The row-products of inverse soft matrices in multicriteria decision making

Abstract

In this paper, we first describe the inverse soft matrices corresponding to the inverse soft sets over both the universal set and the parameter set. For these matrices, we derive some row-products such as And, Or and some column-products such as If, Iff. As an endeavor to find the authentic-life applications of these products in multicriteria decision making (MCDM) through inverse soft matrices, we propose two decision making methods called inverse soft distributive sum-max decision making method (ISDSMDM) and inverse soft distributive If-sum decision making method (ISDISDM). The ISDSMDM aims to obtain a solution for the MCDM problem based on the inverse soft structures consisting of multi-disjoint parameter sets and the common universal set. The ISDISDM focuses on the optimal choice for MCDM problem based on the inverse soft structures consisting of multi-disjoint parameter sets and two discrete universal sets. Also, we present the outstanding applications of these methods. In integration, we give Scilab codes of our algorithms to perform the decision making process expeditiously.

Keywords

Soft set inverse soft set inverse soft matrix,products of inverse soft matrices multicriteria decision making

1 Introduction

In 1999, Molodtsov [22] first introduced the notion of a soft set as a generalization of classical set theory. In [1 , 21], the authors studied their fundamental operations such as union, intersection, and complement with the congruous examples and properties. Thereafter, several authors developed the theory of soft sets. It becomes a sublime mathematical implement to resolve several issues with uncertainty in decision making.

In 2010, Çagman and Enginoglu [5] introduced the soft matrices as matrix representations of the soft sets over the common universal set. It is an important pioneer providing the different perspectives to the soft set theory. Moreover, they derived the fundamental set-theoretic operations such as And, Or, And-Not, Or-Not products for the same types of soft matrices. Through these operations, the soft max-min decision making method was developed to solve the decision problems containing ambiguous data.

Note that the products of different types of soft matrices weren’t potential until recent years. In 2018, Atagün et al. [3] generalized the soft matrix products defined in [5] for the reduced soft matrices over the common universal set. These generalizations allow us to multiply two different kinds of soft matrices at an equivalent time. Therefore, it’s terribly simple to find the solutions for the decision making problems that have more than two decision makers. Hence, they improved a soft distributive max-min decision making method. In the same year, Kamacı et al. [16] introduced the notions of row-products of the soft matrices and studied their algebraic properties. So as to point out that the utilization of row-products in decision making issues, they constructed two decision making methods which are referred to as the soft max-row and the multi-soft distributive max-min. One can utilize these methods to obtain an optimum choice in the decision making process. Additionally, they evidenced that the first method will be acclimated to solve the decision problems mentioned in [2, 8]. The second method will be confirmed to solve the decision making problems mentioned in [3]. By the above utilizable methods, they demonstrated that the row-products of soft matrices are going to be used to analyze the decision making scenario involving the multi-disjoint universal sets. In addition, many decision making algorithms were constructed using the operations of soft matrices [14 , 23].

In 2016, Çetkin et al. [7] introduced the inverse soft sets on the common universal set in lieu of soft sets on the common universal set. In 2019, Khalil and Hassan [18] put forward the idea of inverse soft sets on the common parameter set. Through these sets, they proposed new approaches to handling decision making problems. In recent years, the multicriteria decision making and multicriteria group decision making problems in real-world attracted attention and the solutions of them were researched by using various mathematical tools. Relatedly, several studies related to the multicriteria decision making [9, 20] and multicriteria group decision making [10 , 24–29] were published. Kamacı et al. [17] engaged in multicriteria group decision making based on the inverse soft sets defined in [7].

This study aims to introduce the inverse soft matrices which are matrix representations of inverse soft sets defined in [7, 18]. Also, it focuses on the products of these matrices and their applications in the MCDM. Correspondingly, we adapt the products introduced in [5] for the different types of inverse matrices and then construct the inverse soft distributive sum-max decision algorithm. Also, we create some column-products like If and Iff which have not been mentioned so far for the matrices corresponding to the (inverse) soft structures. By employing these column-products, we develop an algorithm for the MCDM problems based on the inverse soft structures consisting of multi-disjoint parameter sets and two discrete universal sets. Thus, we argue that only one object can be chosen from two discrete universal sets. As a result of our literature review, we can say that this choice is a new beginning by using inverse soft sets.

The remaining contents of the paper are organized as follows: In Section 2, the definitions of a soft set and inverse soft sets are briefly reviewed. In Section 3, the soft matrices and their congruous operations are introduced, and furthermore, some algebraic structures are investigated for these operations. In Section 4, an MCDM algorithm that determines the ranking order for the selection of objects in the universal set is proposed. This is implemented to help choose a new CEO of a company. Also, our algorithm and some existing soft decision algorithms are compared, thus the effectiveness of our algorithm is demonstrated. In Section 5, the inverse soft distributive If-sum decision making algorithm is constructed. It is applied to solve two outstanding MCDM problems that we often encounter in daily life. Conclusions are illustrated in Section 6.

2 Preliminaries

In this section, we recall some rudimentary definitions and examples for our discussions.

Definition 2.1. [22] Let U be a set of objects and E be a set of parameters. Also, let $P (U)$ denotes the power set of U. Then, the following set of ordered pairs $F_{E}^{U} = {(e_{i}, f (e_{i})) : e_{i} \in E, f (e_{i}) \in P (U)}$ is called a soft set over U, where $f : E \to P (U)$ is a set-valued mapping.

Example 2.2.Suppose that we tend to determined to shop for a mobile phone from an organization that has six range of cell phones U = {u₁, u₂, u₃, u₄}, and therefore the parameter set of cell phones is E = {e₁, e₂, e₃}, where e₁-14 cm full screen display, e₂-12MP primary camera, e₃-3300 mAh large battery. If we define $f : E \to P (U)$ such that f (e₁) = {u₁, u₃}, f (e₂) = {u₁, u₂, u₃}, f (e₃) = {u₂, u₃}, then we have the soft set $F_{E}^{U} = {(e_{1}, {u_{1}, u_{3}}), (e_{2}, {u_{1}, u_{2}, u_{3}}), (e_{3}, {u_{2}, u_{3}})} .$

Definition 2.3. [7] Let U be a set of objects and E_t be a set of parameters. Also, let $P (E_{t})$ denotes the power set of E_t. Then, the following set of ordered pairs ${\bar{F}}_{E_{t}}^{U} = {(u_{j}, \bar{f} (u_{j})) : u_{j} \in U, \bar{f} (u_{j}) \in P (E_{t})}$ is called an inverse soft set over U, where $\bar{f} : U \to P (E_{t})$ is a set-valued mapping.

Definition 2.4. [18] Let U_r be a set of objects and E be a set of parameters. Also, let $P (E)$ denotes the power set of E. Then, the following set of ordered pairs ${\bar{F}}_{U_{r}}^{E} = {(u_{j}^{r}, \bar{f} (u_{j}^{r})) : u_{j}^{r} \in U_{r}, \bar{f} (u_{j}^{r}) \in P (E)}$ is called an inverse soft set over E, where $\bar{f} : U_{r} \to P (E)$ is a set-valued mapping.

In the soft set, all objects which owned some concrete characters in the universal set were described by determining the set of objects having any parameter (attribute) e_i ∈ E. In the inverse soft set, all parameters in the universal parameter set were described by determining the set of parameter which corresponds to any object u_j ∈ U. For the soft set, we should ask: Which objects have the specified parameter? For the inverse soft set, we should ask: What are the attributes (parameters) in the universal parameter set of the specified object? This brings to light the difference between the soft set and the inverse soft set. For instance, we consider Example 2.2. If we ask: Which cell phones in the universal set U have a 12MP primary camera? then we obtain a soft set. If we ask: What are the attributes (parameters) in the universal parameter set E of the cell phone u_j? then we obtain an inverse soft set.

Even though the concepts of inverse soft sets in Definitions 2.3 and 2.4 appear to be the same, they are different. This difference can be easily seen with the following example.

Example 2.5. (i) Let $U_{1} = {u_{1}^{1}, u_{2}^{1}, u_{3}^{1}, u_{4}^{1}}$ be a set of objects. Also, let’s take the parameter sets as $E_{1} = {e_{1}^{1}, e_{2}^{1}, e_{3}^{1}}$ and $E_{2} = {e_{1}^{2}, e_{2}^{2}, e_{3}^{2}, e_{4}^{2}}$ .Then, we can write two inverse soft sets over U₁ with respect to the sets E₁ and E₂ as follows:

${\bar{F}}_{E_{1}}^{U_{1}} = {(u_{1}^{1}, {e_{1}^{1}, e_{2}^{1}}), (u_{2}^{1}, {e_{2}^{1}, e_{3}^{1}}), (u_{3}^{1}, E_{1})$ , $(u_{4}^{1}, \emptyset)}$ ,

${\bar{F}}_{E_{2}}^{U_{1}} = {(u_{1}^{1}, E_{2}), (u_{2}^{1}, {e_{1}^{2}, e_{4}^{2}}), (u_{3}^{1}, \emptyset), (u_{4}^{1}, {e_{3}^{2}})}$ .

(ii) Let $U_{1} = {u_{1}^{1}, u_{2}^{1}, u_{3}^{1}, u_{4}^{1}}$ , $U_{2} = {u_{1}^{2}, u_{2}^{2}}$ and $U_{3} = {u_{1}^{3}, u_{2}^{3}, u_{3}^{3}, u_{4}^{3}}$ be three different sets of objects. Also, let’s take the parameter set as $E_{1} = {e_{1}^{1}, e_{2}^{1}, e_{3}^{1}}$ . Then, we can write three inverse soft sets over E₁ with respect to the sets U₁, U₂ and U₃ as follows:

${\bar{F}}_{U_{1}}^{E_{1}} = {(u_{1}^{1}, {e_{1}^{1}, e_{2}^{1}}), (u_{2}^{1}, {e_{2}^{1}, e_{3}^{1}}), (u_{3}^{1}, E_{1}), (u_{4}^{1},$ ${e_{1}^{1}, e_{3}^{1}})}$ ,

${\bar{F}}_{U_{2}}^{E_{1}} = {(u_{1}^{2}, {e_{1}^{1}, e_{2}^{1}}), (u_{2}^{2}, {e_{2}^{1}, e_{3}^{1}})}$ ,

${\bar{F}}_{U_{3}}^{E_{1}} = {(u_{1}^{3}, {e_{1}^{1}, e_{2}^{1}}), (u_{2}^{3}, {e_{2}^{1}, e_{3}^{1}), (u_{3}^{3}, E_{1})$ , $(u_{4}^{3}, {e_{1}^{1}, e_{3}^{1}})}$ .

In Example 2.5 (i), it can be taken as E₁∩ E₂ = ∅ or E₁∩ E₂ ≠ ∅. Likewise, in Example 2 (ii), it can be taken as U_r∩ U_s = ∅ or U_r∩ U_s ≠ ∅ for r, s ∈ 1, 2, 3.

3 Inverse soft matrices and their operations

Until now, in many studies such as [3 , 16], the authors have dealt with the matrices corresponding to soft sets over U. Now, we introduce the inverse soft matrices corresponding to the inverse soft sets (resp. Definitions 2.3 and 2.4) over U and E. Thus, we will deal with the matrices corresponding to the soft structures defined not only on U but also on E.

Definition 3.1. Let $U_{r} = {u_{j}^{r} : j \in J}$ and $E_{t} = {e_{i}^{t} : i \in I}$ , where I and J be the index sets, then the inverse soft matrix for the inverse soft set ( ${\bar{F}}_{E_{t}}^{U_{r}}$ or ${\bar{F}}_{U_{r}}^{E_{t}}$ ) is $[a_{ij}^{(t, r)}]$ , where for all i and j $a_{ij}^{(t, r)} = {\begin{matrix} 1, if e_{i}^{t} \in \bar{f} (u_{j}^{r}) \\ 0, if e_{i}^{t} \notin \bar{f} (u_{j}^{r}) \end{matrix} .$

Then, this inverse soft matrix corresponds to both the inverse soft set over U described in [7] and the inverse soft set over E described in [18].

Example 3.2. The inverse soft matrices corresponding to the inverse soft sets ${\bar{F}}_{E_{1}}^{U_{1}}$ and ${\bar{F}}_{E_{2}}^{U_{1}}$ given in Example 2.5 (i) are respectively $[a_{ij}^{(1, 1)}] = [\begin{matrix} 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \end{matrix}] and [a_{pj}^{(2, 1)}] = [\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 \end{matrix}] .$ The inverse soft matrices corresponding to the inverse soft sets ${\bar{F}}_{U_{1}}^{E_{1}}$ , ${\bar{F}}_{U_{2}}^{E_{1}}$ and ${\bar{F}}_{U_{3}}^{E_{1}}$ given in Example 2.5 (ii) are respectively $[b_{ij}^{(1, 1)}] = [\begin{matrix} 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \end{matrix}], [b_{ik}^{(1, 2)}] = [\begin{matrix} 1 & 0 \\ 1 & 1 \\ 0 & 1 \end{matrix}] and$ $[b_{ij}^{(1, 3)}] = [\begin{matrix} 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \end{matrix}]$ Note 1. There is a one to one correspondence between the inverse soft sets and the inverse soft matrices. The set of all inverse soft matrices over the common universal set U is denoted by ISM (U). Similarly, the set of all inverse soft matrices over the common parameter set E is denoted by ISM (E).

Note 2. If |E| = m and |U| = n, then the set of all m × n inverse soft matrices over U and E is denoted by ISM_m×n.

Considering Example 3.2, it is easy to see that $[a_{ij}^{(1, 1)}], [a_{pj}^{(2, 1)}] \in ISM (U_{1})$ and $[b_{ij}^{(1, 1)}]$ , $[b_{ik}^{(1, 2)}]$ , $[b_{ij}^{(1, 3)}] \in ISM (E_{1})$ . On the other hand, it is obvious that $[a_{ij}^{(1, 1)}], [b_{ij}^{(1, 1)}], [b_{ij}^{(1, 3)}] \in {ISM}_{3 \times 4}$ , $[a_{pj}^{(2, 1)}] \in {ISM}_{4 \times 4}$ and $[b_{ik}^{(1, 2)}] \in {ISM}_{3 \times 2}$ . Then, we have $[a_{ij}^{(1, 1)}], [b_{ij}^{(1, 1)}] \in {ISM}_{3 \times 4}$ while $[a_{ij}^{(1, 1)}] \in ISM (U_{1})$ and $[b_{ij}^{(1, 1)}] \in ISM (E_{1})$ . This is expected result because these two matrices are created by using the common sets U₁ and E₁. However, it is highlighted that $[a_{ij}^{(1, 1)}], [b_{ij}^{(1, 3)}] \in {ISM}_{3 \times 4}$ while $[a_{ij}^{(1, 1)}] \in ISM (U_{1})$ and $[b_{ij}^{(1, 3)}] \in ISM (E_{1})$ .

Definition 3.3. Let $[a_{ij}^{(t_{1}, r_{1})}], [b_{ij}^{(t_{2}, r_{2})}] \in {ISM}_{m \times n}$ .

Let t₁ = t₂ = t. Then, the matrix $[a_{ij}^{(t, r_{1})}]$ is called a t-inverse soft submatrix of $[b_{ij}^{(t, r_{2})}]$ if $a_{ij}^{(t, r_{1})} \leq b_{ij}^{(t, r_{2})}$ for all i and j. It is denoted by $[a_{ij}^{(t, r_{1})}] \subseteq_{t} [b_{ij}^{(t, r_{2})}]$ .

Let r₁ = r₂ = r. Then, the matrix $[a_{ij}^{(t_{1}, r)}]$ is called an r-inverse soft submatrix of $[b_{ij}^{(t_{2}, r)}]$ if $a_{ij}^{(t_{1}, r)} \leq b_{ij}^{(t_{2}, r)}$ for all i and j. It is denoted by $[a_{ij}^{(t_{1}, r)}] \subseteq_{r} [b_{ij}^{(t_{2}, r)}]$ .

Let t₁ = t₂ = t and r₁ = r₂ = r. Then, the matrix $[a_{ij}^{(t, r)}]$ is called an inverse soft submatrix of $[b_{ij}^{(t, r)}]$ if $a_{ij}^{(t, r)} \leq b_{ij}^{(t, r)}$ for all i and j. It is denoted by $[a_{ij}^{(t, r)}] \subseteq [b_{ij}^{(t, r)}]$ .

Definition 3.4. Let $[a_{ij}^{(t_{1}, r_{1})}], [b_{ij}^{(t_{2}, r_{2})}] \in {ISM}_{m \times n}$ .

Let t₁ = t₂ = t. Then, the matrices $[a_{ij}^{(t, r_{1})}]$ and $[b_{ij}^{(t, r_{2})}]$ are called t-inverse soft equal matrices if $a_{ij}^{(t, r_{1})} = b_{ij}^{(t, r_{2})}$ for all i and j. It is denoted by $[a_{ij}^{(t, r_{1})}] =_{t} [b_{ij}^{(t, r_{2})}]$ .

Let r₁ = r₂ = r. Then, the matrices $[a_{ij}^{(t_{1}, r)}]$ and $[b_{ij}^{(t_{2}, r)}]$ are called r-inverse soft equal matrices if $a_{ij}^{(t_{1}, r)} = b_{ij}^{(t_{2}, r)}$ for all i and j. It is denoted by $[a_{ij}^{(t_{1}, r)}] =_{r} [b_{ij}^{(t_{2}, r)}]$ .

Let t₁ = t₂ = t and r₁ = r₂ = r. Then, the matrices $[a_{ij}^{(t, r)}]$ and $[b_{ij}^{(t, r)}]$ are called inverse soft equal matrices if $a_{ij}^{(t, r)} = b_{ij}^{(t, r)}$ for all i and j. It is denoted by $[a_{ij}^{(t, r)}] = [b_{ij}^{(t, r)}]$ .

Definition 3.5. Let $[a_{ij}^{(t, r)}] \in {ISM}_{m \times n}$ . Then, the complement of inverse soft matrix $[a_{ij}^{(t, r)}]$ is defined as $[a_{ij}^{(t, r)}]^{'} = [c_{ij}^{(t, r)}]$ , where $c_{ij}^{(t, r)} = 1 - a_{ij}^{(t, r)}$ for all i and j.

Example 3.6. Let’s consider the inverse soft matrices given in Example 3.2. Then, we have $[a_{ij}^{(1, 1)}] \subseteq [b_{ij}^{(1, 1)}]$ , $[a_{ij}^{(1, 1)}] \subseteq_{t} [b_{ij}^{(1, 3)}]$ and $[b_{ij}^{(1, 1)}] =_{t} [b_{ij}^{(1, 3)}]$ . Also, we obtain that $[a_{ij}^{(1, 1)}]^{'} = [\begin{matrix} 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \end{matrix}], [a_{pj}^{(2, 1)}]^{'} = [\begin{matrix} 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{matrix}]$ $and [b_{ik}^{(1, 2)}]^{'} = [\begin{matrix} 0 & 1 \\ 0 & 0 \\ 1 & 0 \end{matrix}] .$ Definition 3.7. Let $[a_{ij}^{(t_{1}, r_{1})}], [b_{ij}^{(t_{2}, r_{2})}] \in {ISM}_{m \times n}$ .

Let t₁ = t₂ = t. Then, the t-intersection of inverse soft matrices $[a_{ij}^{(t, r_{1})}]$ and $[b_{ij}^{(t, r_{2})}]$ is defined as $[a_{ij}^{(t, r_{1})}] \cap_{t} [b_{ij}^{(t, r_{2})}] = [c_{ij}] \in {ISM}_{m \times n}$ , where $c_{ij} = \min {a_{ij}^{(t, r_{1})}, b_{ij}^{(t, r_{2})}}$ for all i and j.

Let r₁ = r₂ = r. Then, the r-intersection of inverse soft matrices $[a_{ij}^{(t_{1}, r)}]$ and $[b_{ij}^{(t_{2}, r)}]$ is defined as $[a_{ij}^{(t_{1}, r)}] \cap_{r} [b_{ij}^{(t_{2}, r)}] = [c_{ij}] \in {ISM}_{m \times n}$ , where $c_{ij} = \min {a_{ij}^{(t_{1}, r)}, b_{ij}^{(t_{2}, r)}}$ for all i and j.

Let t₁ = t₂ = t and r₁ = r₂ = r. Then, the intersection of inverse soft matrices $[a_{ij}^{(t, r)}]$ and $[b_{ij}^{(t, r)}]$ is defined as $[a_{ij}^{(t, r)}] \cap [b_{ij}^{(t, r)}] = [c_{ij}] \in {ISM}_{m \times n}$ , where $c_{ij} = \min {a_{ij}^{(t, r)}, b_{ij}^{(t, r)}}$ for all i and j.

Definition 3.8. Let $[a_{ij}^{(t_{1}, r_{1})}], [b_{ij}^{(t_{2}, r_{2})}] \in {ISM}_{m \times n}$ .

Let t₁ = t₂ = t. Then, the t-union of inverse soft matrices $[a_{ij}^{(t, r_{1})}]$ and $[b_{ij}^{(t, r_{2})}]$ is defined as $[a_{ij}^{(t, r_{1})}] \cup_{t} [b_{ij}^{(t, r_{2})}] = [c_{ij}] \in {ISM}_{m \times n}$ , where $c_{ij} = \max {a_{ij}^{(t, r_{1})}, b_{ij}^{(t, r_{2})}}$ for all i and j.

Let r₁ = r₂ = r. Then, the r-union of inverse soft matrices $[a_{ij}^{(t_{1}, r)}]$ and $[b_{ij}^{(t_{2}, r)}]$ is defined as $[a_{ij}^{(t_{1}, r)}] \cup_{r} [b_{ij}^{(t_{2}, r)}] = [c_{ij}] \in {ISM}_{m \times n}$ , where $c_{ij} = \max {a_{ij}^{(t_{1}, r)}, b_{ij}^{(t_{2}, r)}}$ for all i and j.

Let t₁ = t₂ = t and r₁ = r₂ = r. Then, the union of inverse soft matrices $[a_{ij}^{(t, r)}]$ and $[b_{ij}^{(t, r)}]$ is defined as $[a_{ij}^{(t, r)}] \cup [b_{ij}^{(t, r)}] = [c_{ij}] \in {ISM}_{m \times n}$ , where $c_{ij} = \max {a_{ij}^{(t, r)}, b_{ij}^{(t, r)}}$ for all i and j.

How can the t-operations of inverse soft matrices be used in real life? Let’s show this with the following example.

Example 3.9. Assume that an expert wants to compare the automobiles in the same segmentof the companies X and Y under the parameters: $e_{1}^{1} -$ performance, $e_{2}^{1} -$ aesthetic-comfort, $e_{3}^{1} -$ price, $e_{4}^{1} -$ safety. Suppose that the set of automobiles of the company X is $U_{1} = {u_{1}^{1} = automobile in A segment of X, u_{2}^{1} = automobile in B segment of X, u_{3}^{1} = automobile in C segment of X}$ and the set of automobiles of the company Y is $U_{2} = {u_{1}^{2} = automobile in A segment of Y, u_{2}^{2} = automobile in B segment of Y, u_{3}^{2} = automobile in C segment of Y}$ . Then, it is seen that U₁∩ U₂ = ∅.

According to these data, if the expert constructs the following inverse soft matrices: $[a_{ij}^{(1, 1)}] = [\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{matrix}] and [b_{ik}^{(1, 2)}] = [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{matrix}],$ then the expert asserts that the automobiles of company Y are better than automobiles of company X in all segments since $[a_{ij}^{(1, 1)}] \subseteq_{t} [b_{ik}^{(1, 2)}]$ .

If the expert constructs the following inverse soft matrices: $[a_{ij}^{(1, 1)}] = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}] and [b_{ik}^{(1, 2)}] = [\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \end{matrix}],$ then the expert obtains the t-intersection of inverse soft matrices $[a_{ij}^{(1, 1)}]$ and $[b_{ik}^{(1, 2)}]$ as follows: $[a_{ij}^{(1, 1)}] = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{matrix}] .$ Thus, the expert asserts that the automobiles in A segment of the companies X and Y have the same safety, the automobiles in B segment have the same aesthetic-comfort and the automobiles in C segment have the same performance and safety.

If the expert finds t-intersection of inverse soft matrices $[a_{ij}^{(1, 1)}]^{'}$ and $[b_{ik}^{(1, 2)}]$ , then he/she can say that the performances of automobiles in A and B segments of company Y are better than those of company X and the price of an automobile in C segment of company Y is better than that of company X.

If the expert can interpret the t-intersection of inverse soft matrices $[a_{ij}^{(1, 1)}]$ and $[b_{ik}^{(1, 2)}]^{'}$ in a similar way.

In real life, there are many congruous examples which the r-operations of inverse soft matrices can be used.

Definition 3.10. Let $[a_{pj}^{(t_{1}, r)}] \in {ISM}_{m_{1} \times n}$ and $[b_{qj}^{(t_{2}, r)}] \in {ISM}_{m_{2} \times n}$ .

The And row-product (And r-product) of inverse soft matrices $[a_{pj}^{(t_{1}, r)}]$ and $[b_{qj}^{(t_{2}, r)}]$ is defined as $[a_{pj}^{(t_{1}, r)}] \land_{r} [b_{qj}^{(t_{2}, r)}] = [c_{sj}] \in {ISM}_{m_{1} m_{2} \times n}$ , where $c_{sj} = \min {a_{pj}^{(t_{1}, r)}, b_{qj}^{(t_{2}, r)}}$ for s = (p - 1) m₂ + q.

The Or row-product (Or r-product) of inverse soft matrices $[a_{pj}^{(t_{1}, r)}]$ and $[b_{qj}^{(t_{2}, r)}]$ is defined as $[a_{pj}^{(t_{1}, r)}] \lor_{r} [b_{qj}^{(t_{2}, r)}] = [c_{sj}] \in {ISM}_{m_{1} m_{2} \times n}$ , where $c_{sj} = \max {a_{pj}^{(t_{1}, r)}, b_{qj}^{(t_{2}, r)}}$ for s = (p - 1) m₂ + q.

The And-Not row-product (And-Not r-product) of inverse soft matrices $[a_{pj}^{(t_{1}, r)}]$ and $[b_{qj}^{(t_{2}, r)}]$ is defined as $[a_{pj}^{(t_{1}, r)}] \cap_{r} [b_{qj}^{(t_{2}, r)}] = [c_{sj}] \in {ISM}_{m_{1} m_{2} \times n}$ , where $c_{sj} = \min {a_{pj}^{(t_{1}, r)}, 1 - b_{qj}^{(t_{2}, r)}}$ for s = (p - 1) m₂ + q.

The Or-Not row-product (Or-Not r-product) of inverse soft matrices $[a_{pj}^{(t_{1}, r)}]$ and $[b_{qj}^{(t_{2}, r)}]$ is defined as $[a_{pj}^{(t_{1}, r)}] ⊻_{r} [b_{qj}^{(t_{2}, r)}] = [c_{sj}] \in {ISM}_{m_{1} m_{2} \times n}$ , where $c_{sj} = \max {a_{pj}^{(t_{1}, r)}, 1 - b_{qj}^{(t_{2}, r)}}$ for s = (p - 1) m₂ + q.

Example 3.11. Let’s consider the inverse soft matrices $[a_{qj}^{(2, 1)}] \in {ISM}_{4 \times 4}$ , $[b_{ij}^{(1, 1)}] \in {ISM}_{3 \times 4}$ and $[b_{ij}^{(1, 3)}] \in {ISM}_{3 \times 4}$ given in Example 3.2. Then, we obtain that $[a_{pj}^{(2, 1)}] \land_{r} [b_{ij}^{(1, 1)}] = [\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}] and$ $[b_{ij}^{(1, 1)}] \land_{r} [a_{pj}^{(2, 1)}] = [\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{matrix}] .$ Then, we have that $[a_{pj}^{(2, 1)}] \land_{r} [b_{ij}^{(1, 1)}] \neq_{r} [b_{ij}^{(1, 1)}] \land_{r} [a_{pj}^{(2, 1)}]$ . Thus, we can say that the r-products of inverse soft matrices is not commutative.

Also, it is highlighted that the row-product (r-product) of $[a_{pj}^{(2, 1)}] \in {ISM}_{4 \times 4}$ and $[b_{ij}^{(1, 3)}] \in {ISM}_{3 \times 4}$ is not possible although $[b_{ij}^{(1, 1)}] =_{r} [b_{ij}^{(1, 3)}]$ .

Note 3. If it is taken as t₁ = t₂ = t in Definition 3.10, then the transpose of r-product (row-product) of inverse matrices are equal to the column-product of soft matrices defined by Çagman and Enginoglu [5]. Here, the transpose of inverse soft matrix is obtained by interchanging the rows and columns of the inverse soft matrix.

Note 4. The relation form of the row-products of inverse soft matrices is a subset of (E₁ × E₂, U) while the relation form of the column-products of soft matrices defined by Çagman and Enginoglu [5] is a subset of (U, E × E) and the relation form of the column-products of reduced soft matrices defined by Atagün et al. [3] is a subset of (U, A × B) where A, B ⊆ E.

Proposition 3.12.Let $[a_{pj}^{(t_{1}, r)}] \in {ISM}_{m_{1} \times n}$ and $[b_{qj}^{(t_{2}, r)}] \in {ISM}_{m_{2} \times n}$ . Then, the DeMorgan’s Laws are hold for the row-products of inverse soft matrices.

$([a_{pj}^{(t_{1}, r)}] \land_{r} [b_{qj}^{(t_{2}, r)}])^{'} = [a_{pj}^{(t_{1}, r)}]^{'} \lor_{r} [b_{qj}^{(t_{2}, r)}]^{'}$ .

$([a_{pj}^{(t_{1}, r)}] \lor_{r} [b_{qj}^{(t_{2}, r)}])^{'} = [a_{pj}^{(t_{1}, r)}]^{'} \land_{r} [b_{qj}^{(t_{2}, r)}]^{'}$ .

$([a_{pj}^{(t_{1}, r)}] \cap_{r} [b_{qj}^{(t_{2}, r)}])^{'} = [a_{pj}^{(t_{1}, r)}]^{'} ⊻_{r} [b_{qj}^{(t_{2}, r)}]^{'}$ .

$([a_{pj}^{(t_{1}, r)}] ⊻_{r} [b_{qj}^{(t_{2}, r)}])^{'} = [a_{pj}]^{'} \cap_{r} [b_{qj}^{(t_{2}, r)}]^{'}$ .

Theorem 3.13. Let $[a_{pj}^{(t_{1}, r)}] \in {ISM}_{m_{1} \times n}$ , $[b_{qj}^{(t_{2}, r)}] \in {ISM}_{m_{2} \times n}$ and $[c_{vj}^{(t_{3}, r)}] \in {ISM}_{m_{3} \times n}$ .

$([a_{pj}^{(t_{1}, r)}] \land_{r} [b_{qj}^{(t_{2}, r)}]) \land_{r} [c_{vj}^{(t_{3}, r)}] = [a_{pj}^{(t_{1}, r)}] \land_{r} ([b_{qj}^{(t_{2}, r)}] \land_{r} [c_{vj}^{(t_{3}, r)}])$ .

$([a_{pj}^{(t_{1}, r)}] \lor_{r} [b_{qj}^{(t_{2}, r)}]) \lor_{r} [c_{vj}^{(t_{3}, r)}] = [a_{pj}^{(t_{1}, r)}] \lor_{r} ([b_{qj}^{(t_{2}, r)}] \lor_{r} [c_{vj}^{(t_{3}, r)}])$

Proposition 3.14. Let

[a_{pj}^{(t_{1}, r)}] \in {ISM}_{m_{1} \times n}

$[a_{pj}^{(t_{1}, r)}] \land_{r} [1]_{1 \times n} = [a_{pj}^{(t_{1}, r)}]$ and [1] _1×n ∧ _r $[a_{pj}^{(t_{1}, r)}] = [a_{pj}^{(t_{1}, r)}]$ , where [1] _1×n represents the 1 × n inverse matrix with all components 1.

$[a_{pj}^{(t_{1}, r)}] \lor_{r} [0]_{1 \times n} = [a_{pj}^{(t_{1}, r)}]$ and [0] _1×n ∨ _r $[a_{pj}^{(t_{1}, r)}] = [a_{pj}^{(t_{1}, r)}]$ , where [0] _1×n represents the 1 × n inverse matrix with all components 0.

Theorem 3.15. The set ISM_m×n is a monoid with respect to each of And row-product ∧_r and Or row-product ∨_r.

The proofs of Propositions 3.12 and 3.14, Theorems 3.13 and 3.15 are comparable to Propositions 1 and 2, Theorems 1 and 2 in [16], respectively. Thus, we avoid them.

In [16], Kamacı et al. presented these theoretical results for the soft matrices corresponding to the tabular forms of (U₁, E) and (U₂, E). We achieve the above theoretical results for the inverse soft matrices corresponding to the tabular forms of (U, E₁) and (U, E₂). But, the proofs of the above theoretical findings are still similar to those in [16].

Now, we define right-If, left-If and Iff column-products of inverse soft matrices.

Definition 3.16. Let $[a_{ij}^{(t, r_{1})}] \in {ISM}_{m \times n_{1}}$ and $[b_{ik}^{(t, r_{2})}] \in {ISM}_{m \times n_{2}}$ .

The right-If column-product (right-If t-product) of inverse soft matrices $[a_{ij}^{(t, r_{1})}]$ and $[b_{ik}^{(t, r_{2})}]$ is defined as $[a_{ij}^{(t, r_{1})}] \Rightarrow_{t} [b_{ik}^{(t, r_{2})}] = [c_{il}] \in {ISM}_{m \times n_{1} n_{2}}$ , where $c_{il} = {\begin{matrix} 1, & if a_{ij}^{(t, r_{1})} \leq b_{ik}^{(t, r_{2})} \\ 0, & if otherwise \end{matrix}$ for l = (j - 1) n₂ + k.

The left-If column-product (left-If t-product) of inverse soft matrices $[a_{ij}^{(t, r_{1})}]$ and $[b_{ik}^{(t, r_{2})}]$ is defined as $[a_{ij}^{(t, r_{1})}] \Leftarrow_{t} [b_{ik}^{(t, r_{2})}] = [c_{il}] \in {ISM}_{m \times n_{1} n_{2}}$ , where $c_{il} = {\begin{matrix} 1, & if a_{ij}^{(t, r_{1})} \geq b_{ik}^{(t, r_{2})} \\ 0, & if otherwise \end{matrix}$ for l = (j - 1) n₂ + k.

The Iff column-product (Iff t-product) of inverse soft matrices $[a_{ij}^{(t, r_{1})}]$ and $[b_{ik}^{(t, r_{2})}]$ is defined as $[a_{ij}^{(t, r_{1})}] \Leftrightarrow_{t} [b_{ik}^{(t, r_{2})}] = [c_{il}] \in {ISM}_{m \times n_{1} n_{2}}$ , wherever $c_{il} = \min {a_{pj}^{(t_{1}, r)}, b_{qj}^{(t_{2}, r)}}$ where $c_{il} = {\begin{matrix} 1, & if a_{ij}^{(t, r_{1})} = b_{ik}^{(t, r_{2})} \\ 0, & if otherwise \end{matrix}$ for l = (j - 1) n₂ + k.

Proposition 3.17. Let $[a_{ij}^{(t, r_{1})}] \in {ISM}_{m \times n_{1}}$ and $[b_{ik}^{(t, r_{2})}] \in {ISM}_{m \times n_{2}}$ . Then, we have $([a_{ij}^{(t, r_{1})}] \Rightarrow_{t} [b_{ik}^{(t, r_{2})}]) \cap ([a_{ij}^{(t, r_{1})}] \Leftarrow_{t} [b_{ik}^{(t, r_{2})}]) = [a_{ij}^{(t, r_{1})}] \Leftrightarrow_{t} [b_{ik}^{(t, r_{2})}]$ .

Proof. Let’s take $[c_{il}] = [a_{ij}^{(t, r_{1})}] \Rightarrow_{t} [b_{ik}^{(t, r_{2})}]$ and $[d_{il}] = [a_{ij}^{(t, r_{1})}] \Leftarrow_{t} [b_{ik}^{(t, r_{2})}]$ . By Definition 3.16 (a) and (b), we write, for all i, j and k

$c_{il} = {\begin{matrix} 1, & if a_{ij}^{(t, r_{1})} \leq b_{ik}^{(t, r_{2})} \\ 0, & if a_{ij}^{(t, r_{1})} > b_{ik}^{(t, r_{2})} \end{matrix} and$ $d_{il} = {\begin{matrix} 1, & if a_{ij}^{(t, r_{1})} \geq b_{ik}^{(t, r_{2})} \\ 0, & if a_{ij}^{(t, r_{1})} < b_{ik}^{(t, r_{2})} \end{matrix} .$ Then, we have $[g_{il}] = [a_{ij}^{(t, r_{1})}] \Rightarrow_{t} [b_{ik}^{(t, r_{2})}] \cap [a_{ij}^{(t, r_{1})}] \Leftarrow_{t} [b_{ik}^{(t, r_{2})}]$ such that for all i, j and k

$g_{il} = {\begin{matrix} \min {1, 1}, & if a_{ij}^{(t, r_{1})} = b_{ik}^{(t, r_{2})} \\ \min {1, 0}, & if a_{ij}^{(t, r_{1})} < b_{ik}^{(t, r_{2})} \\ \min {0, 1}, & if a_{ij}^{(t, r_{1})} > b_{ik}^{(t, r_{2})} \end{matrix}$ $= {\begin{matrix} 1, & if a_{ij}^{(t, r_{1})} = b_{ik}^{(t, r_{2})} \\ 0, & if otherwise \end{matrix} .$ By Definition 3.16 (c), it is seen that $([a_{ij}^{(t, r_{1})}] \Rightarrow_{t}$ $[b_{ik}^{(t, r_{2})}]) \cap ([a_{ij}^{(t, r_{1})}] \Leftarrow_{t} [b_{ik}^{(t, r_{2})}]) = [a_{ij}^{(t, r_{1})}] \Leftrightarrow_{t} [b_{ik}^{(t, r_{2})}]$ .□

Example 3.18. Let’s take the following inverse soft matrices $[a_{ij}^{(1, 1)}] \in {ISM}_{4 \times 3}$ , $[b_{ik}^{(1, 2)}] \in {ISM}_{4 \times 2}$ .

$[a_{ij}^{(1, 1)}] = [\begin{matrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] and [b_{ik}^{(1, 2)}] = [\begin{matrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{matrix}],$ Then, we obtain that

$[a_{ij}^{(1, 1)}] \Rightarrow_{t} [b_{ik}^{(1, 2)}] = [\begin{matrix} 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}],$

$[a_{ij}^{(1, 1)}] \Leftarrow_{t} [b_{ik}^{(1, 2)}] = [\begin{matrix} 1 & 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{matrix}] and$

$[a_{ij}^{(1, 1)}] \Leftrightarrow_{t} [b_{ik}^{(1, 2)}] = [\begin{matrix} 1 & 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{matrix}] .$

Thus, we see that $([a_{ij}^{(1, 1)}] \Rightarrow_{t} [b_{ik}^{(1, 2)}]) \cap ([a_{ij}^{(1, 1)}] \Leftarrow_{t} [b_{ik}^{(1, 2)}]) = [a_{ij}^{(1, 1)}] \Leftrightarrow_{t} [b_{ik}^{(1, 2)}]$ .

Note that the right-If, left-If and Iff column-products of inverse soft matrices is not commutative.

Example 3.19. Lets’s consider $[a_{ij}^{(1, 1)}]$ and $[b_{ik}^{(1, 2)}]$ as in Example 3.18.

Then, we find that $[b_{ik}^{(1, 2)}] \Rightarrow_{t} [a_{ij}^{(1, 1)}] = [\begin{matrix} 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \end{matrix}]$ ,

$[b_{ik}^{(1, 2)}] \Leftarrow_{t} [a_{ij}^{(1, 1)}]$ = $[\begin{matrix} 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}]$ and

$[b_{ik}^{(1, 2)}] \Leftrightarrow_{t} [a_{ij}^{(1, 1)}]$ = $[\begin{matrix} 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \end{matrix}]$ .

Then, we say that $[a_{ij}^{(1, 1)}] \Rightarrow_{t} [b_{ik}^{(1, 2)}] \neq [b_{ik}^{(1, 2)}] \Rightarrow_{t} [a_{ij}^{(1, 1)}]$ , $[a_{ij}^{(1, 1)}] \Leftarrow_{t} [b_{ik}^{(1, 2)}] \neq [b_{ik}^{(1, 2)}] \Leftarrow_{t} [a_{ij}^{(1, 1)}]$ and $[a_{ij}^{(1, 1)}] \Leftrightarrow_{t} [b_{ik}^{(1, 2)}] \neq [b_{ik}^{(1, 2)}] \Leftrightarrow_{t} [a_{ij}^{(1, 1)}]$ .

Also, it is seen that $[a_{ij}^{(1, 1)}] \Leftarrow_{t} [b_{ik}^{(1, 2)}] \neq [b_{ik}^{(1, 2)}] \Rightarrow_{t} [a_{ij}^{(1, 1)}]$ and $([a_{ij}^{(1, 1)}] \Rightarrow_{t} [b_{ik}^{(1, 2)}]) \cap ([b_{ik}^{(1, 2)}] \Rightarrow_{t} [a_{ij}^{(1, 1)}]) \neq [a_{ij}^{(1, 1)}] \Leftrightarrow_{t} [b_{ik}^{(1, 2)}]$ .

4 The inverse soft distributive sum-max decision making method

In this section, we define the left-distributive, right-distributive and distributive sum-max decision inverse soft fuzzification matrices for the row-products (r-products) of two inverse soft matrices. Then, we conferred an algorithm to search out an optimum set of the given universe set U. Finally, we solve a real-life drawback by victimization our algorithm.

Definition 4.1. Let $[a_{pj}^{(t_{1}, r)}] \in {ISM}_{m_{1} \times n}$ and $[b_{qj}^{(t_{2}, r)}] \in {ISM}_{m_{2} \times n}$ . If $[a_{pj}^{(t_{1}, r)}] ★ [b_{qj}^{(t_{2}, r)}] = [c_{sj}] \in {ISM}_{m_{1} m_{2} \times n}$ , then the matrix [α_1j] is called a left-distributive sum-max decision inverse soft fuzzification matrix, where $α_{1 j} = \frac{1}{m_{1}} \sum_{k = 1}^{m_{1}} v_{kj}$ (0 ≤ α_1j ≤ 1) and v_kj = max {c_sj : (k - 1) m₂ + 1 ≤ s ≤ km₂} for each k = 1, 2, …, m₁.

Definition 4.2. Let $[a_{pj}^{(t_{1}, r)}] \in {ISM}_{m_{1} \times n}$ and $[b_{qj}^{(t_{2}, r)}] \in$ ISM_m₂×n. If $[b_{qj}^{(t_{2}, r)}] ★ [a_{pj}^{(t_{1}, r)}] = [d_{tj}] \in {ISM}_{m_{2} m_{1} \times n}$ .Then, the matrix [β_1j] is called a right-distributive sum-max decision inverse soft fuzzification matrix, where $β_{1 j} = \frac{1}{m_{2}} \sum_{k = 1}^{m_{2}} w_{kj}$ (0 ≤ β_1j ≤ 1) and w_kj = max {d_tj : (k - 1) m₁ + 1 ≤ t ≤ km₁} for each k = 1, 2, …, m₂.

Definition 4.3. Let [α_1j] ∈ ISM_1×n and [β_1j] ∈ ISM_1×n be the left and right-distributive sum-max decision inverse soft fuzzification matrices, respectively. Then, the matrix [γ_1j] is called a distributive sum-max decision inverse soft fuzzification matrix, where $γ_{1 j} = \frac{1}{2} (α_{1 j} + β_{1 j})$ (0 ≤ γ_1j ≤ 1).

Definition 4.4. Let $[γ_{1 j}^{1}], [γ_{1 j}^{2}], . . ., [γ_{1 j}^{ℓ}] \in {ISM}_{1 \times n}$ be the distributive sum-max decision inverse soft fuzzification matrices. Then, the matrix [θ_1j] is called an aggregate-distributive sum-max decision inverse soft fuzzification matrix, where $θ_{1 j} = \frac{1}{ℓ} \sum_{i = 1}^{ℓ} γ_{1 j}^{i}$ (0 ≤ θ_1j ≤ 1).

Definition 4.5. Let $U_{1} = {u_{1}^{1}, u_{2}^{1}, . . ., u_{n}^{1}}$ be a universal set. By using the aggregate-distributive sum-max decision inverse soft fuzzification matrix [θ_1j], we get the ranking order of objects as below: $u_{j_{1}}^{1} ≻ u_{j_{2}}^{1} ≻ . . . ≻ u_{j_{n}}^{1} if θ_{1 j_{1}} > θ_{1 j_{2}} > . . . > θ_{1 j_{n}} .$ Then, the optimum set of U is obtained as follows: ${Opt}_{[θ_{1 j}]} (U_{1}) = {u_{j}^{1} : u_{j}^{1} \in U_{1} and θ_{1 j} > θ_{1 j^{'}} for \forall j^{'} \neq j} .$

By using the above Definitions, we construct the subsequent algorithm to search out an optimum set through the inverse soft sets created by considering the universal set U₁.

Algorithm 1.

Step 1. Create the inverse soft matrices (represented as $[a_{pj}^{(t_{1}, r)}]$ , $[b_{qj}^{(t_{2}, r)}]$ and $[c_{vj}^{(t_{3}, r)}]$ ) from the given three inverse soft sets.

Step 2. Select the suitable product (symbolized as ★) based on the real scenario of the problem. It is going to be one in all And row-product, Or row-product, And-Not row-product and Or-Not row product.

Step 3. Obtain the row-products $[a_{pj}^{(t_{1}, r)}] ★ [b_{qj}^{(t_{2}, r)}]$ , $[b_{qj}^{(t_{2}, r)}] ★ [a_{pj}^{(t_{1}, r)}]$ , $[b_{qj}^{(t_{2}, r)}]$ $★ [c_{vj}^{(t_{3}, r)}]$ , $[c_{vj}^{(t_{3}, r)}] ★ [b_{qj}^{(t_{2}, r)}]$ , $[a_{pj}^{(t_{1}, r)}] ★ [c_{vj}^{(t_{3}, r)}]$ and $[c_{vj}^{(t_{3}, r)}] ★ [a_{pj}^{(t_{1}, r)}]$ .

Step 4. Find the left-distributive sum-max decision inverse soft fuzzification matrices $[α_{1 j}^{1}]$ , $[α_{1 j}^{2}]$ and $[α_{1 j}^{3}]$ for $[a_{pj}^{(t_{1}, r)}] ★ [b_{qj}^{(t_{2}, r)}]$ , $[b_{qj}^{(t_{2}, r)}] ★ [c_{vj}^{(t_{3}, r)}]$ , and $[a_{pj}^{(t_{1}, r)}] ★ [c_{vj}^{(t_{3}, r)}]$ respectively.

Step 5. Find the right-distributive sum-max decision inverse soft fuzzification matrices $[β_{1 j}^{1}]$ , $[β_{1 j}^{2}]$ and $[β_{1 j}^{3}]$ for $[b_{qj}^{(t_{2}, r)}] ★ [a_{pj}^{(t_{1}, r)}]$ , $[c_{vj}^{(t_{3}, r)}] ★ [b_{qj}^{(t_{2}, r)}]$ , and $[c_{vj}^{(t_{3}, r)}] ★ [a_{pj}^{(t_{1}, r)}]$ respectively.

Step 6. Find the distributive sum-max decision inverse soft fuzzification matrices $[γ_{1 j}^{1}], [γ_{1 j}^{2}]$ and $[γ_{1 j}^{3}]$ where $γ_{1 j}^{1} = \frac{1}{2} (α_{1 j}^{1} + β_{1 j}^{1})$ , $γ_{1 j}^{2} = \frac{1}{2} (α_{1 j}^{2} + β_{1 j}^{2})$ , $γ_{1 j}^{3} = \frac{1}{2} (α_{1 j}^{3} + β_{1 j}^{3})$ .

Step 7. Find the aggregate-distributive sum-max decision inverse soft fuzzification matrix [θ_1j], where $θ_{1 j} = \frac{1}{3} \sum_{i = 1}^{3} γ_{1 j}^{i}$ .

Step 8. Obtain an optimum set Opt_{[θ_1j]} (U₁) with respect to the aggregate-distributive sum-max decision inverse soft fuzzification matrix.

We give a congruous application of Algorithm 1 in multicriteria decision making.

Example 4.6. Suppose that a company decided to select a brand new Chief Executive Officer (CEO) so as to tackle the key challenges of the company faces nowadays and over the next few years. To execute this task, the company constituted a board of well-experienced members. Based on the key challenges of the company, the board designed the key responsibilities of the next CEO. According to the key responsibilities of the next CEO, the board created 3 types of parameters which the next CEO can Possess. The first type of parameters, named as “Skills”, $E_{1} = {e_{1}^{1}, e_{2}^{1}, \dots, e_{9}^{1}}$ , where $e_{1}^{1} -$ Ability to learn from the past, $e_{2}^{1} -$ Strong communication skills, $e_{3}^{1} -$ Building relationships, $e_{4}^{1} -$ Understanding, $e_{5}^{1} -$ Listening skills, $e_{6}^{1} -$ Reading people and adapting to necessary management styles, $e_{7}^{1} -$ Coaching employees effectively, $e_{8}^{1} -$ Thinking outside the box, $e_{9}^{1} -$ Realistic optimism. The second type of parameters, named as “Experience”, $E_{2} = {e_{1}^{2}, e_{2}^{2}, \dots, e_{6}^{2}}$ , where $e_{1}^{2} -$ Having experience in running public companies, $e_{2}^{2} -$ International business experience, $e_{3}^{2} -$ Relevant Industry Experience, $e_{4}^{2} -$ Prior Experience in the CEO Role, $e_{5}^{2} -$ Experience on how is the product made? $e_{6}^{2} -$ Experience on how will the client buy? The third type of parameters named as “Characteristics”, $E_{3} = {e_{1}^{3}, e_{2}^{3}, \dots, e_{9}^{3}}$ , where $e_{1}^{3} -$ Passion, $e_{2}^{3} -$ Vision, $e_{3}^{3} -$ Grit and Courage, $e_{4}^{3} -$ Decisiveness, $e_{5}^{3} -$ Self-Confidence, $e_{6}^{3} -$ A Connection with the culture, $e_{7}^{3} -$ No ego, $e_{8}^{3} -$ Curiosity, $e_{9}^{3} -$ Emotional Intelligence. For that purpose, the company advertised the requirements through the right channel. It had received about 40 application forms through the advertising. After 3 rounds of basic screening tests, the board had eliminated 34 applications and shortlisted 5 candidates $u_{1}^{1}, u_{2}^{1}, u_{3}^{1}, u_{4}^{1}$ , and $u_{5}^{1}$ for the final selection. In order to select the most effective candidate according to their requirements, the board had been instructed to analyze the above 5 shortlisted candidates based on the parameter sets, Skills, Experience, and Characteristics. The board had submitted the subsequent questionnaire, collectively, to take the final decision.

Then, we are ready assist the board to take the final decision by identifying the foremost effective new CEO for the concern company using our Algorithm 1. In order to use our algorithm, we need to create the inverse soft sets from the above table. With respect to the three types of parameters Skills, Experience and Characteristics, it is clear that the inverse soft sets are respectively

${\bar{F}}_{E_{1}}^{U_{1}} = {(u_{1}^{1}, {e_{1}^{1}, e_{2}^{1}, e_{3}^{1}, e_{5}^{1}, e_{6}^{1}, e_{7}^{1}, e_{8}^{1}}), (u_{2}^{1}, {e_{2}^{1}, e_{3}^{1},$ $e_{4}^{1}, e_{5}^{1}, e_{6}^{1}, e_{7}^{1}, e_{9}^{1}}), (u_{3}^{1}, {e_{1}^{1}, e_{2}^{1}, e_{4}^{1}, e_{5}^{1}, e_{7}^{1}, e_{9}^{1}}),$ $(u_{4}^{1}, {e_{1}^{1}, e_{2}^{1}, e_{3}^{1}, e_{5}^{1}, e_{6}^{1}, e_{7}^{1}, e_{8}^{1}}), (u_{5}^{1}, {e_{2}^{1}, e_{3}^{1}, e_{4}^{1}, e_{5}^{1},$ $e_{7}^{1}, e_{9}^{1}})}$ ,

${\bar{F}}_{E_{2}}^{U_{1}} = {(u_{1}^{1}, {e_{1}^{2}, e_{2}^{2}, e_{6}^{2}}), (u_{2}^{1}, {e_{1}^{2}, e_{2}^{2}, e_{3}^{2}, e_{6}^{2}}),$ $(u_{3}^{1}, {e_{1}^{2}, e_{2}^{2}, e_{4}^{2}, e_{6}^{2}}), (u_{4}^{1}, {e_{2}^{2}, e_{3}^{2}, e_{5}^{2}, e_{6}^{2}}), (u_{5}^{1}, {e_{2}^{2}, e_{5}^{2}, e_{6}^{2}})}$ and

${\bar{F}}_{E_{3}}^{U_{1}} = {(u_{1}^{1}, {e_{1}^{3}, e_{2}^{3}, e_{3}^{3}, e_{4}^{3}, e_{6}^{3}, e_{8}^{3}, e_{9}^{3}}), (u_{2}^{1}, {e_{1}^{3}, e_{2}^{3},$

$e_{3}^{3}, e_{5}^{3}, e_{6}^{3}, e_{8}^{3}}), (u_{3}^{1}, {e_{1}^{3}, e_{2}^{3}, e_{3}^{3}, e_{4}^{3}, e_{5}^{3}, e_{6}^{3}, e_{7}^{3}, e_{8}^{3},$ $e_{9}^{3}}), (u_{4}^{1}, {e_{1}^{3}, e_{2}^{3}, e_{3}^{3}, e_{5}^{3}, e_{6}^{3}, e_{7}^{3}, e_{9}^{3}}), (u_{5}^{1}, {e_{1}^{3}, e_{2}^{3}, e_{4}^{3},$ $e_{5}^{3}, e_{6}^{3}, e_{7}^{3}, e_{8}^{3}})}$ .

Now we are able to use our Algorithm 1.

Step 1. The inverse soft matrices corresponding to the inverse soft sets ${\bar{F}}_{E_{1}}^{U_{1}}$ , ${\bar{F}}_{E_{2}}^{U_{1}}$ and ${\bar{F}}_{E_{3}}^{U_{1}}$ are respectively $[a_{pj}^{(1, 1)}] = [\begin{matrix} 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \end{matrix}], [b_{qj}^{(2, 1)}] = [\begin{matrix} 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{matrix}] and$ $[c_{vj}^{(3, 1)}] = [\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 \end{matrix}] .$

		Skills									Experience						Characteristics
		$e_{1}^{1}$	$e_{2}^{1}$	$e_{3}^{1}$	$e_{4}^{1}$	$e_{5}^{1}$	$e_{6}^{1}$	$e_{7}^{1}$	$e_{8}^{1}$	$e_{9}^{1}$	$e_{1}^{2}$	$e_{2}^{2}$	$e_{3}^{2}$	$e_{4}^{2}$	$e_{5}^{2}$	$e_{6}^{2}$	$e_{1}^{3}$	$e_{2}^{3}$	$e_{3}^{3}$	$e_{4}^{3}$	$e_{5}^{3}$	$e_{6}^{3}$	$e_{7}^{3}$	$e_{8}^{3}$	$e_{9}^{3}$
Candidates	$u_{1}^{1}$	✓	✓	✓	×	✓	✓	✓	✓	×	✓	✓	×	×	×	✓	✓	✓	✓	✓	×	✓	×	✓	✓
	$u_{2}^{1}$	×	✓	✓	✓	✓	✓	✓	×	✓	✓	✓	✓	×	×	✓	✓	✓	✓	×	✓	✓	×	✓	×
	$u_{3}^{1}$	✓	✓	×	✓	✓	×	✓	×	✓	✓	✓	×	✓	×	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
	$u_{4}^{1}$	✓	✓	✓	×	✓	✓	✓	✓	×	×	✓	✓	×	✓	✓	✓	✓	✓	×	✓	✓	✓	×	✓
	$u_{5}^{1}$	×	✓	✓	✓	✓	×	✓	×	✓	×	✓	×	×	✓	✓	✓	✓	×	✓	✓	✓	✓	✓	×

Step 2. To find the optimum set from the universal set $U_{1} = {u_{1}^{1}, u_{2}^{1}, u_{3}^{1}, u_{4}^{1}, u_{5}^{1}}$ , we use And row-product of inverse soft matrices in this decision making.

Step 3. Using the And row-product ∧_r, we obtainthe inverse soft matrices $[a_{pj}^{(1, 1)}] \land_{r}$ , $[b_{qj}^{(2, 1)}] \land_{r} [a_{pj}^{(1, 1)}], [b_{qj}^{(2, 1)}] \land_{r} [c_{vj}^{(3, 1)}], [c_{vj}^{(3, 1)}] \land_{r} [b_{qj}^{(2, 1)}], [a_{pj}^{(1, 1)}] \land_{r} [c_{vj}^{(3, 1)}]$ and $[c_{vj}^{(3, 1)}] \land_{r} [a_{pj}^{(1, 1)}]$ . Since the dimensions of these inverse soft matrices are large, we can obtain them using the Scilab codes in Appendix. So they are omitted.

Step 4. The left-distributive sum-max decision inverse soft fuzzification matrices are $[α_{1 j}^{1}] = [\begin{matrix} 0.778 & 0.778 & 0.667 & 0.778 & 0.667 \end{matrix}]$ , $[α_{1 j}^{2}] = [\begin{matrix} 0.5 & 0.667 & 0.667 & 0.667 & 0.5 \end{matrix}]$ and $[α_{1 j}^{3}] = [\begin{matrix} 0.778 & 0.778 & 0.667 & 0.778 & 0.667 \end{matrix}]$

Step 5. The right-distributive sum-max decisioninverse soft fuzzification matrices are $[β_{1 j}^{1}] = [0.5$ 0.667 0.667 0.667 0.5], $[β_{1 j}^{2}] = [0.778$ 0.667 1 0.778 0.778] and $[β_{1 j}^{3}] = [0.778 0.667 1 0.778 0.778]$

Step 6. The distributive sum-max decisioninverse soft fuzzification matrices are $[γ_{1 j}^{1}] = [0.639 0.723 0.667 0.723 0.584]$ , $[γ_{1 j}^{2}] =$ [0.639 0.667 0.834 0.723 0.639] and $[γ_{1 j}^{3}] =$ [0.778 0.723 0.834 0.778 0.723]

Step 7. The aggregate-distributive sum-max decision inverse soft fuzzification matrix is [θ_1j] = [0.685 0.704 0.778 0.741 0.649]

Step 8. Then, we obtain the ranking order of objects as $u_{3}^{1} ≻ u_{4}^{1} ≻ u_{2}^{1} ≻ u_{1}^{1} ≻ u_{5}^{1}$ . Thus, the optimum set of U₁ is ${Opt}_{[θ_{1 j}]} (U_{1}) = {u_{3}^{1}}$ .

Hence we propose to the board to select the candidate u₃ as a new Chief Executive Officer(CEO) according to the concern company current and future situations.

Comparison. The authors in [17] gave u₂≻u₄ = u₆ ≻ u₅ ≻ u₁ ≻ u₃ as a solution of the multicriteria decision making problem in Example 6.3 by using their soft sum-row decision method based on the inverse soft sets. By using our Algorithm 1, we also obtain u₂ ≻ u₄ = u₆ ≻ u₅ ≻ u₁ ≻ u₃ as the ranking order for the selection of objects in this decision making problem. On the other hand, Çetkin et al. in [7] proposed {x₄, x₁₃, x₂₁, x₂₈, x₃₆, x₄₂} as the optimal choice for two soft sets consisting of forty-eight objects in the first part of Example 3.1 by using their decision making algorithm based on the inverse soft sets. Furthermore, they proposed {x₃, x₄, x₁₃, x₂₈, x₃₆, x₄₂} as the optimal choice for three soft sets in the second part of Example 3.1. By using our Algorithm 1, we also have x₁₃ = x₃₆ ≻ x₂₁ = x₂₈ = x₄₂ ≻ x₄ for two soft sets in the first part of Example 3.1 and x₁₃ = x₃₆ ≻ x₂₈ ≻ x₃ = x₄ = x₄₂ for three soft sets in the second part of Example 3.1. These support the consistency of Algorithm 1.

Also, we know that a soft set can be uniquely represented as an inverse soft set, vice versa. Thus, we compare our Algorithm 1 with some of the existing decision making algorithms based on the soft sets and soft matrices. Table 1 shows that our Algorithm 1 gives more convincing results than the existing soft decision algorithms.

Table 1
The comparison results of Algorithm 1 with some soft decision making algorithms

Paper Problem Result of their decision algorithm Result of our Algorithm 1

[3] Example 5.1 [3] {u₁, u₃} u₃ ≻ u₁ ≻ u₄ ≻ u₅ ≻ u₂

[4] Example 5.1 [4] {d₂, d₃} d₃ ≻ d₂ ≻ d₁ ≻ d₄

[6] Example 4 [6] {u₄, u₁₃, u₂₁, u₃₆, u₄₂} u₁₃ = u₃₆ ≻ u₂₁ = u₂₈ = u₄₂ ≻ u₄

[11] Application [11] u₁ > u₃ > u₂ u₁ ≻ u₃ ≻ u₂

[12] Example 5.17 [12] {h₁, h₂, h₃} h₁ = h₂ = h₃ ≻ h₄ = h₅

[12] Example 5.19 [12] {u₁₃, u₃₆} u₁₃ = u₃₆ ≻ u₂₁ = u₂₈ = u₄₂ ≻ u₄

[13] Example 3.3 [13] {h₁, h₂, h₃} h₁ = h₂ = h₃ ≻ h₄ = h₅

[19] Example 31 [19] {u₃} u₃ ≻ u₂ ≻ u₄ ≻ u₅ ≻ u₁

[23] Example 3.10 [23] {m₃} m₃ ≻ m₂ = m₅ ≻ m₁ ≻ m₄

[23] Example 3.12 [23] {m₃} m₃ ≻ m₁ ≻ m₂ = m₅ ≻ m₄

Paper	Problem	Result of their decision algorithm	Result of our Algorithm 1
[3]	Example 5.1 [3]	{u₁, u₃}	u₃ ≻ u₁ ≻ u₄ ≻ u₅ ≻ u₂
[4]	Example 5.1 [4]	{d₂, d₃}	d₃ ≻ d₂ ≻ d₁ ≻ d₄
[6]	Example 4 [6]	{u₄, u₁₃, u₂₁, u₃₆, u₄₂}	u₁₃ = u₃₆ ≻ u₂₁ = u₂₈ = u₄₂ ≻ u₄
[11]	Application [11]	u₁ > u₃ > u₂	u₁ ≻ u₃ ≻ u₂
[12]	Example 5.17 [12]	{h₁, h₂, h₃}	h₁ = h₂ = h₃ ≻ h₄ = h₅
[12]	Example 5.19 [12]	{u₁₃, u₃₆}	u₁₃ = u₃₆ ≻ u₂₁ = u₂₈ = u₄₂ ≻ u₄
[13]	Example 3.3 [13]	{h₁, h₂, h₃}	h₁ = h₂ = h₃ ≻ h₄ = h₅
[19]	Example 31 [19]	{u₃}	u₃ ≻ u₂ ≻ u₄ ≻ u₅ ≻ u₁
[23]	Example 3.10 [23]	{m₃}	m₃ ≻ m₂ = m₅ ≻ m₁ ≻ m₄
[23]	Example 3.12 [23]	{m₃}	m₃ ≻ m₁ ≻ m₂ = m₅ ≻ m₄

5 The inverse soft distributive If-sum decision making method

In this section, we propose a multicriteria decision making algorithm based on the If column-products of inverse soft matrices. With the help of this algorithm, we can obtain the ranking order for the selection of objects of two discrete universal sets U₁ and U₂ with respect to the determined parameters.

Definition 5.1. Let $[e_{ij}^{(t, r_{1})}] \in {ISM}_{m \times n_{1}}$ , $[f_{ik}^{(t, r_{2})}] \in {ISM}_{m \times n_{2}}$ .

If $[e_{ij}^{(t, r_{1})}] \Leftarrow_{t} [f_{ik}^{(t, r_{2})}] = [g_{il}] \in {ISM}_{m \times n_{1} n_{2}}$ , then the matrix $[λ_{1 ℓ}^{1}]$ is called a left-distributive left-If-sum decision inverse soft fuzzification matrix, where

$λ_{1 ℓ}^{1} = \frac{1}{n_{2}} \sum_{y = (ℓ - 1) n_{2} + 1}^{ℓ n_{2}} v_{1 y}$ , $(0 \leq λ_{1 ℓ}^{1} \leq 1)$ for each ℓ=1, 2, …, n₁ and $v_{1 y} = \frac{1}{m} \sum_{i = 1}^{m} g_{iy}$ .

If $[e_{ij}^{(t, r_{1})}] \Rightarrow_{t} [f_{ik}^{(t, r_{2})}] = [h_{il}] \in {ISM}_{m \times n_{1} n_{2}}$ , then the matrix $[μ_{1 ℓ}^{1}]$ is called a left-distributive right-If-sum decision inverse soft fuzzification matrix, where

$μ_{1 ℓ}^{1} = \frac{1}{n_{2}} \sum_{y = (ℓ - 1) n_{2} + 1}^{ℓ n_{2}} v_{1 y}^{'}$ , $(0 \leq μ_{1 ℓ}^{1} \leq 1)$ for each ℓ=1, 2, …, n₁ and $v_{1 y}^{'} = \frac{1}{m} \sum_{i = 1}^{m} h_{iy}$ .

Definition 5.2. Let $[e_{ij}^{(t, r_{1})}] \in {ISM}_{m \times n_{1}}$ , $[f_{ik}^{(t, r_{2})}] \in {ISM}_{m \times n_{2}}$ .

If $[f_{ik}^{(t, r_{2})}] \Leftarrow_{t} [e_{ij}^{(t, r_{1})}] = [g_{il}^{'}] \in {ISM}_{m \times n_{2} n_{1}}$ , then the matrix $[λ_{1 ℓ}^{2}]$ is called a right-distributive left-If-sum decision inverse soft fuzzification matrix, where

$λ_{1 ℓ}^{2} = \frac{1}{n_{1}} \sum_{z = (ℓ - 1) n_{1} + 1}^{ℓ n_{1}} w_{1 z}$ , $(0 \leq λ_{1 ℓ}^{2} \leq 1)$ for each ℓ=1, 2, …, n₂ and $w_{1 z} = \frac{1}{m} \sum_{i = 1}^{m} g_{iz}^{'}$ .

If $[f_{ik}^{(t, r_{2})}] \Rightarrow_{t} [e_{ij}^{(t, r_{1})}] = [h_{il}^{'}] \in {ISM}_{m \times n_{2} n_{1}}$ , then the matrix $[μ_{1 ℓ}^{2}]$ is called a right-distributive right-If-sum decision inverse soft fuzzification matrix, where

$μ_{1 ℓ}^{2} = \frac{1}{n_{1}} \sum_{z = (ℓ - 1) n_{1} + 1}^{ℓ n_{1}} w_{1 z}^{'}$ , $(0 \leq μ_{1 ℓ}^{2} \leq 1)$ for each ℓ=1, 2, …, n₂ and $w_{1 z}^{'} = \frac{1}{m} \sum_{i = 1}^{m} h_{iz}^{'}$ .

Definition 5.3. Let $[λ_{1 ℓ}^{1}]$ , $[μ_{1 ℓ}^{1}]$ be left-distributive left and right-If-sum decision inverse soft fuzzification matrices and $[λ_{1 ℓ}^{2}]$ , $[μ_{1 ℓ}^{2}]$ be right-distributive left and right-If-sum decision inverse soft fuzzification matrices, respectively. Then, the decision score of ℓth object $(u_{ℓ}^{r})$ of the universal set U_r (r = 1, 2) is defined and denoted by $S_{ℓ}^{r} = λ_{1 ℓ}^{r} - μ_{1 ℓ}^{r} .$

Definition 5.4. Let $U_{1} = {u_{1}^{1}, u_{2}^{1}, . . ., u_{n_{1}}^{1}}$ and $U_{2} = {u_{1}^{2}, u_{2}^{2}, . . ., u_{n_{2}}^{2}}$ be two discrete universal sets. By using decision scores of objects of these universal sets, we get the ranking order of two objects $u_{ℓ_{1}}^{r_{1}}$ and $u_{ℓ_{2}}^{r_{2}}$ as below: $u_{ℓ_{1}}^{r_{1}} ≻ u_{ℓ_{2}}^{r_{2}} if S_{ℓ_{1}}^{r_{1}} > S_{ℓ_{2}}^{r_{2}},$ where it can be r₁ = r₂ or r₁ ≠ r₂ and ℓ₁ = ℓ ₂ or ℓ₁ ≠ ℓ ₂.

Then, the an optimum set of U₁ and U₂ is obtained as follows: ${Opt}_{S} (U_{1}, U_{2}) = {u_{j}^{r_{1}} : u_{j}^{r_{1}} \in U_{1} or u_{j}^{r_{1}} \in U_{2} and$ $S_{j}^{r_{1}} > S_{j^{'}}^{r_{1}} for \forall j^{'} \neq j and S_{j}^{r_{1}} > S_{j^{'}}^{r_{2}} for \forall j^{'}} .$

By using the above Definitions, we construct the subsequent algorithm to search out an optimum set through the different kinds of inverse soft sets created by considering the discrete universal sets U₁ and U₂.

Algorithm 2.

Step 1. The experts create the inverse soft matrices (represented as $[a_{pj}^{(t_{1}, r_{1})}]$ , $[b_{qk}^{(t_{2}, r_{2})}]$ , $[c_{pk}^{(t_{1}, r_{2})}]$ and $[d_{qj}^{(t_{2}, r_{1})}]$ ) from their inverse soft sets.

Step 2. Select the suitable row-product (symbolized as ★) based on the real scenario of the problem. It is going to be one in all And row-product, Or row-product, And-Not row-product and Or-Not row product.

Step 3. Obtain the row-products $[e_{sj}^{(〈 t_{1}, t_{2} 〉, r_{1})}] = [a_{pj}^{(t_{1}, r_{1})}] ★ [c_{qj}^{(t_{2}, r_{1})}]$ for common r₁ and $[f_{sk}^{(〈 t_{1}, t_{2} 〉, r_{2})}] = [b_{pk}^{(t_{1}, r_{2})}] ★ [d_{qk}^{(t_{2}, r_{2})}]$ for common r₂.

Step 4. Obtain If column-products $[g_{sl}] = [e_{sj}^{(〈 t_{1}, t_{2} 〉, r_{1})}] \Leftarrow_{t} [f_{sk}^{(〈 t_{1}, t_{2} 〉, r_{2})}]$ , $[h_{sl}] = [e_{sj}^{(〈 t_{1}, t_{2} 〉, r_{1})}] \Rightarrow_{t} [f_{sk}^{(〈 t_{1}, t_{2} 〉, r_{2})}]$ and $[g_{sl}^{'}] = [f_{sk}^{(〈 t_{1}, t_{2} 〉, r_{2})}] \Leftarrow_{t} [e_{sj}^{(〈 t_{1}, t_{2} 〉, r_{1})}]$ , $[h_{sl}^{'}] = [f_{sk}^{(〈 t_{1}, t_{2} 〉, r_{2})}] \Rightarrow_{t} [e_{sj}^{(〈 t_{1}, t_{2} 〉, r_{1})}]$ .

Step 5. Find the left-distributive left and right-If-sum decision inverse soft fuzzification matrices $[λ_{1 ℓ}^{1}]$ , $[μ_{1 ℓ}^{1}]$ for [g_sl], [h_sl], and the right-distributive left and right-If-sum decision inverse soft fuzzification matrices $[λ_{1 ℓ}^{2}]$ , $[μ_{1 ℓ}^{2}]$ for $[g_{sl}^{'}]$ , $[h_{sl}^{'}]$ , respectively.

Step 6. Calculate decision scores for each object in the discrete universal sets U₁ and U₂.

Step 7. Obtain an optimum set ${Opt}_{S} (U_{1}, U_{2})$ with respect to the decision scores for the objects.

We present two different applications of Algorithm 2 in multicriteria decision making.

Example 5.5. Assume that a married couple, Mr. and Mrs. X are planning to buy a car or an SUV (Sports Utility Vehicle). As a result of their preliminary research, they identified three cars $U_{1} = {u_{1}^{1}, u_{2}^{1}, u_{3}^{1}}$ and two SUVs $U_{2} = {u_{1}^{2}, u_{2}^{2}}$ that could be purchased. Mrs. X states that she will consider the comfort and security during the evaluation process and therefore selects the parameter set as $E_{1} = {e_{1}^{1} - comfort, e_{2}^{1} - safety}$ . Mr. X states that he will consider the technical specifications during the evaluation process and therefore selects the parameter set as $E_{2} = {e_{1}^{2} - engine capacity, e_{2}^{2} - fuel efficient, e_{3}^{2} - automatic transmission e_{4}^{2} - motor thrust force}$ .

After the couple create the inverse soft sets, we are ready to apply Algorithm 2 for their optimal choice.

Mrs. X generates the following inverse soft sets:

${\bar{F}}_{U_{1}}^{E_{1}} = {(u_{1}^{1}, {e_{1}^{1}}), (u_{3}^{1}, {e_{2}^{1}})}$ and ${\bar{F}}_{U_{2}}^{E_{1}} = {(u_{1}^{2},$ ${e_{1}^{1}}),$ $(u_{2}^{2}, {e_{2}^{1}})}$ .

Mr. X generates the following inverse soft sets:

${\bar{F}}_{U_{1}}^{E_{2}} = {(u_{1}^{1}, {e_{1}^{2}, e_{3}^{2}}), (u_{2}^{1}, {e_{2}^{2}, e_{3}^{2}}), (u_{3}^{1}, {e_{1}^{2}, e_{2}^{2}, e_{4}^{2}})}$ and ${\bar{F}}_{U_{2}}^{E_{2}} = {(u_{1}^{2}, {e_{1}^{2}, e_{3}^{2}, e_{4}^{2}}), (u_{2}^{2}, {e_{4}^{2}})}$ .

Now we are able to use our Algorithm 2.

Step 1. The inverse soft matrices corresponding to the inverse soft sets of Mrs. X are $[a_{pj}^{(1, 1)}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}] and [b_{pk}^{(1, 2)}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] .$ The inverse soft matrices corresponding to the inverse soft sets of Mr. X are $[c_{qj}^{(2, 1)}] = [\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] and [d_{qk}^{(2, 2)}] = [\begin{matrix} 1 & 0 \\ 0 & 0 \\ 1 & 0 \\ 1 & 1 \end{matrix}] .$ Step 2. To find the optimum set from the discrete universal sets $U_{1} = {u_{1}^{1}, u_{2}^{1}, u_{3}^{1}}$ and $U_{2} = {u_{1}^{2}, u_{2}^{2}}$ , we use the Or row-product of inverse soft matrices in this decision problem.

Step 3. The Or row-products of inverse soft matrices are $[e_{sj}^{(〈 1, 2 〉, 1)}] = [a_{pj}^{(1, 1)}] \lor_{r} [c_{qj}^{(2, 1)}] = [\begin{matrix} 1 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}] and$ $[f_{sk}^{(〈 1, 2 〉, 2)}] = [b_{pk}^{(1, 2)}] \lor_{r} [d_{qk}^{(2, 2)}] = [\begin{matrix} 1 & 0 \\ 1 & 0 \\ 1 & 0 \\ 1 & 1 \\ 1 & 1 \\ 0 & 1 \\ 1 & 1 \\ 1 & 1 \end{matrix}]$ Step 4. Using the If column-products of inverse soft matrices, the inverse soft matrices $[g_{sl}] = [e_{sj}^{(〈 1, 2 〉, 1)}] \Leftarrow_{t} [f_{sk}^{(〈 1, 2 〉, 2)}]$ , $[h_{sl}] = [e_{sj}^{(〈 1, 2 〉, 1)}] \Rightarrow_{t} [f_{sk}^{(〈 1, 2 〉, 2)}]$ and $[g_{sl}^{'}] = [f_{sk}^{(〈 1, 2 〉, 2)}] \Leftarrow_{t} [e_{sj}^{(〈 1, 2 〉, 1)}]$ , $[h_{sl}^{'}] = [f_{sk}^{(〈 1, 2 〉, 2)}] \Rightarrow_{t} [e_{sj}^{(〈 1, 2 〉, 1)}]$ are respectively obtained as follows: $[g_{sl}] = [\begin{matrix} 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{matrix}], [h_{sl}] = [\begin{matrix} 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}] and$ $[g'_{s l}] = [\begin{array}{l} 1 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{array}], [h'_{s l}] = [\begin{array}{l} 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 & 1 \end{array}] .$

Step 5. The left-distributive left and right-If-sum decision inverse soft fuzzification matrices are $[λ_{1 ℓ}^{1}] = [\begin{matrix} 0.813 & 0.563 & 0.938 \end{matrix}]$ and $[μ_{1 ℓ}^{1}] = [\begin{matrix} 0.813 & 0.813 & 0.813 \end{matrix}]$ .

The right-distributive left and right-If-sum decision inverse soft fuzzification matrices are $[λ_{1 ℓ}^{2}] = [\begin{matrix} 0.917 & 0.708 \end{matrix}]$ and $[μ_{1 ℓ}^{2}] = [\begin{matrix} 0.75 & 0.792 \end{matrix}]$ .

Step 6. The decision score for each object in the discrete universal sets U₁ and U₂ is calculated as below: $S_{1}^{1} = 0, S_{2}^{1} = - 0.25, S_{3}^{1} = 0.125, S_{1}^{2} = 0.167$ $and S_{2}^{2} = - 0.084 .$ Step 7. Then, we obtain the ranking order of objects in U₁ and U₂ as $\begin{matrix} u_{1}^{2} ≻ u_{3}^{1} ≻ u_{1}^{1} ≻ u_{2}^{2} ≻ u_{2}^{1} since S_{1}^{2} > S_{3}^{1} > \\ S_{1}^{1} > S_{2}^{2} > S_{2}^{1} . \end{matrix}$

Hence, the optimum set of U₁ and U₂ is ${Opt}_{S} (U_{1}, U_{2}) = {u_{1}^{2}}$ . Therefore, we recommend that the couple buy the SUV $u_{1}^{2}$ according to these data.

Example 5.6. Assume that an investment company wants to invest some money in the best option from different types of companies. There are two discrete universal sets which each of them involves possible options to invest the money: $U_{1} = {u_{1}^{1}, u_{2}^{1}, u_{3}^{1}, u_{4}^{1}}$ is a set of commercial finance companies and $U_{2} = {u_{1}^{2}, u_{2}^{2}}$ is a set of media companies. A decision committee consisting of two experts is composed of the members X₁ and X₂. X₁ is a well-informed expert about the existing commercial finance companies. X₂ is a well-informed expert about the existing media companies. Therefore, according to the following parameter sets, the investment company proposes that the expert X₁ should evaluate the companies in U₁ and the expert X₂ should evaluate the companies in U₂. There are two parameter sets: $E_{1} = {e_{1}^{1} - environmental risks, e_{2}^{1} - political risks}$ is a set of the (low) economic risk parameters and $E_{2} = {e_{1}^{2} - financial fluctuations, e_{2}^{2} - exchange rate fluctuations, e_{3}^{2} - cyclical$ fluctuations} is a set of the (low) economic fluctuation parameters.

After the experts create the inverse soft sets, we are ready to implement Algorithm 2 in determining the optimal company to invest the existing money of the investment company.

Suppose that the inverse soft sets of the expert X₁ are respectively ${\bar{F}}_{E_{1}}^{U_{1}} = {(u_{1}^{1}, {e_{1}^{1}}), (u_{2}^{1}, E_{1}), (u_{3}^{1}, {e_{2}^{1}}), (u_{4}^{1}, {e_{1}^{1}})}$ and ${\bar{F}}_{E_{2}}^{U_{1}} = {(u_{1}^{1}, {e_{2}^{2}}), (u_{2}^{1}, {e_{1}^{2}, e_{3}^{2}}), (u_{3}^{1}, E_{2}}), (u_{4}^{1},$ ${e_{2}^{2}, e_{3}^{2}})}$ .

The inverse soft sets of the expert X₂ are respectively ${\bar{F}}_{E_{1}}^{U_{2}} = {(u_{1}^{2}, \emptyset), (u_{2}^{2}, E_{1})$ and ${\bar{F}}_{E_{2}}^{U_{2}} = {(u_{1}^{2}, {e_{2}^{2}, e_{3}^{2}}),$ $(u_{2}^{2}, {e_{1}^{2}, e_{2}^{2}})}$ .

Now we are able to use our Algorithm 2.

Step 1. The inverse soft matrices corresponding tothe inverse soft sets of the expert X₁ are $[a_{pj}^{(1, 1)}] = [\begin{matrix} 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{matrix}] and [b_{qj}^{(2, 1)}] = [\begin{matrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \end{matrix}] .$ The inverse soft matrices corresponding to the inverse soft sets of the expert X₂ are $[c_{pk}^{(1, 2)}] = [\begin{matrix} 0 & 1 \\ 0 & 1 \end{matrix}] and [d_{qk}^{(2, 2)}] = [\begin{matrix} 0 & 1 \\ 1 & 1 \\ 1 & 0 \end{matrix}] .$ Step 2. To find the optimum set from the discrete universal sets $U_{1} = {u_{1}^{1}, u_{2}^{1}, u_{3}^{1}, u_{4}^{1}}$ and $U_{2} = {u_{1}^{2}, u_{2}^{2}}$ , we use the And row-product of inverse soft matrices in this decision problem.

Step 3. The And row-products of inverse soft matrices are $[e_{sj}^{(〈 1, 2 〉, 1)}] = [a_{pj}^{(1, 1)}] \land_{r} [b_{qj}^{(2, 1)}] = [\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \end{matrix}] and$ $[f_{sk}^{(〈 1, 2 〉, 2)}] = [c_{pk}^{(1, 2)}] \land_{r} [d_{qk}^{(2, 2)}] = [\begin{matrix} 0 & 1 \\ 0 & 1 \\ 0 & 0 \\ 0 & 1 \\ 0 & 1 \\ 0 & 0 \end{matrix}]$ Step 4. Using the If column-products of inverse soft matrices, the inverse soft matrices $[g_{sl}] = [e_{sj}^{(〈 1, 2 〉, 1)}] \Leftarrow_{t} [f_{sk}^{(〈 1, 2 〉, 2)}]$ , $[h_{sl}] = [e_{sj}^{(〈 1, 2 〉, 1)}] \Rightarrow_{t} [f_{sk}^{(〈 1, 2 〉, 2)}]$ and $[g_{sl}^{'}] = [f_{sk}^{(〈 1, 2 〉, 2)}] \Leftarrow_{t} [e_{sj}^{(〈 1, 2 〉, 1)}]$ , $[h_{sl}^{'}] = [f_{sk}^{(〈 1, 2 〉, 2)}] \Rightarrow_{t} [e_{sj}^{(〈 1, 2 〉, 1)}]$ are respectively obtained as follows: $[g_{sl}] = [\begin{matrix} 1 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}], [h_{sl}] = [\begin{matrix} 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix}] and$ $[g'_{s l}] = [\begin{array}{l} 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \end{array}], [h'_{s l}] = [\begin{array}{l} 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{array}] .$

Step 5. The left-distributive left and right-If-sum decision inverse soft fuzzification matrices are $[λ_{1 ℓ}^{1}] = [\begin{matrix} 0.75 & 0.833 & 0.833 & 0.75 \end{matrix}]$ and $[μ_{1 ℓ}^{1}] = [\begin{matrix} 0.917 & 0.5 & 0.667 & 0.75 \end{matrix}]$ .

The right-distributive left and right-If-sum decision inverse soft fuzzification matrices are $[λ_{1 ℓ}^{2}] = [\begin{matrix} 0.583 & 0.833 \end{matrix}]$ and $[μ_{1 ℓ}^{2}] = [\begin{matrix} 1 & 0.583 \end{matrix}]$ .

Step 6. The decision score for each element in the discrete universal sets U₁ and U₂ is calculated as below: $\begin{matrix} S_{2}^{1} = 0.333, S_{3}^{1} = 0.166, S_{4}^{1} = 0, \\ and S_{2}^{2} = 0.25 . \end{matrix}$

Step 7. Then, we obtain the ranking order of objects in U₁ and U₂ as

$\begin{matrix} u_{2}^{1} ≻ u_{2}^{2} ≻ u_{3}^{1} ≻ u_{4}^{1} ≻ u_{1}^{1} ≻ u_{1}^{2} since \\ S_{2}^{1} > S_{2}^{2} > S_{3}^{1} > S_{4}^{1} > S_{1}^{1} > S_{1}^{2} . \end{matrix}$

Thus, the optimum set of U₁ and U₂ is ${Opt}_{S} (U_{1}, U_{2}) = {u_{2}^{1}}$ .

Hence, we suggest that the investment company can invest the money in the commercial finance company $u_{2}^{1}$ according to these data.

It is seen that the inverse soft sets in Example 5.5 are in the same structure as the inverse soft set defined by Khalil and Hassan [18] while the inverse soft sets in Examples 4.6 and 5.6 are in the same structure as the inverse soft set defined by Çetkin et al. [7].

6 Conclusion

In this work, we defined the inverse soft matrices corresponding to the inverse soft sets and their some products. Then, we created two valuable multicriteria decision making methods by using these products. We gave various examples intended for their practices in real life. Considering the usefulness and practicality of the products presented in this study, it is foreseen that they will be the focus of interest in decision making based on soft structures in the near future. In addition, the If and Iff column-products of the inverse soft matrices can be adapted to many matrix types such as the fuzzy soft matrices, intuitionistic fuzzy soft matrices. Thus, we hope that many methods will be developed for decision making involving the discrete universal sets and multi-disjoint parameter sets. In conclusion, this study is just the tip of the iceberg about the products of inverse soft matrices and their applications in decision making.

Footnotes

Appendix

Scilab Codes:

We give the following Scilab codes for Algorithm 1 and Algorithm 2, respectively.

Inverse soft distributive sum-max decision making method (Algorithm 1)

for And row-product (∧_r)	for [α_1j],[β_1j] and [γ_1j]
For $[e_{s j}] = [a_{p j}^{(1, 1)}] \land_{r} [b_{q j}^{(2, 1)}] :$ function e12=androwprod(a11,b21) [m1,n]=size(a11); [m2,n]=size(b21); e12=zeros(m1m2,n); for j=1:n for p=1:m1 for q=1:m2 s=(p-1)m2+q; e12(s,j)=min(a11(p,j),b21(q,j)); end end end endfunction For $[e_{s j}] = [b_{q j}^{(2, 1)}] \land_{r} [a_{p j}^{(1, 1)}] :$ function e21=androwprod(b21,a11) [m1,n]=size(a11); [m2,n]=size(b21); e21=zeros(m1m2,n); for j=1:n for q=1:m2 for p=1:m1 s=(q-1)m1+p; e21(s,j)=min(b21(q,j),a11(p,j)); end end end endfunction For Or row-product, the above Scilab codes are written “max" instead of “min".	For [α_1j] : function Alpha=leftsummax(a11,b21,e12) [m1,n]=size(a11); [m2,n]=size(b21); [m1m2,n]=size(e12); v=zeros(m1,n); for p=1:m1 for j=1:n v(p,j)=max(e12((p-1)m2+1:pm2,j)); end end Alpha=sum(v,′r′)/m1; endfunction For [β_1j] : function Beta=rightsummax(b21,a11,e21) [m1,n]=size(a11); [m2,n]=size(b21); [m1m2,n]=size(e21); w=zeros(m1,n); for q=1:m2 for j=1:n w(q,j)=max(e21((q-1)m1+1:qm1,j)); end end Beta=sum(w,′r′)/m2; endfunction For [γ_1j] : function Gamma=summax(Alpha,Beta) Gamma=(Alpha+Beta)/2; endfunction

for And row-product (∧_r)

for [α_1j],[β_1j] and [γ_1j]

For

[e_{s j}] = [a_{p j}^{(1, 1)}] \land_{r} [b_{q j}^{(2, 1)}] :

function e12=androwprod(a11,b21)
[m1,n]=size(a11);
[m2,n]=size(b21);
e12=zeros(m1*m2,n);
for j=1:n
for p=1:m1
for q=1:m2
s=(p-1)*m2+q;
e12(s,j)=min(a11(p,j),b21(q,j));
end
end
end
endfunction
For

[e_{s j}] = [b_{q j}^{(2, 1)}] \land_{r} [a_{p j}^{(1, 1)}] :

function e21=androwprod(b21,a11)
[m1,n]=size(a11);
[m2,n]=size(b21);
e21=zeros(m1*m2,n);
for j=1:n
for q=1:m2
for p=1:m1
s=(q-1)*m1+p;
e21(s,j)=min(b21(q,j),a11(p,j));
end
end
end
endfunction
For Or row-product, the above Scilab codes are written “max" instead of “min".

For [α_1j] :
function Alpha=leftsummax(a11,b21,e12)
[m1,n]=size(a11);
[m2,n]=size(b21);
[m1m2,n]=size(e12);
v=zeros(m1,n);
for p=1:m1
for j=1:n
v(p,j)=max(e12((p-1)*m2+1:p*m2,j));
end
end
Alpha=sum(v,′r′)/m1; endfunction
For [β_1j] :
function Beta=rightsummax(b21,a11,e21)
[m1,n]=size(a11);
[m2,n]=size(b21);
[m1m2,n]=size(e21);
w=zeros(m1,n);
for q=1:m2
for j=1:n
w(q,j)=max(e21((q-1)*m1+1:q*m1,j));
end
end
Beta=sum(w,′r′)/m2; endfunction
For [γ_1j] :
function Gamma=summax(Alpha,Beta)
Gamma=(Alpha+Beta)/2;
endfunction

Inverse soft distributive If -sum decision making method (Algorithm 2)

for if column-products	for $[λ_{1 l}^{1}]$ and $[μ_{1 l}^{1}]$	for $[λ_{2 l}^{1}]$ and $[μ_{2 l}^{1}]$
For $[g_{u j}] = [e_{i j}^{(1, 1)}] \Leftarrow [f_{i k}^{(1, 2)}]$ functiong=leftif(e1,f2) [m,n1]=size(e1); [m,n2]=size(f2); g=zeros(m,n1n2); for i=1:m for j=1:n1 for k=1:n2 l=(j-1)n2+k; ife1(i,j)greaterthan=f2(i,k) g(i,l)=1; else g(i,l)=0; end end end end endfunction For $[h_{u j}] = [e_{i j}^{(1, 1)}] \Rightarrow [f_{i k}^{(1, 2)}]$ function h=rightif(e1,f2) [m,n1]=size(e1); [m,n2]=size(f2); h=zeros(m,n1n2); for i=1:m for j=1:n1 for k=1:n2 l=(j-1)n2+k; if e1(i,j)lessthan=f2(i,k) h(i,l)=1; else h(i,l)=0; end end end end endfunction	For $[λ_{1 l}^{1}] :$ function Lambda1=leftsum(e1,f2,g1) [m,n1]=size(e1); [m,n2]=size(f2); [m,n1n2]=size(g1); v=zeros(m,n1); for l=1:n1 for i=1:m v(i,l)=sum(g1(i,(l-1)n2+1:ln2)); end end Lambda1=sum(v,′r′)/(n2m); endfunction For $[μ_{1 l}^{1}] :$ function Mu1=leftsum(e1,f2,h1) [m,n1]=size(e1); [m,n2]=size(f2); [m,n1n2]=size(h1); v=zeros(m,n1); for l=1:n1 for i=1:m v(i,l)=sum(h1(i,(l-1)n2+1:ln2)); end end Mu1=sum(v,′r′)/(n2m); endfunction	For $[λ_{2 l}^{1}] :$ function Lambda2=rightsum(f2,e1,g2) [m,n1]=size(e1); [m,n2]=size(f2); [m,n2n1]=size(g2); w=zeros(m,n2);for l=1:n2 for i=1:m w(i,l)=sum(g2(i,(l-1)n1+1:ln1)); end end Lambda2=sum(w,′r′)/(n1m); endfunction For $[μ_{2 l}^{1}] :$ function Mu2=rightsum(f2,e1,h2) [m,n1]=size(e1); [m,n2]=size(f2); [m,n2n1]=size(h2); w=zeros(m,n2); for l=1:n2 for i=1:m w(i,l)=sum(h2(i,(l-1)n1+1:ln1)); end end Mu_2=sum(w,′r′)/(n1m); endfunction

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