana

and Herrera-Viedma

, Computing with words in decision making: Foundations, trends and prospects, Fuzzy Optimization and Decision Making 8(4) (2009), 337–364.

Hoang

Le.

, Measuring analogousness in picture fuzzy sets: From picture distance measures to picture association measures, Fuzzy Optimization and Decision Making (2016), 1–20.

Mahmood

, On intuitionistic fuzzy soft b - closed sets in intuitionistic fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Informatics 10(2) (2015), 221–233.

10.

Mahmood

, Dissimilarity Fuzzy Soft Points and their Applications, Fuzzy Information and Engineering 8 (2016), 281–294.

11.

Mahmood

, Generation the Regular Pictures and Irregular Pictures from the Sound by Using Effect Matrix

\overset{\land}{E} (R, p_{n}, q_{n})

, published in, Journal of Mathematical and Computational Science 3(2) (2013), 668–693.

12.

Mahmood

, Soft Regular Generalized b-Closed Sets in Soft Topological Spaces, Journal of Linear and Topological Algebra 3(4) (2014), 195–204.

13.

Mahmood

and Muhamad

, Soft BCL-Algebras of the Power Sets, International Journal of Algebra 11(7) (2017), 329–341.

14.

Mahmood

, Soft Sequentially Absolutely Closed Spaces, International Journal of Applications of Fuzzy Sets and Artificial Intelligence 7 (2017), 73–92.

15.

Mahmood

, Tychonoff Spaces in Soft Setting and their Basic Properties, International Journal of Applications of Fuzzy Sets and Artificial Intelligence 7 (2017), 93–112.

16.

Mahmood

and Hameed

, An algorithm for generating permutations in symmetric groups using soft spaces with general study and basic properties of permutations spaces, J Theor Appl Inform Technol 96(9) (2018), 2445–2457.

17.

Mahmood

and Hameed

, An algorithm for generating permutation algebras using soft spaces, Journal of Taibah University for Science 12(3) (2018), 299–308.

18.

Mahmood

and Abd Ulrazaq

, SoftAlgebras of the Power Sets, American Journal of Mathematics and Statistics 8(1) (2018), 1–7.

19.

Mahmood

and Alradha

, Soft Edge ρ-Algebras of the power sets, International Journal of Applications of Fuzzy Sets and Artificial Intelligence 7 (2017), 231–243.

20.

Mahmood

and Abd Ulrazaq

, New Category of Soft Topological Spaces, American, Journal of Computational and Applied Mathematics 8(1) (2018), 20–25.

21.

Mahmood

, New category of the fuzzy d-algebras, Journal of Taibah University for Science 12(2) (2018), 143–149.

22.

Mahmood

, Fuzzy Lindelof Closed Spaces and Their Operatios, Journal of General Mathematics Notes 21(1) (2014), 28–136.

23.

Mahmood

and Al-Batat

, Intuitionistic Fuzzy Soft LA- Semigroups and Intuitionistic Fuzzy Soft Ideals, International Journal of Applications of Fuzzy Sets and Artificial Intelligence 6 (2016), 119–132.

24.

Mahmood

, Ulrazaq

, Abdul-Ghani

and Al-Musawi

, s–Algebra and s–Baire in Fuzzy Soft Setting, Advances in Fuzzy Systems 2018 (2018), Article ID 5731682, 10 pages.

25.

Maji

P.K.

, Roy

A.R.

and Biswas

, Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 589–602.

26.

Maji

P.K.

, Biswas

and Roy

A.R.

, “Soft set theory,”, Computers & Mathematics with Applications 45(4-5) (2003), 555–562.

27.

Meng

, Jiang

, Xu

and Priananda

, Support top irrelevant machine: Learning similarity measures to maximize top precision for image retrieval, Neural Computing and Applications (2016), 1–10.

28.

Molodtsov

, Soft set theory first results, Computers & Mathematics with Applications 37(4-5) (1999), 19–31.

29.

Monin

, Recent methods for predicting quality of whole meat,, Meat Science 49(1) (1998), S231–S243.

30.

Pree

, Herwig

, Gruber

, Sick

, David

and Lukowicz

, On general purpose time series similarity measures and their use as kernel functions in support vector machines, Information Sciences 281 (2014), 478–495.

31.

Samaher

, Shuker

, Mayadah

and Abu Firas

, New Branch of Intuitionistic Fuzzification in Algebras with Their Applications, International Journal of Mathematics and Mathematical Sciences 2018 (2018), Article ID 5712676, 6 pages.

32.

Yang

, A not on soft set theory, Computers and Mathematics with Applications 56 (2008), 1899–1900.

33.

Yue

, An avoiding information loss approach to group decision making, Applied Mathematical Modelling 37 (2013), 112–126.

34.

Yue

, A group decision making approach based on aggregating interval data into interval-valued intuitionistic fuzzy information, Applied Mathematical Modelling 38 (2014), 683–698.

λ	0.12890625
λ/2	0.064453125
3λ/2	0.193359375
5λ/2	0.322265625
7λ/2	0.4511718
9λ/2	0.580078125
11λ/2	0.708984375
13λ/2	0.837890625
15λ/2	0.966796875
17λ/2	1.095703125

λ	0.100390625
λ/2	0.0501953125
3λ/2	0.1505859375
5λ/2	0.2509765625
7λ/2	0.3513671875
9λ/2	0.4517578125
11λ/2	0.5521484375
13λ/2	0.6525390625
15λ/2	0.752929875
17λ/2	0.8533203125

λ	0.006298828125
λ/2	0.0031494140625
3λ/2	0.0094482421875
5λ/2	0.0157470703125
7λ/2	0.0220458984375
9λ/2	0.0283447265625
11λ/2	0.0346435546875
13λ/2	0.0409423828125
15λ/2	0.0472412109375
17λ/2	0.0535400390625

Decision making using algebraic operations on soft effect matrix as new category of similarity measures and study their application in medical diagnosis problems

Abstract

Keywords

1 Introduction

2 Preliminaries and basic results

2.1 Definition ([28, 32])

2.2 Definition ([25])

2.3 Definition ([9])

2.4 Effect graph ([11])

2.6 Definition ([11])

3 Algebraic operations on effect matrix of type 2-tuple

3.1 Definition

3.2 Lemma

3.3 Lemma

3.4 Remark

3.5 Definition

3.6 Example

3.7 Definition

3.8 Definition

3.9 Definition

3.10 Steps of the work

3.11 Remarks

3.12 Remark

3.13 Example

Table 1 λ 0.12890625 λ/2 0.064453125 3λ/2 0.193359375 5λ/2 0.322265625 7λ/2 0.4511718 9λ/2 0.580078125 11λ/2 0.708984375 13λ/2 0.837890625 15λ/2 0.966796875 17λ/2 1.095703125

Table 2 λ 0.100390625 λ/2 0.0501953125 3λ/2 0.1505859375 5λ/2 0.2509765625 7λ/2 0.3513671875 9λ/2 0.4517578125 11λ/2 0.5521484375 13λ/2 0.6525390625 15λ/2 0.752929875 17λ/2 0.8533203125

3.17 Remark

3.18 Example

Table 3 λ 0.006298828125 λ/2 0.0031494140625 3λ/2 0.0094482421875 5λ/2 0.0157470703125 7λ/2 0.0220458984375 9λ/2 0.0283447265625 11λ/2 0.0346435546875 13λ/2 0.0409423828125 15λ/2 0.0472412109375 17λ/2 0.0535400390625

3.20 Example

4 Conclusion

References

Table 1

λ 0.12890625

λ/2 0.064453125

3λ/2 0.193359375

5λ/2 0.322265625

7λ/2 0.4511718

9λ/2 0.580078125

11λ/2 0.708984375

13λ/2 0.837890625

15λ/2 0.966796875

17λ/2 1.095703125

Table 2

λ 0.100390625

λ/2 0.0501953125

3λ/2 0.1505859375

5λ/2 0.2509765625

7λ/2 0.3513671875

9λ/2 0.4517578125

11λ/2 0.5521484375

13λ/2 0.6525390625

15λ/2 0.752929875

17λ/2 0.8533203125

Table 3

λ 0.006298828125

λ/2 0.0031494140625

3λ/2 0.0094482421875

5λ/2 0.0157470703125

7λ/2 0.0220458984375

9λ/2 0.0283447265625

11λ/2 0.0346435546875

13λ/2 0.0409423828125

15λ/2 0.0472412109375

17λ/2 0.0535400390625