Cardinality inverse soft matrix theory and its applications in multicriteria group decision making

Abstract

The main objective of this paper is to present a novel decision making algorithm using matrix representation of the inverse soft set defined in [10]. Therefore, we first introduce cardinality inverse soft matrix theory and its operations, products and algebraic structures in detail. Afterwards, an algorithmic solution employing the cardinality inverse soft matrix to find the optimum object and the ranking order of objects is proposed. The performance of algorithm named soft sum-row decision making algorithm is demonstrated by solving various decision problems. Also, we compare it with existing algorithms based on the soft set theory, soft matrix theory and inverse soft set theory. Moreover, we give Scilab codes of the algorithm and argue that this codes make the process of decision making composed of many objects, criteria and decision makers faster and easier.

Keywords

Inverse soft set cardinality inverse soft matrix operations of cardinality inverse soft matrix soft sum-row decision making

1 Introduction

Decision making problems are often encountered in various forms in our private, professional and social life. Because these problems are one of the unchanging elements of real life filled with uncertainties. In some cases, it is impossible for us to reach a conclusion without solving them. To remove these difficulties which involve uncertainties and prevent from reaching a conclusion, many decision making methods have been constructed by using various mathematical tools such as fuzzy set theory [41], intuitionistic fuzzy set theory [4], vague set theory [18] and rough set theory [30]. Also, many studies investigating the relationships between these mathematical tools have been presented [2 , 7].

In recent years, soft set theory defined in [29] is proposed as a new mathematical tool to solve the decision making problems. Many researchers focused on the soft set theory to construct various decision methods according to the type of uncertainty in the decision problems. Maji et al. [27, 28] gave representations in the form of binary information table of soft sets. They showed that these representations can be used to solve the decision making problems, and thus they pioneered the soft set theory based on decision making. Afterwards, Çaḡman and Enginoḡlu [9] redefined some operations of soft sets with a different approach to make them more useful for improving several new results and constructed a uni - int decision making method. They pointed out that two decision makers can choose an optimum alternative from the alternatives via this method. Feng et al. [16] presented the decision making schemes called uni - int^k, $uni - {int}_{s}^{t}$ and int^m - intⁿ which improved and extended uni - int decision making method. Han and Geng [20] developed a pruning method which has higher efficiency in computing the optimal solutions of int^m - intⁿ decision making scheme. Zou and Xiao [47] argued that there may be unknown, missing or nonexistent data in the process of collecting data. Therefore, standard soft sets under incomplete information must be taken into consideration, which calls for the check of incomplete soft sets. Later on, Han et al. [21] and even Qin et al. [35] introduced other interesting approaches to incomplete soft set based on decision making. However, in some cases in which there is uncertainty about the real value of the missing data, a decision-making procedure that avoids estimations can be applied. Relatedly, Alcantud and Santos-García [1] proposed a new algorithmic solution applying the soft set for the decision making problem under incomplete information. Kharal [24] introduced the notions of core, support, and inversion of a soft set and used these notions to describe soft approximations granulating the soft space. He suggested a novel conjecture based on soft approximations to solve an optimum choice problem. Furthermore, many decision making methods are also constructed employing special soft sets which are newly defined or existed. Gong et al. [19] introduced bijective soft set which its each alternative has only one parameter. Also, they showed that this set can easily be used for the solutions of some decision problems. Xiao et al. [39] defined exclusive disjunctive soft set based on the bijective soft set and gave an application of this concept. Zhang [46] investigated interval soft set and the tabular representation of it in detail. He also introduced the notion of interval choice value and applied the interval soft set theory in handling the decision making problem. Fatimah et al. [15] constructed novel algorithms for decision making based the probabilistic soft set theory and dual probabilistic soft set theory. Moreover, many authors established various decision making algorithms by using the neutrosophic soft set, level soft set, interval-valued fuzzy soft set, interval valued hesitant fuzzy soft set, hybrid soft set and soft rough fuzzy set [26 , 44], Çaḡman and Enginoḡlu introduced the notion of soft matrix which is a matrix representation of the soft set and its related operations. They pioneered the soft matrix based on decision making by constructing a soft max-min decision making method. Inspired by the soft matrices, Vijayabalaji and Ramesh [37] proposed a new decision making algorithm which can be used by three decision makers and they analyzed the performance of algorithm by two experimental study. Additionally, Basu et al. [5] introduced the matrices in different types in soft set theory, and also they argued that these matrices are more functional to solve some decision problems. Atagün et al. [3] generalized products defined in [8] for soft matrices in different types and presented two monoid for these generalized products. Also, they constructed a soft distributive max-min decision making method improving the decision method proposed in [8]. Feng and Zhou [17] introduced the concepts of soft discernibility matrix and weighted soft discernibility matrix which contribute to decision making. In [22, 23], the researchers concerned with decision making of patients suspected influenza and tried to reach a solution by using soft structures. In [12, 45], the decision making methods based on the soft set and soft matrix are reviewed in detail.

Qin et al. [36] emphasized that there are two approaches for the decision making problems, namely, choice value based approach that select the optimum object and comparison score based approach that obtain the ranking order of objects. Also, the group decision making procedures are generally based on the solutions of two types of decision problems which one of them involves only one parameter set and the other involves at least two different parameter sets.

Çetkin et al. [10] initiated the concept of inverse soft set theory and brought a new perspective to the computation of the decision making problems applying the inverse soft sets instead of the soft sets. This study inspired by the inverse soft sets aims to construct a new group decision making method using matrix representation of the inverse soft set for finding the optimum object and the ranking order of objects in the decision problems which involve one or more parameter sets. The contributions of this study are listed below:

By introducing the cardinality inverse soft matrix and related operations, it improves the theory of inverse soft set. Thus, it gives a new perspective for the notions of soft set and inverse soft set.

By presenting the soft sum-row decision making model, it suggests a new approach for the decision problems that are frequently encountered in daily life. This decision making model is very practical and gives fast result since it uses the matrices. While most of the existing soft decision making models only find the optimum object, this model finds both the optimum object and the ranking order of objects. Also, we can collect assessments under same or different criteria from many decision makers in this model. So, it is not only effective for different perspective evaluation, but also effective for group decision making involving multicriteria.

The rest of this study is organized as follows. Second 2 gives some basic principles of soft set theory and inverse soft set theory. Section 3 introduces the notion of cardinality inverse soft matrix which is a matrix representation of the inverse soft set and its operations. Section 4 is devoted four products of cardinality inverse soft matrix and their algebraic structures. Section 5 presents a novel decision making method called soft sum-row decision making method, and its Scilab algorithm which makes computations faster and easier and also provides convenience to solve the decision making problems involving many parameters, alternatives and decision makers. Section 6 gives applications of the method in four different decision making problems based on the inverse soft set. Also, its performance with previous decision methods based on the soft set, soft matrix and inverse soft set is compared. Section 7 presents some concluding comments.

2 Preliminaries

Molodtsov [29] and Çetkin et al. [10] introduced the concepts of soft set and inverse soft set respectively as follows:

Definition 2.1. [29] U = {u₁, u₂, . . . , u_n} be an initial universe set, E = {e₁, e₂, . . . , e_m} be a set of parameters, P (U) be the power set of U. The set of ordered pairs $S = {(e_{j}, F (e_{j})) : e_{j} \in E, F (e_{j}) \in P (U)}$ is called a soft set over U where F is a mapping given by F : E → P (U).

Definition 2.2. [10] U = {u₁, u₂, . . . , u_n} be an initial universe set, E = {e₁, e₂, . . . , e_m} be a set of parameters, P (E) be the power set of E. The set of ordered pairs $\bar{S} = {(u_{i}, \bar{F} (u_{i})) : u_{i} \in U, \bar{F} (u_{i}) \in P (E)}$ is called an inverse soft set (in short is-set) over U where $\bar{F}$ is a mapping given by $\bar{F} : U \to P (E)$ .

In other words, an inverse soft set over U is a collection of subsets of the parameter set E.

Notation: The set of all inverse soft sets over U is denoted by ISS (U).

Remark 2.3. Let $S$ and $\bar{S}$ be a soft set and an inverse soft set over common U, respectively. By using the mapping F : E → P (U), we can write $\bar{S} = {(u_{i}, \bar{F} (u_{i})) : \bar{F} (u_{i}) = ⋃ e_{j}$ such that u_i ∈ F (e_j)}. Similarly, by using the mapping $\bar{F} : U \to P (E)$ , we can write $S = {(e_{j}, F (e_{j})) : F (e_{j}) = ⋃ u_{i}$ such that $e_{j} \in \bar{F} (u_{i})}$ . Therefore, a soft set can be uniquely represented as an inverse soft set, vice versa.

Example 2.4. Suppose that U = {u₁, u₂, u₃, u₄} is a universal set and E = {e₁, e₂, e₃, e₄, e₅, e₆} is a set of all parameters. If F : E → P (U) and F (e₁) = {u₄}, F (e₂) = {u₁, u₃}, F (e₃) = {u₃}, F (e₅) = {u₃, u₄}, F (e₆) = U, then we write the soft set $\begin{matrix} S & = & {(e_{1}, {u_{4}}), (e_{2}, {u_{1}, u_{3}}), (e_{3}, {u_{3}}), \\ (e_{4}, \emptyset), (e_{5}, {u_{3}, u_{4}}), (e_{6}, U)} . \end{matrix}$

If $\bar{F} : U \to P (E)$ and $\bar{F} (u_{1}) = {e_{2}, e_{6}}$ , $\bar{F} (u_{2}) = {e_{6}}$ , $\bar{F} (u_{3}) = {e_{2}, e_{3}, e_{5}, e_{6}}$ , $\bar{F} (u_{4}) = {e_{1}, e_{5}, e_{6}}$ , then we write the inverse soft set $\begin{matrix} \bar{S} & = & {(u_{1}, {e_{2}, e_{6}}), (u_{2}, {e_{6}}), \\ (u_{3}, {e_{2}, e_{3}, e_{5}, e_{6}}), (u_{4}, {e_{1}, e_{5}, e_{6}})} . \end{matrix}$

Definition 2.5. [10] Let ${\bar{S}}_{1} = {(u_{i}, {\bar{F}}_{1} (u_{i})) : u_{i} \in U, {\bar{F}}_{1} (u_{i}) \in P (E)}$ , ${\bar{S}}_{2} = {(u_{i}, {\bar{F}}_{2} (u_{i})) : u_{i} \in U, {\bar{F}}_{2} (u_{i}) \in P (E)}$ be two inverse soft sets. Then,

${\bar{S}}_{1}$ is a subset of ${\bar{S}}_{2}$ , denoted by ${\bar{S}}_{1} ⊑ {\bar{S}}_{2}$ if ${\bar{F}}_{1} (u_{i}) \subseteq {\bar{F}}_{2} (u_{i})$ for each u_i ∈ U.

the intersection of ${\bar{S}}_{1}$ and ${\bar{S}}_{2}$ denoted by $⊓_{k = 1}^{2} {\bar{S}}_{k} = {\bar{S}}_{1} ⊓ {\bar{S}}_{2}$ is an inverse soft set defined by $\begin{matrix} {(u_{i}, {\bar{F}}_{1} (u_{i}) \cap {\bar{F}}_{2} (u_{i})) : u_{i} \in U, \\ {\bar{F}}_{1} (u_{i}), {\bar{F}}_{2} (u_{i}) \in P (E)} . \end{matrix}$

the union of ${\bar{S}}_{1}$ and ${\bar{S}}_{2}$ denoted by $⊔_{k = 1}^{2} {\bar{S}}_{k} = {\bar{S}}_{1} ⊔ {\bar{S}}_{2}$ is an inverse soft set defined by $\begin{matrix} {(u_{i}, {\bar{F}}_{1} (u_{i}) \cup {\bar{F}}_{2} (u_{i})) : u_{i} \in U, \\ {\bar{F}}_{1} (u_{i}), {\bar{F}}_{2} (u_{i}) \in P (E)} . \end{matrix}$

the complement of ${\bar{S}}_{1}$ denoted by ${\bar{S}}_{1}^{c}$ is an inverse soft set defined by ${(u_{i}, E \ {\bar{F}}_{1} (u_{i})) : u_{i} \in U, {\bar{F}}_{1} (u_{i}) \in P (E)} .$

Example 2.6. Assume that U = {u₁, u₂, u₃, u₄} is a universal set and E = {e₁, e₂, e₃, e₄, e₅} is a set of all parameters. Let ${\bar{F}}_{1} : U \to P (E)$ , ${\bar{F}}_{1} (u_{1}) = {e_{1}, e_{3}}$ , ${\bar{F}}_{1} (u_{2}) = {e_{3}}$ , ${\bar{F}}_{1} (u_{3}) = {e_{1}, e_{3}, e_{5}}$ , ${\bar{F}}_{1} (u_{4}) = {e_{2}, e_{4}}}$ , and also $\begin{matrix} {\bar{S}}_{1} & = & {(u_{1}, {e_{1}, e_{3}}), (u_{2}, {e_{3}}), \\ (u_{3}, {e_{1}, e_{3}, e_{5}}), (u_{4}, {e_{2}, e_{4}})} . \end{matrix}$

Let ${\bar{F}}_{2} : U \to P (E)$ , ${\bar{F}}_{2} (u_{1}) = {e_{1}, e_{3}}$ , ${\bar{F}}_{2} (u_{2}) = {e_{3}, e_{4}}$ , ${\bar{F}}_{2} (u_{3}) = {e_{1}, e_{2}, e_{3}, e_{5}}$ , ${\bar{F}}_{2} (u_{4}) = U}$ , and also $\begin{matrix} {\bar{S}}_{2} & = & {(u_{1}, {e_{1}, e_{3}}), (u_{2}, {e_{3}, e_{4}}), \\ (u_{3}, {e_{1}, e_{2}, e_{3}, e_{5}}), (u_{4}, U)} . \end{matrix}$

Then, we obtain ${\bar{S}}_{1} ⊑ {\bar{S}}_{2}$ and $⊓_{k = 1}^{2} {\bar{S}}_{k} = {\bar{S}}_{1} ⊓ {\bar{S}}_{2} = {(u_{1}, {e_{1}, e_{3}}), (u_{2}, {e_{3}}), (u_{3}, {e_{1}, e_{3}, e_{5}}), (u_{4}, {e_{2}, e_{4}})}$ , $⊔_{k = 1}^{2} {\bar{S}}_{k} = {\bar{S}}_{1} ⊔ {\bar{S}}_{2} = {(u_{1}, {e_{1}, e_{3}}), (u_{2}, {e_{3}, e_{4}}), (u_{3}, {e_{1}, e_{2}, e_{3}, e_{5}}), (u_{4}, U)}$ , ${\bar{S}}_{1}^{c} = {(u_{1}, {e_{2}, e_{4}, e_{5}}), (u_{2}, {e_{1}, e_{2}, e_{4}, e_{5}}), (u_{3}, {e_{2}, e_{4}}), (u_{4}, {e_{1}, e_{3}, e_{5}})}$ .

3 Cardinality inverse soft matrices

Let U = {u₁, u₂, . . . , u_n} be an initial universe set, E = {e₁, e₂, . . . , e_m} be a set of parameters and ℵ_E = {1, 2, . . . , |E|} where |E| is the cardinality of the set E.

Definition 3.1. Let $\bar{S}$ be an inverse soft set over U. Then, a subset of U × ℵ _E is uniquely defined by ${CR}_{\bar{S}} = {(u_{i}, p) : p \in ℵ_{E} and p \leq | \bar{F} (u_{i}) |} .$ which is called a cardinality relation form of $\bar{S}$ . The characteristic function of ${CR}_{\bar{S}}$ is written by $\begin{matrix} χ_{{CR}_{\bar{S}}} : U \times ℵ_{E} \to {0, 1}, \\ χ_{{CR}_{\bar{S}}} (u_{i}, p) = {\begin{matrix} 1; & (u_{i}, p) \in {CR}_{\bar{S}} \\ 0; & (u_{i}, p) \notin {CR}_{\bar{S}} \end{matrix} \end{matrix}$

Then, ${CR}_{\bar{S}}$ can be given as a binary form in Table 1.

If $x_{ip} = χ_{{CR}_{\bar{S}}} (u_{i}, p)$ , we can define a matrix $[x_{ip}]_{n \times m} = [\begin{matrix} x_{11} & x_{12} & . & . & . & x_{1 m} \\ x_{21} & x_{22} & . & . & . & x_{2 m} \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ x_{n 1} & x_{n 2} & . & . & . & x_{nm} \end{matrix}]$ which is called a cardinality inverse soft matrix (in short cis-matrix) of $\bar{S}$ over U.

Table 1
The tabular form of ${CR}_{\bar{S}}$ t1

${CR}_{\bar{S}}$ 1 2 . . . m

u ₁ $χ_{{CR}_{\bar{S}}} (u_{1}, 1)$ $χ_{{CR}_{\bar{S}}} (u_{1}, 2)$ . . . $χ_{{CR}_{\bar{S}}} (u_{1}, m)$

u ₂ $χ_{{CR}_{\bar{S}}} (u_{2}, 1)$ $χ_{{CR}_{\bar{S}}} (u_{2}, 2)$ . . . $χ_{{CR}_{\bar{S}}} (u_{2}, m)$

. . . . .

. . . . .

. . . . .

u _n $χ_{{CR}_{\bar{S}}} (u_{n}, 1)$ $χ_{{CR}_{\bar{S}}} (u_{n}, 2)$ . . . $χ_{{CR}_{\bar{S}}} (u_{n}, m)$

${CR}_{\bar{S}}$	1	2	.	.	.	m
u ₁	$χ_{{CR}_{\bar{S}}} (u_{1}, 1)$	$χ_{{CR}_{\bar{S}}} (u_{1}, 2)$	.	.	.	$χ_{{CR}_{\bar{S}}} (u_{1}, m)$
u ₂	$χ_{{CR}_{\bar{S}}} (u_{2}, 1)$	$χ_{{CR}_{\bar{S}}} (u_{2}, 2)$	.	.	.	$χ_{{CR}_{\bar{S}}} (u_{2}, m)$
.	.	.	.	.
.	.	.	.	.
.	.	.	.	.
u _n	$χ_{{CR}_{\bar{S}}} (u_{n}, 1)$	$χ_{{CR}_{\bar{S}}} (u_{n}, 2)$	.	.	.	$χ_{{CR}_{\bar{S}}} (u_{n}, m)$

According to the definition of n × m cis-matrix, an is-set $\bar{S}$ is uniquely characterized by the matrix [x_ip]. This means that an is-set $\bar{S}$ is formally equal to its cis-matrix [x_ip].

Notation: The set of all cis-matrices corresponding to the is-sets in ISS (U) will be denoted by CISM (U).

Example 3.2. Consider is-set $\bar{S}$ in Example 2.4. Then the cardinality relation form of $\bar{S}$ is $\begin{matrix} {CR}_{\bar{S}} & = & (u_{1}, 1), (u_{1}, 2), (u_{2}, 1), (u_{3}, 1), (u_{3}, 2), \\ (u_{3}, 3), (u_{3}, 4), (u_{4}, 1), (u_{4}, 2), (u_{4}, 3)} . \end{matrix}$

Thus, we obtain cis-matrix $[x_{ip}]_{4 \times 6} = [\begin{matrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \end{matrix}] .$

Definition 3.3. Let [x_ip] ∈ CISM_n×m. Then [x_ip] is called

a zero cis-matrix if x_ip = 0 for all i, p and it is denoted by [0].

a universal cis-matrix if x_ip = 1 for all i, p and it is denoted by [1].

Example 3.4. Assume that U = {u₁, u₂, u₃} is a universe set, E = {e₁, e₂, e₃, e₄, e₅} is a set of parameters and [x_ip] , [y_ip] ∈ CISM_3×5.

If $\bar{F} (u_{1}) = \emptyset$ , $\bar{F} (u_{2}) = \emptyset$ and $\bar{F} (u_{3}) = \emptyset$ , then [x_ip] = [0] is a zero cis-matrix given as $[0]_{3 \times 5} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] .$

If $\bar{F} (u_{1}) = E$ , $\bar{F} (u_{2}) = E$ and $\bar{F} (u_{3}) = E$ , then [y_ip] = [1] is a zero cis-matrix given as $[1]_{3 \times 5} = [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{matrix}] .$

Definition 3.5. Let [x_ip] , [y_ip] ∈ CISM_n×m. Then

[x_ip] is a cis-submatrix of [y_ip] if x_ip ≤ y_ip for all i, p and it is denoted by [x_ip] ⊑ [y_ip].

[x_ip] is a proper cis-submatrix of [y_ip] if x_ip ≤ y_ip for at least one term x_ip < y_ip for all i, p and it is denoted by [x_ip] ⊏ [y_ip].

[x_ip] and [y_ip] are a cis-equal matrices if x_ip = y_ip for all i, p and it is denoted by [x_ip] = [y_ip].

Definition 3.6. Let [x_ip] , [y_ip] ∈ CISM_n×m. Then, the cis-matrix [v_ip] is called

union of [x_ip] and [y_ip] if v_ip = max {x_ip, y_ip} for all i, p and it is denoted by [x_ip] ⊔ [y_ip].

intersection of [x_ip] and [y_ip] if v_ip = min {x_ip, y_ip} for all i, p and it is denoted by [x_ip] ⊓ [y_ip].

complement of [x_ip] if $v_{ip} = {\begin{matrix} 1; & if p \leq m - \sum_{l = 1}^{m} x_{il} \\ 0; & if p > m - \sum_{l = 1}^{m} x_{il} \end{matrix}$ for all i and it is denoted by [x_ip] ^c.

Example 3.7. Assume that $[x_{ip}]_{4 \times 3} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix}] and [y_{ip}]_{4 \times 3} = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix}] .$

Then we obtain [x_ip] ⊏ [y_ip] and $\begin{matrix} [x_{ip}] ⊔ [y_{ip}] & = & [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix}], \\ [x_{ip}] ⊓ [y_{ip}] & = & [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix}], \\ [x_{ip}]^{c} & = & [\begin{matrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}] . \end{matrix}$

Proposition 3.8. Let [x_ip] ∈ CISM_n×m. Then,

[[x_ip] ^c] ^c = [x_ip]

[0] ^c = [1]

[1] ^c = [0]

Proposition 3.9. Let [x_ip] , [y_ip] , [z_ip] ∈ CISM_n×m.

[x_ip] ⊑ [x_ip]

[0] _n×m ⊑ [x_ip]

[x_ip] ⊑ [1] _n×m

[x_ip] ⊑ [y_ip] and [y_ip] ⊑ [x_ip] ⇒ [x_ip] = [y_ip]

[x_ip] ⊑ [y_ip] and [y_ip] ⊑ [z_ip] ⇔ [x_ip] ⊑ [z_ip]

[x_ip] = [y_ip] and [y_ip] = [z_ip] ⇔ [x_ip] = [z_ip]

Proposition 3.10. Let [x_ip] , [y_ip] , [z_ip] ∈ CISM_n×m.

[x_ip] ⊔ [x_ip] = [x_ip]

[x_ip] ⊔ [0] _n×m = [x_ip]

[x_ip] ⊔ [1] _n×m = [1] _n×m

[x_ip] ⊔ [y_ip] = [y_ip] ⊔ [x_ip]

([x_ip] ⊔ [y_ip]) ⊔ [z_ip] = [x_ip] ⊔ ([y_ip] ⊔ [z_ip])

Proposition 3.11. Let [x_ip] , [y_ip] , [z_ip] ∈ CISM_n×m.

[x_ip] ⊓ [x_ip] = [x_ip]

[x_ip] ⊓ [0] _n×m = [0] _n×m

[x_ip] ⊓ [1] _n×m = [x_ip]

[x_ip] ⊓ [y_ip] = [y_ip] ⊓ [x_ip]

([x_ip] ⊓ [y_ip]) ⊓ [z_ip] = [x_ip] ⊓ ([y_ip] ⊓ [z_ip])

Proposition 3.12. Let [x_ip] , [y_ip] , [z_ip] ∈ CISM_n×m.

[x_ip] ⊔ ([y_ip] ⊓ [z_ip]) = ([x_ip] ⊔ [y_ip]) ⊓ ([x_ip] ⊔ [z_ip])

[x_ip] ⊓ ([y_ip] ⊔ [z_ip]) = ([x_ip] ⊓ [y_ip]) ⊔ ([x_ip] ⊓ [z_ip])

Proposition 3.13. Let [x_ip] , [y_ip] , [z_ip] ∈ CISM_n×m.

([x_ip] ⊔ [y_ip]) ^c = ([x_ip]) ^c ⊓ ([y_ip]) ^c

([x_ip] ⊓ [y_ip]) ^c = ([x_ip]) ^c ⊔ ([y_ip]) ^c

Proof. i) For all i, $\begin{matrix} ([x_{ip}] ⊔ [y_{ip}])^{c} = [\max {x_{ip}, y_{ip}}]^{c} \\ = [{\begin{matrix} 1; & if p \leq m - \sum_{l = 1}^{m} \max {x_{il}, y_{il}} \\ 0; & if p > m - \sum_{l = 1}^{m} \max {x_{il}, y_{il}} \end{matrix}] \\ = \min \begin{matrix} [{\begin{matrix} 1; & if p \leq m - \sum_{l = 1}^{m} x_{il} \\ 0; & if p > m - \sum_{l = 1}^{m} x_{il} \end{matrix} \\ {\begin{matrix} 1; & if p \leq m - \sum_{l = 1}^{m} y_{il} \\ 0; & if p > m - \sum_{l = 1}^{m} y_{il} \end{matrix}] \end{matrix} \\ = ([x_{ip}])^{c} ⊓ ([y_{ip}])^{c} \end{matrix}$

The part (ii) can be proved similarly. □

Example 3.14. Let U = {u₁, u₂, u₃}, E₁ = {e₁, e₂, e₃, e₄, e₅} and also the is-sets be $\begin{matrix} {\bar{S}}_{1} & = & {(u_{i}, {\bar{F}}_{1} (u_{i})) : u_{i} \in U, {\bar{F}}_{1} (u_{i}) \in P (E_{1})} \\ = & {(u_{1}, {e_{1}, e_{2}, e_{4}, e_{5}}), (u_{2}, {e_{1}, e_{4}}), \\ (u_{3}, {e_{2}, e_{4}})}, \\ {\bar{S}}_{2} & = & {(u_{i}, {\bar{F}}_{2} (u_{i})) : u_{i} \in U, {\bar{F}}_{2} (u_{i}) \in P (E_{1})} \\ = & {(u_{1}, {e_{1}, e_{4}}), (u_{2}, {e_{4}}), (u_{3}, E_{1})} . \end{matrix}$

Then, the cis-matrices corresponding to the is-sets ${\bar{S}}_{1}$ and ${\bar{S}}_{2}$ , respectively are $\begin{matrix} [x_{ip}]_{3 \times 5} & = & [\begin{matrix} 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \end{matrix}] and \\ [y_{ip}]_{3 \times 5} & = & [\begin{matrix} 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \end{matrix}] . \end{matrix}$

It is easily seen that $\begin{matrix} ([x_{ip}] ⊔ [y_{ip}])^{c} & = & [\begin{matrix} 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ = & ([x_{ip}])^{c} ⊓ ([y_{ip}])^{c} . \end{matrix}$

4 Products of cardinality inverse soft matrices

In this section, we introduce four special products of the cis-matrices to construct a novel decision making method. We emphasize that these cis-matrices can be of the same type or different types. Here, we give products of the cis-matrices of the different types. If it is taken m₂ = m₁, then these products are also valid for the cis-matrices of the same type.

Definition 4.1. and Let [x_ip] ∈ CISM_n×m₁ and [y_ir] ∈ CISM_n×m₂. Then, And-product of [x_ip] and [y_ir] denoted by ⋏ is defined by

$\begin{matrix} λ : C I S M_{n \times m 1} \times C I S M_{n \times m 2} \to C I S M_{n \times m 1 m 2} \\ [x_{i p}], [y_{i r}] \to [x_{i p}] λ [y_{i r}] = [z_{i t}] \end{matrix}$ where z_it = min {x_ip, y_ir} such that p = γ, t = (γ - 1) m₂ + r and γ is the smallest positive integer which satisfies t ≤ γm₂.

Definition 4.2. or Let [x_ip] ∈ CISM_n×m₁ and [y_ir] ∈ CISM_n×m₂. Then, Or-product of [x_ip] and [y_ir] denoted by ⋎ is defined by

$\begin{matrix} Υ : C I S M_{n \times m 1} \times C I S M_{n \times m 2} \to C I S M_{n \times m 1 m 2} \\ [x_{i p}], [y_{i r}] \to [x_{i p}] Υ [y_{i r}] = [z_{i t}] \end{matrix}$ where z_it = max {x_ip, y_ir} such that p = γ, t = (γ - 1) m₂ + r and γ is the smallest positive integer which satisfies t ≤ γm₂.

Definition 4.3. and-not Let [x_ip] ∈ CISM_n×m₁ and [y_ir] ∈ CISM_n×m₂. Then, And-Not-product of [x_ip] and [y_ir] denoted by ƛ is defined by

$\begin{matrix} ƛ : C I S M_{n \times m 1} \times C I S M_{n \times m 2} \to C I S M_{n \times m 1 m 2} \\ [x_{i p}], [y_{i r}] \to [x_{i p}] ƛ [y_{i r}] = [z_{i t}] \end{matrix}$ where z_it = min {x_ip, 1 - y_ir} such that p = γ, t = (γ - 1) m₂ + r and γ is the smallest positive integer which satisfies t ≤ γm₂.

Definition 4.4. and-not Let [x_ip] ∈ CISM_n×m₁ and [y_ir] ∈ CISM_n×m₂. Then, Or-Not-product of [x_ip] and [y_ir] denoted by Ɣ is defined by

$\begin{matrix} Ɣ : C I S M_{n \times m 1} \times C I S M_{n \times m 2} \to C I S M_{n \times m 1 m 2} \\ [x_{i p}], [y_{i r}] \to [x_{i p}] Ɣ [y_{i r}] = [z_{i t}] \end{matrix}$ where z_it = max {x_ip, 1 - y_ir} such that p = γ, t = (γ - 1) m₂ + r and γ is the smallest positive integer which satisfies t ≤ γm₂.

Example 4.5. Consider cis-matrices [x_ip] _3×5 and [y_ip] _3×5 corresponding to the is-sets ${\bar{S}}_{1}$ and ${\bar{S}}_{2}$ given in Example 3.14. Then we obtain $[z_{it}] = [x_{ip} λ y_{ip}] = [\begin{matrix} 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] .$

Here, z₁₍₁₇₎ = min {x₁₄, y₁₂} =1 means $4 \leq | {\bar{F}}_{1} (u_{1}) |$ and $2 \leq | {\bar{F}}_{2} (u_{1}) |$ .

Similarly, z₂₂ = min {x₂₁, y₂₂} =0 means $1 {\bar{F}}_{1} (u_{2}) |$ or $2 {\bar{F}}_{2} (u_{2}) |$ .

If we take cis-matrices [x_ip] _3×5 given in Example 3.14 and the following cis-matrix [y_ir] _3×3 corresponding to the ${\bar{S}}_{2} = {(u_{i}, {\bar{F}}_{2} (u_{i})) : u_{i} \in U, {\bar{F}}_{2} (u_{i}) \in P (E_{2})} = {(u_{1}, E_{2}), (u_{2}, {e_{1}^{'}}), (u_{3}, {e_{1}^{'}, e_{2}^{'}})}$ for $E_{2} = {e_{1}^{'}, e_{2}^{'}, e_{3}^{'}}$ $[y_{ir}]_{3 \times 3} = [\begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix}],$ then we obtain $\begin{matrix} [z_{it}^{'}] & = & [x_{ip} λ y_{ir}] \\ [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}$

Here, the type of cis-matrix $[z_{it}^{'}]$ is 3 × 15.

Remark 4.6. In general, the products of cis-matrices are not commutative.

Example 4.7. Let [x_ip], [y_ir] and $[z_{it}^{'}]$ given in Example 4.5. Then, we obtain $\begin{matrix} [z_{is}^{″}] & = & [y_{ir} λ x_{ip}] \\ [\begin{matrix} 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}$

From here, it appears that the types of cis-matrices $[z_{it}^{'}]$ and $[z_{is}^{″}]$ are same, but $[z_{it}^{'}] \neq [z_{is}^{″}]$ .

Theorem 4.8.Let [x_ip] ∈ CISM_n×m₁, [y_ir] ∈ CISM_n×m₂ and [z_is] ∈ CISM_n×m₃. Then,

([x_ip] ⋏ [y_ir]) ⋏ [z_is] = [x_ip] ⋏ ([y_ir] ⋏ [z_is])

([x_ip] ⋎ [y_ir]) ⋎ [z_is] = [x_ip] ⋎ ([y_ir] ⋎ [z_is])

Proof. i) Let [x_ip] ∈ CISM_n×m₁, [y_ir] ∈ CISM_n×m₂ and [z_is] ∈ CISM_n×m₃. From Definition 4.1, we can write [x_ip] ⋏ [y_ir] = [v_iq] ∈ CISM_n×m₁m₂ where v_iq = min {x_ip, y_ir} such that p = γ₁, q = (γ₁ - 1) m₂ + r and γ₁ is the smallest positive integer which satisfies q ≤ γ₁m₂. Likewise, we can write [v_iq] ⋏ [z_is] = [w_it] ∈ CISM_{n×m₁m₂m₃} where w_it = min {v_iq, z_is} such that q = γ₂, t = (γ₂ - 1) m₃ + s and γ₂ is the smallest positive integer which satisfies t ≤ γ₂m₃. Then, we obtain

$\begin{matrix} r & = & (γ_{2} - 1) m_{3} + s \\ = & (q - 1) m_{3} + s \\ = & ((γ_{1} - 1) m_{2} + r - 1) m_{3} + s \\ = & (γ_{1} - 1) m_{2} m_{3} + (r - 1) m_{3} + s \\ = & (p - 1) m_{2} m_{3} + (r - 1) m_{3} + s \end{matrix}$ (1)

Similarly, we can write $[y_{ir} λ z_{is}] = [v_{{iq}^{'}}^{'}] \in {CISM}_{n \times m_{2} m_{3}}$ where $v_{{iq}^{'}}^{'} = \min {y_{ir}, z_{is}}$ such that r = γ₃, q′ = (γ₃ - 1) m₃ + s and γ₃ is the smallest positive integer which satisfies q′ ≤ γ₃m₃. Likewise, we can write $[x_{ip} λ v_{{iq}^{'}}] = [w_{{it}^{'}}^{'}] \in {CISM}_{n \times m_{1} m_{2} m_{3}}$ where w_it′ = min {x_ip, v_iq′} such that p = γ₄, t′ = (γ₄ - 1) m₂m₃ + q′ and γ₄ is the smallest positive integer which satisfies t′ ≤ γ₄m₂m₃. Then, we obtain

$\begin{matrix} t & = & (γ_{4} - 1) m_{2} m_{3} + q^{'} \\ = & (γ_{4} - 1) m_{2} m_{3} + (γ_{3} - 1) m_{3} + s \\ = & (p - 1) m_{2} m_{3} + (r - 1) m_{3} + s \end{matrix}$ (2)

Since min {min {x_ip, y_ir} , z_is} = min {x_ip, min {y_ir, z_is}} and by (1) and (2), $[w_{it}] = [w_{{it}^{'}}^{'}]$ .

The part (ii) can be proved similarly. □

Proposition 4.9.

According to operation And-product, [1] _n×1 is a unit element of CISM (U).

According to operation Or-product, [0] _n×1 is a unit element of CISM (U).

Proof. i) Let [x_ip] ∈ CISM_n×m₁ be a cis-matrix and let [z_ip] = [x_ip] ⋏ [1] _n×1. Since the type of cis-matrix [z_ip] is n × m₁ and z_ip = min {x_ip, 1} = x_ip by Definition 4.1, we obtain [z_ip] = [x_ip].

The part (ii) can be proved similarly. □

Now let’s remember the definition of monoid:

A monoid is a set S together with a binary operation ∗ and an element e ∈ S such that:

(a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ S (associativity)

a ∗ e = e ∗ a for all a ∈ S (identity element)

In other words, a monoid is a set equipped with a binary operation that is associative and has an identity element.

By Theorem 4.8 and Proposition 4.9, we have the following theorem:

Theorem 4.10.

According to operation And-product, CISM (U) is a monoid.

According to operation Or-product, CISM (U) is a monoid.

5 Soft sum-row decision making

In this section, we construct a soft sum-row decision making method (in short SSRDM) by using soft sum-row decision function which is also be defined here. By using this method, one can not only choose optimum alternative from the set of given alternatives but also get the ranking order of alternatives.

Definition 5.1. (i) Let [x_ip] ∈ CISM_n×m be a cis-matrix. Then soft sum-row function denoted by $R_{s}$ is defined as follows: $\begin{matrix} R_{s} : {CISM}_{n \times m} \to ℝ^{n} \\ [x_{ip}] \to R_{s} ([x_{ip}]) = 〈 λ_{1}, λ_{2}, . . ., λ_{n} 〉 \end{matrix}$ where $λ_{i} = \sum_{p = 1}^{m} x_{ip}$ for each i ∈ {1, 2, . . . , n}.

Example 5.2. Assume that [x_ip] ∈ CISM_4×5 is given as follows $[x_{ip}]_{4 \times 5} = [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \end{matrix}] .$

Then, we obtain $R_{s} ([x_{ip}]) = 〈 5, 1, 0, 3 〉$ .

Note: Let $[x_{ip}], [x_{ip}^{'}] \in {CISM}_{n \times m}$ . Also, let $R_{s} ([x_{ip}]) = 〈 λ_{1}, λ_{2}, . . ., λ_{n} 〉$ and $R_{s} ([x_{ip}^{'}]) = 〈 λ_{1}^{'}, λ_{2}^{'}, . . ., λ_{n}^{'} 〉$ .

$R_{s} ([x_{ip}]) \pm R_{s} ([x_{ip}^{'}]) = 〈 λ_{1}, λ_{2}, . . ., λ_{n} 〉 \pm 〈 λ_{1}^{'}, λ_{2}^{'}, . . ., λ_{n}^{'} 〉 = 〈 λ_{1} \pm λ_{1}^{'}, λ_{2} \pm λ_{2}^{'}, . . ., λ_{n} \pm λ_{n}^{'} 〉$ .

$c \times R_{s} ([x_{ip}]) = c \times 〈 λ_{1}, λ_{2}, . . ., λ_{n} 〉 = 〈 c λ_{1}, c λ_{2}, . . ., c λ_{n} 〉$ for $c \in ℝ$ .

$R_{s} ([x_{ip}]) ⪯ R_{s} ([x_{ip}^{'}]) = 〈 λ_{1}, λ_{2}, . . ., λ_{n} 〉 ⪯ 〈 λ_{1}^{'}, λ_{2}^{'}, . . ., λ_{n}^{'} 〉$ means $λ_{i} \leq λ_{i}^{'}$ for all i ∈ {1, 2, . . . , n}.

Let ${\bar{S}}_{k}$ for each k = 1, 2, . . . , ℓ be an inverse soft set over U. Also, let $[x_{ip}^{1}], [x_{ip}^{2}], . . ., [x_{ip}^{k}]$ be n × m cis-matrices corresponding to the inverse soft sets ${\bar{S}}_{k}$ (k = 1, 2, . . . , ℓ), respectively.

Lemma 5.3. Let $\bar{S^{'}} = ⊓_{k = 1}^{ℓ} {\bar{S}}_{k}$ be an intersection of the inverse soft sets ${\bar{S}}_{k}$ for all k = 1, 2, . . . , ℓ. Then, $| F^{'} (u_{i}) | \leq \min {| F^{1} (u_{i}) |, | F^{2} (u_{i}) |, . . ., | F^{ℓ} (u_{i}) |}$ for all i ∈ {1, 2, . . . , n}.

Proof. Let $\bar{S^{'}} = ⊓_{k = 1}^{ℓ} {\bar{S}}_{k}$ . Then $\bar{S^{'}} ⊑ {\bar{S}}_{k}$ for each k = 1, 2, . . . , ℓ. From Definition 2.5, $F^{'} (u_{i}) \subseteq F^{k} (u_{i}) and so | F^{'} (u_{i}) | \leq | F^{k} (u_{i}) |$ for each k = 1, 2, . . . , ℓ. Hence, it is obtained that |F′ (u_i) | ≤ min {|F¹ (u_i) |, |F² (u_i) |, . . . , |F^ℓ (u_i) |} .

Lemma 5.4. Let [t_ip] be a cis-matrix corresponding to the inverse soft set $\bar{S^{'}} = ⊓_{k = 1}^{ℓ} {\bar{S}}_{k}$ . Also, let $R_{s} ([x_{ip}^{k}]) = 〈 λ_{1}^{k}, λ_{2}^{k}, . . ., λ_{n}^{k} 〉$ for each k = 1, 2, . . . , ℓ and $R_{s} ([t_{ip}]) = 〈 σ_{1}, σ_{2}, . . ., σ_{n} 〉$ . Then, $σ_{i} \leq \min {λ_{i}^{1}, λ_{i}^{2}, . . ., λ_{i}^{ℓ}}$ for all i ∈ {1, 2, . . . , n}.

Proof. Since $\bar{S^{'}} ⊑ {\bar{S}}_{k}$ for each k = 1, 2, . . . , ℓ and the is-set is formally equal to its cis-matrix, we can write $[t_{ip}] ⊑ [x_{ip}^{k}]$ for each k = 1, 2, . . . , ℓ. By Definition 3.5, we obtain $t_{ip} \leq x_{ip}^{k}$ (k = 1, 2, . . . , ℓ) for all i and p. Then, we say that $σ_{i} = \sum_{p = 1}^{m} t_{ip} \leq \sum_{p = 1}^{m} x_{ip}^{k} = λ_{i}^{k}$ for all k ∈ {1, 2, . . . , ℓ}. Hence, it is obtained that $σ_{i} \leq \min {λ_{i}^{1}, λ_{i}^{2}, . . ., λ_{i}^{ℓ}}$ for all i ∈ {1, 2, . . . , n}. □

Proposition 5.5. Let $R_{s} ([x_{ip}^{k}]) = 〈 λ_{1}^{k}, λ_{2}^{k}, . . ., λ_{n}^{k} 〉$ for each k = 1, 2, . . . , ℓ.

$R_{s} (⊓_{k = 1}^{ℓ} [x_{ip}^{k}]) = 〈 \min {λ_{1}^{1}, λ_{1}^{2}, . . ., λ_{1}^{ℓ}}, . . ., \min {λ_{k}^{1}, λ_{k}^{2}, . . ., λ_{k}^{ℓ}} 〉$

$R_{s} (⊔_{k = 1}^{ℓ} [x_{ip}^{k}]) = 〈 \max {λ_{1}^{1}, λ_{1}^{2}, . . ., λ_{1}^{ℓ}}, . . ., \max {λ_{k}^{1}, λ_{k}^{2}, . . ., λ_{k}^{ℓ}} 〉$

$R_{s} ([x_{ip}^{k}]^{c}) = 〈 m - λ_{1}^{k}, m - λ_{2}^{k}, . . ., m - λ_{n}^{k} 〉 = 〈 \underset{n}{\underset{︸}{m, m, . . ., m}} 〉 - R_{s} ([x_{ip}^{k}])$

Proof. It is obvious. □

Theorem 5.6. Let $[ν_{ip}] = ⊓_{k = 1}^{ℓ} [x_{ip}^{k}]$ and let [t_ip] be a cis-matrix corresponding to the inverse soft set $\bar{S^{'}} = ⊓_{k = 1}^{ℓ} {\bar{S}}_{k}$ . Then, $R_{s} ([t_{ip}]) ⪯ R_{s} ([ν_{ip}]) .$ for all i ∈ {1, 2, . . . , n}.

Proof. Let $R_{s} ([x_{ip}^{k}]) = 〈 λ_{1}^{k}, λ_{2}^{k}, . . ., λ_{n}^{k} 〉$ for each k = 1, 2, . . . , ℓ and $R_{s} ([t_{ip}]) = 〈 σ_{1}, σ_{2}, . . ., σ_{n} 〉$ . Also, we know that $\begin{matrix} R_{s} ([ν_{ip}]) \\ = 〈 \min {λ_{1}^{1}, λ_{1}^{2}, . . ., λ_{1}^{ℓ}}, . . ., \min {λ_{n}^{1}, λ_{n}^{2}, . . ., λ_{n}^{ℓ}} 〉 \end{matrix}$ by Proposition 5.5 and $σ_{i} \leq \min {λ_{i}^{1}, λ_{i}^{2}, . . ., λ_{i}^{ℓ}}$ for all i ∈ {1, 2, . . . , n} by Lemma 5.4. Thus, we obtain $\begin{matrix} R_{s} ([t_{ip}]) & = & 〈 σ_{1}, σ_{2}, . . ., σ_{n} 〉 ⪯ 〈 \min {λ_{1}^{1}, λ_{1}^{2}, . . ., λ_{1}^{ℓ}}, \\ . . ., \min {λ_{n}^{1}, λ_{n}^{2}, . . ., λ_{n}^{ℓ}} 〉 \\ = & R_{s} ([ν_{ip}]) . □ \end{matrix}$

Theorem 5.7. Let $R_{s} ([x_{ip}^{k}]) = 〈 λ_{1}^{k}, λ_{2}^{k}, . . ., λ_{n}^{k} 〉$ for each k = 1, 2, . . . , ℓ and let [t_ip] be a cis-matrix corresponding to the inverse soft set $\bar{S^{'}} = ⊓_{k = 1}^{ℓ} {\bar{S}}_{k}$ . If $λ_{i}^{1} + λ_{i}^{2} + . . . + λ_{i}^{ℓ} > (ℓ - 1) m$ for any i ∈ {1, 2, . . . , n}, then [t_ip] ≠ [0].

Proof. It is obvious by Lemma 5.4. □

Definition 5.8. Let U = {u₁, u₂, . . . , u_n} be the universe set. If $[x_{ip}^{1}]_{n \times m}, [x_{ip}^{2}]_{n \times m}, . . ., [x_{ip}^{ℓ}]_{n \times m}$ be cis-matrices corresponding to the inverse soft sets ${\bar{S}}_{1}, {\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ}$ generated from same parameter set, $[ν_{ip}] = ⊓_{k = 1}^{ℓ} [x_{ip}^{k}]$ , $[z_{it}] = λ k = 1 ℓ [x_{ip}^{k}]$ and $[w_{it}] = λ k = 1 ℓ [x_{ip}^{k}]^{c}$ then $\begin{matrix} D & = & 〈 α_{1}, α_{2}, . . ., α_{n} 〉 \\ = & R_{s} ([z_{it}]) - R_{s} ([w_{it}]) + ℓ \times R_{s} ([ν_{ip}]) \\ + \frac{Σ_{k = 1}^{ℓ} R_{s} ([x_{ip}^{k}]) - 〈 {\underset{︸}{(ℓ - 1) m, (ℓ - 1) m, . . ., (ℓ - 1) m}}_{n} 〉}{2} \\ + \frac{| Σ_{k = 1}^{ℓ} R_{s} ([x_{ip}^{k}]) - 〈 {\underset{︸}{(ℓ - 1) m, (ℓ - 1) m, . . ., (ℓ - 1) m}}_{n} 〉 |}{2} \end{matrix}$ is called decision vector of U.

If $[x_{ip}^{1}]_{n \times m_{1}}, [x_{ip}^{2}]_{n \times m_{2}}, . . ., [x_{ip}^{ℓ}]_{n \times m_{ℓ}}$ be cis-matrices corresponding to the inverse soft sets ${\bar{S}}_{1}, {\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ}$ generated from different parameter sets, $[z_{it}] = λ k = 1 ℓ [x ip k]$ and $[w_{it}] = λ k = 1 ℓ [x_{ip}^{k}]^{c}$ then $D = 〈 α_{1}, α_{2}, . . ., α_{n} 〉 = R_{s} ([z_{it}]) - R_{s} ([w_{it}]) .$ is called decision vector of U.

Definition 5.9. Let U = {u₁, u₂, . . . , u_n} be a universe set and $D$ be a decision vector of U. By using $D$ , a subset of U can be found as follows: ${opt}_{D} (U) = {u_{i} : u_{i} \in U and α_{i} > α_{j} for \forall j \neq i}$ which is called an optimum set of U.

Also, we can get the ranking order of alternatives as u_{i
₁} ≻ u_{i
₂} ≻ . . . ≻ u_{i
_n} if α_{i
₁} > α_{i
₂} > . . . > α_{i
_n}.

Now, we can construct a soft sum-row decision making method (SSRDM) by using the Definitions 5.1, 5.9. The algorithm and Scilab algorithm of SSRDM are given in Table 2.

Table 2
Tabular form of Algorithm

Algorithm of SSRDM Scilab Algorithm of SSRDM

Step 1: Construct the inverse soft sets ${\bar{S}}_{1}, {\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ}$ . Step 1: Construct the inverse soft sets ${\bar{S}}_{1}, {\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ} .$

Step 2: Construct the cis-matrices $[x_{ip}^{1}], [x_{ip}^{2}], . . ., [x_{ip}^{ℓ}]$ corresponding to the is-sets ${\bar{S}}_{1}$ , ${\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ}$ , respectively. Step 2:x _ 1 = input (squoenter cis - matrix1′) x _ 2 = input (squoenter cis - matrix2′) . . . x _ l = input (squoenter cis - matrixl′)

Step 3: Find intersection cis-matrix $[ν_{ip}] = ⊓_{k = 1}^{ℓ}$ . Step 3: For ⊓-operation: function v = carint (x _ 1, x _ 2) [n, m] = size (x _ 1) ; [n, m] = size (x _ 2) ; v = zeros (n, m) ; for i = 1 : nfor p = 1 : mv (i, p) = min (x _ 1 (i, p) , x _ 2 (i, p)) ; endendendfunction If the number of cis-matrices is more than two, the following code are added: function V = carintmulti (varargin) r = argn (2) ; Q = varargin (1) ; for i = 2 : rQ = carint (Q, varargin (i)) ; endV = Qendfunction

Step 4: Using And-product λ for given decision making problem, find convenient cis-matrices $[z_{it}] = λ k = 1 ℓ [x_{ip}^{k}]$ and $[w_{it}] = λ k = 1 ℓ [x_{ip}^{k}]^{c}$ . Step 4: For [z_it]: function z = carand (x _ 1, x _ 2) [n, m1] = size (x _ 1) ; [n, m2] = size (x _ 2) ; z = zeros (n, m1* m2) ; for i = 1 : nfor p = 1 : m1 for r = 1 : m2 t = (p - 1)* m2 + r ; z (i, t) = min (x _ 1 (i, p) , x _ 2 (i, r)) ; endendendendfunction If the number of cis-matrices is more than two, the following code are added: function Z = carandmulti (varargin) r = argn (2) ; X = carand (varargin (1) , varargin (2)) ; for i = 3 : rX = carand (X, varargin (i)) ; endZ = Xendfunction

For [w_it]: function w1 = carandnot (x _ 1, x _ 2) [n, m1] = size (x _ 1) ; [n, m2] = size (x _ 2) ; w1 = zeros (n, m1* m2) ; for i = 1 : nfor p = 1 : m1 for r = 1 : m2 t = (m1 - (p - 1) -1)* m2 + m2 - (r - 1) ; w1 (i, t) = min (1 - x _ 1 (i, p) , 1 - x _ 2 (i, r)) ; endendendendfunction If the number of cis-matrices is more than two, the following codes are added: function w2 = andnot (x _ 1, x _ 2) [n, m1] = size (x _ 1) ; [n, m2] = size (x _ 2) ; w = zeros (n, m1* m2) ; for i = 1 : nfor p = 1 : m1 for r = 1 : m2 t = (p - 1)* m2 + m2 - (r - 1) ; w2 (i, t) = min (x _ 1 (i, p) , 1 - x _ 2 (i, r)) ; endendendendfunction function W = carandnotdmulti (varargin) r = argn (2) ; X = carandnot (varargin (1) , varargin (2)) ; for i = 3 : rX = andnot (X, varargin (i)) ; endW = Xendfunction

Step 5: Calculate decision vector $D$ according to the constructed cis-matrices. Step 5: For SSRDM: function D = carrowsum (z, w1, v, x _ 1, x _ 2) [n, m] = size (x _ 1) ; D = sum (z, ′c′) - sum (w1, ′c′) +2 * sum (v, ′c′) +1/2 * ((sum (x₁, ′c′) + sum (x _ 2, ′c′) - (2 - 1) * m) + abs ((sum (x _ 1, ′c′) + sum (x _ 2, ′c′) - (2 - 1)* m))) ; endfunction If the number of cis-matrices is more than two, then function D = carrowsum (Z, W, V, x _ 1, x _ 2, . . . , x _ l) [n, m] = size (x _ 1) ; D = sum (Z, ′c′) - sum (W, ′c′) + l * sum (V, ′c′) +1/2 * ((sum (x _ 1, ′c′) + sum (x₂, ′c′) + . . . + sum (x_l, ′c′) - (l - 1) * m) + abs ((sum (x₁, ′c′) + sum (x₂, ′c′)+ . . . + sum (x_l, ′c′) - (l - 1) * m))) ; endfunction In this code, the value l is replaced by the number of cis-matrices given in Step 1.

Step 6: Find ${opt}_{D}$ of U and determine the ranking order of alternatives by using $D$ . Step 6: Find ${opt}_{D}$ of U and determine the ranking order of alternatives by using $D$ .

Remark: If the is-sets ${\bar{S}}_{1}, {\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ}$ generated from E₁, E₂, . . . , E_ℓ such that $\cap_{k = 1}^{ℓ} E_{k} = \emptyset$ then it is taken $D = R_{s} ([z_{it}]) - R_{s} ([w_{it}])$ in Step 5, and also Step 3 is deleted from the algorithm. Remark: In the Scilab algorithm, Step 3 is deleted and Step 5 is taken as function D = carrowsum (z, w1) D = sum (z, ′c′) - sum (w1, ′c′) ; endfunction If the number of cis-matrices is more than two, then function D = carrowsum (Z, W) D = sum (Z, ′c′) - sum (W, ′c′) ; endfunction

Algorithm of SSRDM	Scilab Algorithm of SSRDM
Step 1: Construct the inverse soft sets ${\bar{S}}_{1}, {\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ}$ .	Step 1: Construct the inverse soft sets ${\bar{S}}_{1}, {\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ} .$
Step 2: Construct the cis-matrices $[x_{ip}^{1}], [x_{ip}^{2}], . . ., [x_{ip}^{ℓ}]$ corresponding to the is-sets ${\bar{S}}_{1}$ , ${\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ}$ , respectively.	Step 2:x _ 1 = input (squoenter cis - matrix1′) x _ 2 = input (squoenter cis - matrix2′) . . . x _ l = input (squoenter cis - matrixl′)
Step 3: Find intersection cis-matrix $[ν_{ip}] = ⊓_{k = 1}^{ℓ}$ .	Step 3: For ⊓-operation: function v = carint (x _ 1, x _ 2) [n, m] = size (x _ 1) ; [n, m] = size (x _ 2) ; v = zeros (n, m) ; for i = 1 : nfor p = 1 : mv (i, p) = min (x _ 1 (i, p) , x _ 2 (i, p)) ; endendendfunction If the number of cis-matrices is more than two, the following code are added: function V = carintmulti (varargin) r = argn (2) ; Q = varargin (1) ; for i = 2 : rQ = carint (Q, varargin (i)) ; endV = Qendfunction
Step 4: Using And-product λ for given decision making problem, find convenient cis-matrices $[z_{it}] = λ k = 1 ℓ [x_{ip}^{k}]$ and $[w_{it}] = λ k = 1 ℓ [x_{ip}^{k}]^{c}$ .	Step 4: For [z_it]: function z = carand (x _ 1, x _ 2) [n, m1] = size (x _ 1) ; [n, m2] = size (x _ 2) ; z = zeros (n, m1* m2) ; for i = 1 : nfor p = 1 : m1 for r = 1 : m2 t = (p - 1)* m2 + r ; z (i, t) = min (x _ 1 (i, p) , x _ 2 (i, r)) ; endendendendfunction If the number of cis-matrices is more than two, the following code are added: function Z = carandmulti (varargin) r = argn (2) ; X = carand (varargin (1) , varargin (2)) ; for i = 3 : rX = carand (X, varargin (i)) ; endZ = Xendfunction
	For [w_it]: function w1 = carandnot (x _ 1, x _ 2) [n, m1] = size (x _ 1) ; [n, m2] = size (x _ 2) ; w1 = zeros (n, m1* m2) ; for i = 1 : nfor p = 1 : m1 for r = 1 : m2 t = (m1 - (p - 1) -1)* m2 + m2 - (r - 1) ; w1 (i, t) = min (1 - x _ 1 (i, p) , 1 - x _ 2 (i, r)) ; endendendendfunction If the number of cis-matrices is more than two, the following codes are added: function w2 = andnot (x _ 1, x _ 2) [n, m1] = size (x _ 1) ; [n, m2] = size (x _ 2) ; w = zeros (n, m1* m2) ; for i = 1 : nfor p = 1 : m1 for r = 1 : m2 t = (p - 1)* m2 + m2 - (r - 1) ; w2 (i, t) = min (x _ 1 (i, p) , 1 - x _ 2 (i, r)) ; endendendendfunction function W = carandnotdmulti (varargin) r = argn (2) ; X = carandnot (varargin (1) , varargin (2)) ; for i = 3 : rX = andnot (X, varargin (i)) ; endW = Xendfunction
Step 5: Calculate decision vector $D$ according to the constructed cis-matrices.	Step 5: For SSRDM: function D = carrowsum (z, w1, v, x _ 1, x _ 2) [n, m] = size (x _ 1) ; D = sum (z, ′c′) - sum (w1, ′c′) +2 * sum (v, ′c′) +1/2 * ((sum (x₁, ′c′) + sum (x _ 2, ′c′) - (2 - 1) * m) + abs ((sum (x _ 1, ′c′) + sum (x _ 2, ′c′) - (2 - 1)* m))) ; endfunction If the number of cis-matrices is more than two, then function D = carrowsum (Z, W, V, x _ 1, x _ 2, . . . , x _ l) [n, m] = size (x _ 1) ; D = sum (Z, ′c′) - sum (W, ′c′) + l * sum (V, ′c′) +1/2 * ((sum (x _ 1, ′c′) + sum (x₂, ′c′) + . . . + sum (x_l, ′c′) - (l - 1) * m) + abs ((sum (x₁, ′c′) + sum (x₂, ′c′)+ . . . + sum (x_l, ′c′) - (l - 1) * m))) ; endfunction In this code, the value l is replaced by the number of cis-matrices given in Step 1.
Step 6: Find ${opt}_{D}$ of U and determine the ranking order of alternatives by using $D$ .	Step 6: Find ${opt}_{D}$ of U and determine the ranking order of alternatives by using $D$ .
Remark: If the is-sets ${\bar{S}}_{1}, {\bar{S}}_{2}, . . ., {\bar{S}}_{ℓ}$ generated from E₁, E₂, . . . , E_ℓ such that $\cap_{k = 1}^{ℓ} E_{k} = \emptyset$ then it is taken $D = R_{s} ([z_{it}]) - R_{s} ([w_{it}])$ in Step 5, and also Step 3 is deleted from the algorithm.	Remark: In the Scilab algorithm, Step 3 is deleted and Step 5 is taken as function D = carrowsum (z, w1) D = sum (z, ′c′) - sum (w1, ′c′) ; endfunction If the number of cis-matrices is more than two, then function D = carrowsum (Z, W) D = sum (Z, ′c′) - sum (W, ′c′) ; endfunction

6 Applications

Now, we apply the soft sum-row decision making method for four different decision making problems which two of them involve only one parameter set and the others involve the different parameter sets.

Example 6.1. A marketing company plans to seek for the best cargo company which can be made collaboration in delivering process of products. The company appoints two decision makers for this purpose. Assume that U = {u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈} is the set of six cargo company, and E = {e₁, e₂, e₃, e₄, e₅, e₆} is a set of parameters such that these parameters are e₁= low transportation cost, e₂= safe transport of the product, e₃= carrying the product in a short time, e₄= having good profile, e₅= enough network width, e₆= good communication with the buyer.

Now, we are ready to apply the soft sum-row decision making method as follows:

Step 1: The decision makers construct the following two inverse soft sets over U under the parameters, respectively. $\begin{matrix} {\bar{S}}_{1} & = & {(u_{1}, {e_{1}, e_{2}, e_{5}}), (u_{2}, {e_{1}, e_{3}, e_{6}}), \\ (u_{3}, {e_{4}, e_{6}}), (u_{4}, {e_{1}, e_{2}, e_{3}, e_{4}, e_{6}}), \\ (u_{5}, {e_{2}, e_{3}, e_{4}, e_{5}}), (u_{6}, {e_{2}, e_{4}, e_{5}}), \\ (u_{7}, {e_{1}, e_{3}}), (u_{8}, {e_{2}, e_{3}, e_{4}, e_{6}})} . \\ {\bar{S}}_{2} & = & {(u_{1}, {e_{2}, e_{4}, e_{5}, e_{6}}), (u_{2}, E), \\ (u_{3}, {e_{1}, e_{2}, e_{4}, e_{5}}), (u_{4}, {e_{4}, e_{5}}), \\ (u_{5}, {e_{2}, e_{3}, e_{4}, e_{5}, e_{6}}), (u_{6}, {e_{3}, e_{4}, e_{6}}), \\ (u_{7}, {e_{3}, e_{5}, e_{6}}), (u_{8}, {e_{2}, e_{4}, e_{5}, e_{6}})} . \end{matrix}$

Step 2: Their cis-matrices are constructed as $\begin{matrix} = & [\begin{matrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \end{matrix}] and [x_{ip}^{2}] = [\begin{matrix} 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \end{matrix}] . \end{matrix}$

Step 3: Intersection of these cis-matrices are found as

$[ν_{ip}] = ⊓_{k = 1}^{2} [x_{ip}^{k}] = [\begin{matrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \end{matrix}] .$

Step 4: Using And-product λ, it is obtained the following cis-matrices. $\begin{matrix} [z_{it}] & = & λ k = 1 2 [x_{ip}^{k}] \\ = & [\begin{matrix} 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [w_{it}] & = & λ k = 1 2 [x_{ip}^{k}]^{c} \\ = & [\begin{matrix} 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}$

Step 5: By employing the cis-matrices $[x_{ip}^{1}], [x_{ip}^{2}], [ν_{ip}], [z_{it}]$ and [w_it], it is found the decision vector as $\begin{matrix} D & = & R_{s} ([z_{it}]) - R_{s} ([w_{it}]) + 2 \times R_{s} ([ν_{ip}]) \\ + \frac{Σ_{k = 1}^{2} R_{s} ([x_{ip}^{k}]) - 〈 6, 6, 6, 6, 6, 6, 6, 6 〉}{2} \\ + \frac{| Σ_{k = 1}^{2} R_{s} ([x_{ip}^{k}]) - 〈 6, 6, 6, 6, 6, 6, 6, 6 〉 |}{2} \\ = & 〈 12, 18, 8, 10, 20, 9, 6, 16 〉 \\ - 〈 6, 0, 8, 4, 2, 9, 12, 4 〉 \\ + 〈 6, 6, 4, 4, 8, 6, 4, 8 〉 \\ + 〈 1, 3, 0, 1, 3, 0, 0, 2 〉 \\ = & 〈 13, 27, 4, 11, 29, 6, - 2, 22 〉 \end{matrix}$

Step 6: Finally, we can find an optimum set of U as follows ${opt}_{D} = {u_{5}}$ where u₅ is optimum cargo company which can be made collaboration for delivery of products.

Also, we obtain the ranking order of alternatives as u₅ ≻ u₂ ≻ u₈ ≻ u₁ ≻ u₄ ≻ u₆ ≻ u₃ ≻ u₇.

Example 6.2. Assume that Mr. X wants to buy a car for transportation. Let U = {m₁, m₂, m₃, m₄, m₅} be the set of alternatives (the cars) and the parameters are considered while buying a car (from five cars) as follows: e₁= exterior, e₂= interior, e₃= engine condition, e₄= tires, e₅= suspension, e₆= transmission, e₇= frame, e₈= brakes, e₉= steering, e₁₀= navigation, e₁₁= maxi- mum speed, e₁₂= power windows, e₁₃= seating capacity, e₁₄= sunroof, e₁₅= age of vehicle, e₁₆= engine capacity, e₁₇= mileage traveled, e₁₈= fuel consumption, e₁₉= fuel capacity, e₂₀= reputation of manufacturing company. That’s, the set of parameters is E = {e₁, e₂, . . . , e₂₀}. Also, suppose that Mr. X has an expert group G = {T₁, T₂, . . . , T₈} consisting of 8 specialist to evaluate five cars. When Mr. X collects the assessments of experts, he can apply the soft sum-row decision making method as follows:

Step 1: The experts create the following eight inverse soft sets over U according to the parameters, respectively. $\begin{matrix} {\bar{S}}_{1} & = & {(m_{1}, {e_{1}, e_{2}, e_{4}, e_{5}, e_{6}, e_{9}, e_{11}, e_{12}, e_{15}, \\ e_{16}, e_{17}, e_{18}, e_{19}, e_{20}}), (m_{2}, {e_{1}, e_{2}, e_{4}, \\ e_{8}, e_{10}, e_{16}, e_{19}}), (m_{3}, {e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, \\ e_{7}, e_{9}, e_{14}, e_{15}, e_{16}, e_{17}, e_{19}, e_{20}}), \\ (m_{4}, {e_{2}, e_{5}, e_{6}, e_{9}, e_{12}, e_{16}, e_{20}}), (m_{5}, \\ {e_{1}, e_{2}, e_{7}, e_{8}, e_{9}, e_{11}, e_{13}, e_{15}, e_{17}, e_{18}})}, \\ {\bar{S}}_{2} & = & {(m_{1}, {e_{4}, e_{5}, e_{7}, e_{8}, e_{9}, e_{12}, e_{15}, e_{18}, \\ e_{19}, e_{20}}), (m_{2}, {e_{2}, e_{3}, e_{6}, e_{7}, e_{8}, e_{10}, \\ e_{11}, e_{13}, e_{19}}), (m_{3}, {e_{1}, e_{3}, e_{4}, e_{5}, e_{8}, e_{9}, \\ e_{11}, e_{14}, e_{16}, e_{18}, e_{19}, e_{20}}), (m_{4}, {e_{3}, e_{5}, \\ e_{9}, e_{12}, e_{19}}), (m_{5}, {e_{2}, e_{3}, e_{4}, e_{6}, e_{9}, e_{11}, \\ e_{13}, e_{15}, e_{16}, e_{17}, e_{19}, e_{20}})}, \\ {\bar{S}}_{3} & = & {(m_{1}, {e_{1}, e_{3}, e_{4}, e_{6}, e_{7}, e_{8}, e_{9}, e_{11}, e_{12}, \\ e_{13}, e_{15}, e_{16}, e_{18}, e_{19}, e_{20}}), (m_{2}, {e_{1}, e_{4}, \\ e_{6}, e_{7}, e_{9}, e_{10}, e_{11}, e_{15}, e_{16}, e_{17}, e_{18}}), \\ (m_{3}, {e_{1}, e_{2}, e_{4}, e_{5}, e_{7}, e_{8}, e_{9}, e_{11}, e_{14}, \\ e_{16}, e_{17}, e_{18}, e_{19}}), (m_{4}, {e_{2}, e_{6}, e_{7}, e_{11}, \\ e_{12}, e_{16}}), (m_{5}, {e_{1}, e_{3}, e_{5}, e_{6}, e_{11}, e_{15}, \\ e_{16}, e_{19}, e_{20}})}, \\ {\bar{S}}_{4} & = & {(m_{1}, {e_{1}, e_{2}, e_{3}, e_{5}, e_{7}, e_{8}, e_{9}, e_{11}, e_{12}, \\ e_{13}, e_{15}, e_{16}, e_{17}, e_{18}, e_{19}, e_{20}}), (m_{2}, {e_{1}, \\ e_{2}, e_{4}, e_{7}, e_{10}, e_{11}, e_{13}, e_{16}}), (m_{3}, {e_{2}, \\ e_{4}, e_{5}, e_{6}, e_{7}, e_{11}, e_{14}, e_{16}, e_{17}}), \\ (m_{4}, {e_{2}, e_{3}, e_{7}, e_{12}, e_{16}, e_{17}, e_{20}}), \\ (m_{5}, {e_{1}, e_{2}, e_{3}, e_{5}, e_{7}, e_{8}, e_{9}, e_{13}, \\ e_{15}, e_{16}, e_{17}})}, \\ {\bar{S}}_{5} & = & {(m_{1}, {e_{1}, e_{2}, e_{4}, e_{5}, e_{6}, e_{11}, e_{12}, e_{13}, \\ e_{15}, e_{16}, e_{18}, e_{19}}), (m_{2}, {e_{2}, e_{4}, e_{8}, e_{9}, \\ e_{10}, e_{13}}), (m_{3}, {e_{1}, e_{2}, e_{3}, e_{5}, e_{7}, e_{8}, e_{9}, \\ e_{11}, e_{14}, e_{15}, e_{16}, e_{18}, e_{19}, e_{20}}), (m_{4}, {e_{2}, \\ e_{3}, e_{7}, e_{9}, e_{11}, e_{12}, e_{15}, e_{16}, e_{17}, e_{19}, e_{20}}), \\ (m_{5}, {e_{1}, e_{4}, e_{5}, e_{6}, e_{9}, e_{15}, e_{17}, e_{19}})}, \\ {\bar{S}}_{6} & = & {(m_{1}, {e_{1}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}, e_{9}, e_{12}, \\ e_{13}, e_{16}}), (m_{2}, {e_{1}, e_{2}, e_{4}, e_{5}, e_{7}, e_{8}, e_{9}, \\ e_{10}, e_{11}, e_{13}, e_{16}, e_{18}}), (m_{3}, {e_{1}, e_{3}, e_{4}, \\ e_{5}, e_{6}, e_{7}, e_{8}, e_{9}, e_{11}, e_{13}, e_{14}, e_{15}, e_{16}, \\ e_{17}, e_{18}, e_{19}}), (m_{4}, {e_{2}, e_{3}, e_{4}, e_{6}, e_{7}, e_{8}, \\ e_{11}, e_{12}, e_{13}, e_{15}, e_{16}, e_{17}, e_{18}, e_{19}}), \\ (m_{5}, {e_{1}, e_{2}, e_{4}, e_{5}, e_{7}, e_{8}, e_{9}, \\ e_{11}, e_{15}, e_{17}, e_{18}, e_{19}, e_{20}})}, \\ {\bar{S}}_{7} & = & {(m_{1}, {e_{1}, e_{3}, e_{4}, e_{5}, e_{6}, e_{8}, e_{9}, e_{12}, e_{13}, \\ e_{16}, e_{17}, e_{19}, e_{20}}), (m_{2}, {e_{2}, e_{3}, e_{4}, e_{6}, \\ e_{8}, e_{10}, e_{11}, e_{15}, e_{16}}), (m_{3}, {e_{1}, e_{2}, e_{4}, \\ e_{5}, e_{6}, e_{7}, e_{8}, e_{9}, e_{13}, e_{14}, e_{15}, e_{18}, e_{19}}), \\ (m_{4}, {e_{4}, e_{5}, e_{6}, e_{7}, e_{8}, e_{9}, e_{12}, e_{13}, e_{19}}), \\ (m_{5}, {e_{2}, e_{3}, e_{4}, e_{6}, e_{7}, e_{8}, e_{9}, e_{11}, e_{15}, \\ e_{17}, e_{18}, e_{19}})}, \\ {\bar{S}}_{8} & = & {(m_{1}, {e_{1}, e_{2}, e_{4}, e_{5}, e_{9}, e_{11}, e_{12}, e_{13}, e_{15}, \\ e_{17}, e_{18}, e_{19}, e_{20}}), (m_{2}, {e_{2}, e_{3}, e_{4}, e_{7}, e_{8}, \\ e_{10}, e_{11}, e_{13}, e_{16}, e_{17}, e_{19}}), (m_{3}, {e_{1}, e_{2}, e_{3}, \\ e_{5}, e_{7}, e_{8}, e_{11}, e_{13}, e_{14}, e_{15}, e_{16}, e_{18}, e_{19}, \\ e_{20}}), (m_{4}, {e_{2}, e_{4}, e_{5}, e_{6}, e_{7}, e_{8}, e_{9}, e_{11}, \\ e_{12}, e_{15}, e_{17}, e_{18}, e_{20}}), (m_{5}, {e_{1}, e_{3}, e_{4}, e_{5}, \\ e_{8}, e_{9}, e_{11}, e_{13}, e_{17}, e_{18}})} . \end{matrix}$

Step 2: Mr. X constructs the cis-matrices for the inverse soft sets of experts as follows: $\begin{matrix} [x_{ip}^{1}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [x_{ip}^{2}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [x_{ip}^{3}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [x_{ip}^{4}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [x_{ip}^{5}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [x_{ip}^{6}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [x_{ip}^{7}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [x_{ip}^{8}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}$

Step 3: He finds intersection of these matrices as below: $\begin{matrix} [ν_{ip}] \\ = [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}$

Step 4: Using And-product λ, he obtains the cis-matrices $[z_{it}]_{5 \times 20^{8}} = λ k = 1 8 [x_{ip}^{k}]$ and $[w_{it}]_{5 \times 20^{8}} = λ k = 1 8 [x_{ip}^{k}]^{c}$ .

These cis-matrices can be obtained by using Scilab codes since the type of cis-matrices [z_it] and [ω_it] is 5 × 20⁸. Thus, he computes $R_{s} ([z_{it}])$ and $R_{s} ([w_{it}])$ .

Step 5: According to the constructed cis-matrices $[x_{ip}^{1}], [x_{ip}^{2}], [x_{ip}^{3}], [x_{ip}^{4}], [x_{ip}^{5}], [x_{ip}^{6}], [x_{ip}^{7}], [x_{ip}^{8}], [ν_{ip}], [z_{it}]$ and [w_it], he finds the decision vector as $\begin{matrix} D & = & R_{s} ([z_{it}]) - R_{s} ([w_{it}]) + 8 \times R_{s} ([ν_{ip}]) \\ + \frac{Σ_{k = 1}^{8} R_{s} ([x_{ip}^{k}]) - 〈 140, 140, 140, 140, 140 〉}{2} \\ + \frac{| Σ_{k = 1}^{8} R_{s} ([x_{ip}^{k}]) - 〈 140, 140, 140, 140, 140 〉 |}{2} \\ = & 〈 733824000, 35126784, 744097536, \\ 26486460, 148262400 〉 - 〈 4032000, \\ 186810624, 434649672, 147567420, \\ 53222400 〉 + 〈 80, 48, 72, 40, 72 〉 \\ + 〈 0, 0, 0, 0, 0, 0 〉 = 〈 729792080, \\ - 151683792, 739751112, - 121080920, \\ 95040072 〉 \end{matrix}$

Step 6: Finally, Mr. X has an optimum set of U as follows: ${opt}_{D} = {m_{3}}$ where m₃ is optimum car which can be bought for transportation.

Moreover, he obtains the ranking order of alternatives as m₃ ≻ m₁ ≻ m₅ ≻ m₄ ≻ m₂.

Example 6.3.Suppose that an automotive company Y wants to compare its automobile with automobiles in the same segment of six automotive companies. Thus, it aims to determine the situation of own automobile. The company appoints four decision makers for this purpose.

U = {u₁, u₂, u₃, u₄, u₅, u₆} denotes the universe set which describe the automobiles of six automotive companies. E₁, E₂, E₃ and E₄ denote parameter sets of decision makers, respectively. E₁, E₂, E₃ and E₄ describe performance, image-prestige, economy and after-sales advantage of the automobile, respectively. The parameter sets are E₁ = {a₁ = torque, a₂ = engine power}, E₂ = {b₁ = comfortable, b₂ = design - aesthetic, b₃ = safe}, E₃ = {c₁ = sale price, c₂ = fuel consumption} and E₄ = {d₁ = service facilities, d₂ = tax, d₃ = marketing}. Now, we are ready to apply soft sum-row decision making method to determine the situation of the automobile of company Y as follows:

Step 1: According to the specified parameters, the decision makers construct the following four inverse soft sets over U, respectively as follows: $\begin{matrix} {\bar{S}}_{1} & = & {(u_{1}, {a_{2}}), (u_{2}, E_{1}), (u_{3}, {a_{2}}), (u_{4}, {a_{2}}), \\ (u_{5}, {a_{1}}), (u_{6}, {a_{2}})}, \\ {\bar{S}}_{2} & = & {(u_{1}, {b_{1}, b_{2}}), (u_{2}, {b_{3}}), (u_{4}, E_{2}), \\ (u_{5}, {b_{1}}), (u_{6}, {b_{1}, b_{3}})}, \\ {\bar{S}}_{3} & = & {(u_{1}, {c_{2}}), (u_{2}, {c_{2}}), (u_{3}, {c_{1}}), (u_{4}, {c_{2}}), \\ (u_{5}, E_{3}), (u_{6}, {c_{1}})}, \\ {\bar{S}}_{4} & = & {(u_{1}, {d_{1}}), (u_{2}, {d_{1}, d_{2}}), (u_{3}, {d_{1}, d_{2}}), \\ (u_{4}, {d_{3}}), (u_{5}, {d_{3}}), (u_{6}, {d_{1}, d_{2}})} . \end{matrix}$

In the inverse soft set ${\bar{S}}_{1}$ , the first decision maker determine that u₁ has higher engine power than the automobile of company Y but the automobile of company Y has higher torque than u₁. The others are similar.

Step 2: Their cis-matrices are constructed as $\begin{matrix} [x_{{ip}_{1}}^{1}] & = & [\begin{matrix} 1 & 0 \\ 1 & 1 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \end{matrix}], [x_{{ip}_{2}}^{2}] = [\begin{matrix} 1 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix}], \\ [x_{{ip}_{3}}^{3}] & = & [\begin{matrix} 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 1 \\ 1 & 0 \end{matrix}] and [x_{{ip}_{4}}^{4}] = [\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{matrix}] . \end{matrix}$

Step 3: Using And-product λ, the cis-matrices are obtained $\begin{matrix} [z_{it}] & = & λ k = 1 4 [x_{{ip}_{k}}^{k}] \\ = & [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ [w_{it}] & = & λ k = 1 4 [x_{{ip}_{k}}^{k}]^{c} \\ = & [\begin{matrix} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] . \end{matrix}$

Step 4: It is obtained the decision vector by using the cis-matrices [z_it] and [w_it] as below: $D = R_{s} ([z_{it}]) - R_{s} ([w_{it}]) = 〈 0, 4, - 3, 3, 2, 3 〉 .$

Step 5: Finally, it can be found the ranking order of automobiles as u₂ ≻ u₄ = u₆ ≻ u₅ ≻ automobile of the company Y = u₁ ≻ u₃ .

Example 6.4.A retail company Z has a retail stores in major cities of Serbian such as Beograd, Čačak, Šabac, Niš, Valjevo, Obrenovac and Zaječar. In order to support further expansion on the market, the company aims to set up a distribution center (DC) from which retail stores will be supplied. With doing so, preselected locations for setting up a DC are: Šimanovci, Novi Sad, Novi Beograd, Niš, (denoted by l₁, l₂, l₃, l₄, respectively). Each of these locations has some advantages and disadvantages. The company agrees that they should be all taken into consideration since the locations have many different and specific attributes. The company determines criteria (attributes) that should be used to evaluate the locations as follows:

E₁ and E₂ are sets of criteria, and describe quantitative (cost) factors and qualitative factors, respectively. Tables 3 and 4 present the sub-criteria of the quantitative (cost) factors and qualitative factors.

Now, we are ready to apply soft sum-row decision making method to find the optimum location where the company can set up a distribution center (DC) as follows:

Step 1: The company obtains the following six inverse soft sets after it evaluates the locations according to each of the sets of sub-criteria. $\begin{matrix} {\bar{S}}_{1} & = & {(l_{1}, {a_{11}, a_{13}}), (l_{2}, {a_{11}, a_{12}}), \\ (l_{3}, {a_{12}, a_{13}}), (l_{4}, A_{1})}, \\ {\bar{S}}_{2} & = & {(l_{1}, {a_{21}, a_{22}}), (l_{2}, {a_{22}}), \\ (l_{3}, {a_{21}, a_{22}}), (l_{4}, {a_{21}})}, \\ {\bar{S}}_{3} & = & {(l_{1}, A_{3}), (l_{2}, {a_{31}}), (l_{3}, {a_{32}}), (l_{4}, A_{3})}, \\ {\bar{S}}_{4} & = & {(l_{1}, {b_{11}, b_{13}, b_{14}, b_{15}}), (l_{2}, {b_{14}, b_{15}}), \\ (l_{3}, {b_{11}, b_{13}, b_{14}, b_{15}}), (l_{4}, {b_{11}, b_{14}})}, \\ {\bar{S}}_{5} & = & {(l_{1}, {b_{22}, b_{25}, b_{26}}), (l_{2}, {b_{21}, b_{24}, b_{25}, b_{26}}), \\ (l_{3}, {b_{21}, b_{22}, b_{24}, b_{26}}), (l_{4}, {b_{21}, b_{24}, b_{25}})}, \\ {\bar{S}}_{6} & = & {(l_{1}, {b_{32}, b_{34}}), (l_{2}, {b_{32}, b_{33}}), \\ (l_{3}, {b_{31}, b_{32}, b_{35}}), (l_{4}, {b_{31}, b_{32}, b_{33}, b_{35}})} . \end{matrix}$

Step 2: We find the cis-matrices of inverse soft sets, respectively as follows: $\begin{matrix} = & [\begin{matrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{matrix}], [x_{{ip}_{2}}^{2}] = [\begin{matrix} 1 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}], \\ [x_{{ip}_{3}}^{3}] & = & [\begin{matrix} 1 & 1 \\ 1 & 0 \\ 1 & 0 \\ 1 & 1 \end{matrix}], [x_{{ip}_{4}}^{4}] = [\begin{matrix} 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 \end{matrix}], \\ [x_{{ip}_{5}}^{5}] & = & [\begin{matrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \end{matrix}], \\ [x_{{ip}_{6}}^{6}] & = & [\begin{matrix} 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 \end{matrix}] . \end{matrix}$

Step 3: Using And-product λ, we obtain the cis-matrices $[z_{it}]_{4 \times 2700} = λ k = 1 6 [x_{{ip}_{k}}^{k}]$ and $[w_{it}]_{4 \times 2700} = λ k = 1 6 [x_{{ip}_{k}}^{k}]^{c}$ .

Since the type of cis-matrices [z_it] and [ω_it] is 4 × 2700 we obtain $R_{s} ([z_{it}]) = 〈 192, 32, 192, 144 〉$ and $R_{s} ([w_{it}]) = 〈 0, 36, 4, 0 〉$ by using Scilab codes.

Step 4: Then, we compute the decision vector as $D = R_{s} ([z_{it}]) - R_{s} ([w_{it}]) = 〈 192, - 4, 188, 144 〉 .$

Step 5: Finally, we find an optimum location for distribution center (DC) as follows: ${opt}_{D} = {l_{1}} .$

Additionally, we find the ranking order of locations as l₁ ≻ l₃ ≻ l₄ ≻ l₂.

Comment: Existing soft decision making methods in literature are insufficient to solve the decision problems that involve multicriteria or a group of decision makers. With the existing soft decision making methods, it is not possible to solve the decision making problems in Examples 6.2, 6.3 and 6.4. The SSRDM using the inverse soft matrix solves such decision making problems since there are no limitations on the numbers of criteria, alternatives and decision makers. On the other hand, the decision problems addressed in Examples 6.3 and 6.4 are frequently encountered in many areas. In order to solve this kind of decision problems, many studies have been made with various mathematical tools such as fuzzy set, rough set, vague set [11 , 43]. However, these decision problems containing different parameter sets have not been addressed using the soft sets and inverse soft sets up to now. Our decision making method (SSRDM) is especially useful and convenient to handle such decision making problems.

Comparison: The decision problems, which can be solved by decision making methods constructed by using the soft sets and soft matrices, can be also solved by our decision making method (SSRDM) by transforming the soft sets into the inverse soft sets. Moreover, our method can give more accurate results for some of the decision making problems. It can be clearly seen in Table 5.

Table 3
Sub-criteria of the quantitative (cost) factors 5

E₁ = Quantitative (Cost) Factors

A₁ = Investment costs A₂ = Operating costs A₃ = Transportation costs

a₁₁ = construction cost a₂₁ = handling cost a₃₁ = total inbound

a₁₂ = equipment cost a₂₂ = storage cost a₃₂ = total outbound

a₁₃ = land acquisition a₂₃ = administrative expense

E₁ = Quantitative (Cost) Factors
A₁ = Investment costs	A₂ = Operating costs	A₃ = Transportation costs
a₁₁ = construction cost	a₂₁ = handling cost	a₃₁ = total inbound
a₁₂ = equipment cost	a₂₂ = storage cost	a₃₂ = total outbound
a₁₃ = land acquisition	a₂₃ = administrative expense

Table 4

Sub-criteria of the qualitative factors 6

E₂=Qualitative Factors
B₁ = Supply chain and logistics factors	B₂ = Strategic factors	B₃ = Other factors
b₁₁ = optimizing supply chain	b₂₁ = competitive environment	b₃₁ = human resources availability
b₁₂ = impacts on supply chain robustness	b₂₂ = development of new demand	b₃₂ = safety and security
b₁₃ = impact on supply chain responsiveness	b₂₃ = alignment of supply chain	b₃₃ = community attractiveness
b₁₄ = traffic and transportation	b₂₄ = product placement	b₃₄ = infrastructure availability
b₁₅ = capability to expand warehouse capacity	b₂₅ = better procurement	b₃₅ = impact on company visibility
b₂₆ = economic attractiveness of location

Table 5

Comparison of SSRDM with the existing soft decision making methods in the literature 7

Paper	Problem in the paper	Result of decision	Result of the SSRDM for same problem
method in the paper
Çaḡman et al. [9]	Example 4 in [9]	u₄, u₁₃, u₂₁, u₃₆, u₄₂	u₁₃ = u₃₆ ≻ u₂₁ = u₄₂ ≻ u₂₈ ≻ u₄
Feng et al. [16]	Example 5.19 in [16]	u₁₃, u₃₆	u₁₃ = u₃₆ ≻ u₂₁ = u₄₂ ≻ u₂₈ ≻ u₄
Han et al. [20]	Example 3.3 in [20]	h₁, h₂, h₃	h₁ = h₂ = h₃ ≻ h₄ = h₅
Kharal [24]	Example 31 in [24]	u ₃	u₃ ≻ u₂ ≻ u₄ ≻ u₅ ≻ u₁
Eraslan [14]	Application in [14]	u₁ > u₃ > u₂	u₁ ≻ u₃ ≻ u₂
Vijayabalaji et al. [37]	Example 3.12 in [37]	m ₃	m₃ ≻ m₁ ≻ m₂ = m₅ ≻ m₄
Basu et al. [5]	Example 5.1 in [5]	d₂, d₃	d₃ ≻ d₂ ≻ d₁ ≻ d₄
Atagün et al. [3]	Example 5.1 in [3]	u₁, u₃	u₃ ≻ u₁ ≻ u₄ ≻ u₅ ≻ u₂

Additionally, for Example 3.1 in [10], the result of their decision making method based on the inverse soft set theory is {x₄, x₁₃, x₂₁, x₂₈, x₃₆, x₄₂}. The result of our decision making method (SSRDM) is also x₁₃ = x₃₆ ≻ x₂₁ = x₄₂ ≻ x₂₈ ≻ x₄.

7 Conclusion

In this paper, we introduced the concept of cardinality inverse soft matrix which is a matrix representation of the inverse soft set defined in [10]. We also proposed a novel decision making method based on this concept. We emphasized that there is no limitation of decision makers in our method while the existing decision making methods based on the soft set theory, soft matrix theory and inverse soft set theory can generally choose the optimum object of two or three decision makers. In addition, the method focuses on not only finding the optimum object but also getting the ranking order of objects. Furthermore, we compared the results of our method with those of previous decision making methods based on the soft set theory and soft matrix theory, and then we demonstrated that our results will be more convincing than others. We analyzed the steps of our algorithm by four examples and obtained the optimum object and the ranking order of objects. Also, we pointed out that the steps of algorithm can be easily transferred to computer via Scilab codes. We hope that the cardinality inverse soft matrices will become a focus of attention as a novel notion to deal with the decision making problems involving multicriteria and a group of decision makers in the near future.

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