Mathematical analysis of parameters belonging to the universe in the soft set theory with new distance measures

Abstract

Introduction:

The soft set theory has drawn the attention of many researchers, particularly for dealing with uncertainty in decision-making problems. Despite its remarkable advantages, the soft set theory has only been used to tackle decision-making problems that aim to choose the best option. However, there exist different forms of decision-making problems that involve different forms of uncertainty.

Methods:

In this study, we present various algorithms based on the soft set theory in order to handle the cases where one has different uncertainty forms in decision-making problems. Some new concepts such as object code, personal object code, parameter significance weight and new distance measures have been introduced to the literature for the construction of these algorithms. Furthermore, we show the application results of those algorithms and provide several examples.

Results and Conclusions:

As a result, a comparison among the application results of the algorithms implies that the best objects might not always yield the most efficient outcomes.

Keywords

Soft set D-metric space parametric distance algorithm decision making

1 Introduction

Vagueness and uncertainty are important characteristics that have to be dealt with during data analysis in order to increase the robustness of the results. However, it is in general not so straightforward to decompose the vagueness of the data. Therefore, many mathematical approaches, which are based on the analysis of certain data, might be inadequate to capture this component. Many theories have been introduced to handle the vagueness involved in data. To name a few, we can think of the theory of fuzzy sets [23], the theory of rough sets [21] and the theory of intuitionistic fuzzy sets [1]. Among these theories, Zadeh’s fuzzy set theory [23] is the most popular one. Although these theories have brought several novelties into the classical theories, they all have some kind of drawbacks. In 1999, Molodtsov [19] introduced the soft set theory and he further stated that this new theory is exempt from the difficulties seen in other theories since it has sufficient parametrization tools. Since the soft set theory was introduced, many different mathematical models of soft set have been developed such as vague parameterized vague soft sets [25], generalized picture fuzzy soft sets [27], bipolar fuzzy hypersoft set [26], multi-valued picture fuzzy soft sets [28], bipolar soft sets [29], neutrosophic vague soft multiset [30], multi-valued interval neutrosophic soft sets [32], interval-valued complex neutrosophic soft set [33], Q-neutrosophic soft sets [34] and neutrosophic bipolar vague soft set [31]. Currently, the soft set continues to attract the attention of researchers, and accordingly, the field of application of the this theory continues to increase day by day [10 –16].

Multicriteria group decision-making problems aim to determine the best option among all of the possible alternatives. These problems typically contain multiple criteria which serve as a filter to eliminate the alternatives against the best option. There exist various methods to solve multi-criteria decision-making problems. One of the well-known methods introduced by Hwang and Yoon [17] is a technique for order performance by similarity to the ideal solution (TOPSIS). The principle of the TOPSIS hinges on the idea of choosing the alternative which has the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. This method assumes that the criteria consist of crisp values meaning that there is no vagueness involved in the criteria. This can be seen as a drawback of the method since the criteria of real-life problems are typically given by vague values. To overcome this drawback, several methods are developed such as linguistic variables [2 , 24] and Fuzzy TOPSIS method [4]. These methods are adopted to solve the problems in various fields such as engineering, medical sciences, economics and etc. which particularly involve vagueness [35 –38]. In 2011, Feng [7] considered the concept of soft set based group decision making. This study can be seen as a first attempt to utilize the soft rough approximations in multi-criteria group decision-making problems with vagueness. Celik and Yamak [3] applied the fuzzy soft set theory through Sanchez’s well-known approach for medical diagnosis which is based on fuzzy arithmetic operations. Moreover, as another medical application Yüksel et al. [22] introduced a prediction system by using the soft covering based rough sets to estimate the risk of prostate cancer.

The notion of a D-metric space, which is a new class of generalized metric spaces, was originally introduced by Dhage [6] in 1994. He also proved that the D-metrics provide a generalization of ordinary metric functions. Afterwards, Naidu et al. [20] worked on the topology of D-metric spaces.

The application of soft set theory in decision-making problems has traditionally focused on scenarios where the evaluation is contingent on parameter values associated with objects. In essence, the primary emphasis has been on the selection of the optimal choice. However, decision-making encompasses diverse forms of uncertainty, extending beyond the realm of traditional parameter-based evaluations. To illustrate this difference, let us consider a hypothetical scenario: a company’s objective is to hire individuals who will work most effectively together, prioritizing group synergy over the selection of the single best candidate from a pool of applicants. This highlights the need to address a broader spectrum of decision-making problems. Now, we give a more concrete example of this type of uncertainty:

Example 1.1. Let us take three gear wheels and denote them with a set X = {x₁ (r = 2) , x₂ (r = 3) , x₃ (r = 4)}. In the set X, r represents the radius of the gears. Assume that we have the parameter set $\begin{matrix} E = {e_{1} = the radius of the gear (r), \\ e_{2} = number of rotations (n)} \end{matrix}$

Mr. X designs a mechanism and aims to incorporate the fastest rotation of the large gear D into his mechanism. Gear D is located at the end of the mechanism. Furthermore, Mr. X aims to choose the two most suitable gears for the fastest rotation of the gear D. In addition, the gear in the middle of the mechanism has to rotate quickly. To be able to reach this goal, an experiment for each gear is conducted, respectively. As a result, Mr. X observes the gear D by turning each gear ten times. The following table shows the evaluation results:

As shown in Table 1, it is clear that the B and C are the most rapidly rotating gears of the gear D. Thus, Mr. X should have chosen the most appropriate pair of gears B and C. Moreover, the following table evaluates each binary permutation:

Table 1
Evaluation results for each gear

Sequence First Gear Second Gear

A,D 10 2

B,D 10 4

C,D 10 3

According to the results given in Table 2, it is clear that the most suitable gears are the gears B and A. The aim of Mr. X was to build the mechanism in the most efficient way. Hence, the middle and the last gear were required to rotate quickly. However, the results given in Table 2 imply that the gears which rotate the gear D most rapidly do not yield the most efficient results in the mechanism.

Table 2

The result of the evaluations obtained with each binary permutation of the gears

Sequence	First Gear	Second Gear	Third Gear
A,B,D	10	5	2
A,C,D	10	6.667	2
B,C,D	10	13.33	4
B,A,D	10	20	4
C,A,D	10	15	3
C,B,D	10	7.5	3

In the present study, we introduce a novel approach that extends beyond conventional methods. We develop algorithms utilizing the D-metric [6] to select alternatives that optimize group dynamics, recognizing that the most suitable alternative may not always align with the goal of creating the most effective collaborative team. This approach represents a departure from the customary focus on individual excellence and shifts the attention toward collective performance.

While this innovative approach offers the potential for more effective group decision-making, it is essential to acknowledge its advantages and disadvantages. On one hand, it opens up new avenues for addressing decision problems characterized by interdependencies and collaboration requirements. On the other hand, it may not be suitable for scenarios where selecting the single best option is paramount, potentially limiting its applicability in certain contexts. Hence, this study endeavors to provide a comprehensive analysis of the method’s strengths and limitations, thus contributing substantively to the discourse on enhancing decision-making processes within a broader spectrum of applications.

2 Preliminaries

In this section, we recall some basic notions about Molodtsov’s soft sets and D-metric spaces upon which our algorithms are built.

Throughout the paper, U is an initial universe, E is a set of parameters, P (U) is the power set of U and A is a non-empty subset of E.

Definition 2.1. A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P (U). In other words, a soft set over U is a parameterized family of subsets of the universe U.

Definition 2.2. [18] A soft set (F, A) over U is said to be a null soft set denoted by Φ, if for all ɛ ∈ A, F (ɛ) =∅ (null-set).

Definition 2.3. [18] A soft set (F, A) over U is said to be an absolute soft set denoted by $\tilde{A}$ , if for all ɛ ∈ A, F (ɛ) = U.

Definition 2.4. [6] A real function D on U × U × U is said to be a D-metric on U if

D (x, y, z) ≥0 for all x, y, z ∈ U (non-negativity),

D (x, y, z) =0 if and only if x = y = z (coincidence),

D (x, y, z) = D (p (x, y, z)) for every x, y, z ∈ U and for any permutation p (x, y, z) of x, y, z (symmetry),

D (x, y, z) ≤ D (x, y, u) + D (x, u, z) + D (u, y, z) for every x, y, z, u ∈ U (tetrahedral inequality).

A D-metric space is a pair (U, D), where D is a D-metric on U.

Example 2.5. [6] For $x, y, z \in ℝ$ define $D_{1} (x, y, z) = | x - y | + | x - z | + | y - z |$ $D_{\infty} (x, y, z) = \max {| x - y |, | x - z |, | y - z |}$ then $(ℝ, D_{1})$ and $(ℝ, D_{\infty})$ are D-metric spaces.

3 Proposed methodology

In this section, we present some technical details which are necessary for developing the algorithms. We set n to be an element of the initial universe U and s (E) to denote the number of elements of the parameter set E.

Definition 3.1. Let (F, E) be a soft set over U. Here, F is a mapping defined by F : E → P (U). Furthermore, let $f : U \to 𝕀^{s (E)}$ be another mapping which maps each element of the initial universe U to the parameter set E with $𝕀 = {0, 1}$ . The mapping f is defined for all u ∈ U as follows: (1 ≤ k ≤ n and 1 ≤ i ≤ s (E)) $f (u_{k}) = (f (u_{k})_{1}, f (u_{k})_{2}, . . ., f (u_{k})_{s (E)})$ (3.1) $f (u_{k})_{i} = {\begin{matrix} 1, & uk ∈ F(ei) \\ 0, & uk ∉ F(ei) \end{matrix}$ (3.2) Here, the value f (u) is called the “object code” for the objects and “OC” is used throughout the study as an abbreviation for the “object code”.

Definition 3.2. It is possible to assign different significance weight for each parameter. In our toy example given in 1.1, Mr. X might determine a significance weight for each parameter. Therefore, the "parameter significance weight" of Mr. X is expressed with the help of a vector in the $ℝ^{s (E)}$ . This vector is given as follows: ${PSW}_{X} = (p_{1}^{X}, p_{2}^{X}, . . . p_{s (E)}^{X})$ (3.3)

Throughout the study, we denote the "parameter significance weight" in short with "PSW". For example, the value of "PSW" for Mr. X is denoted by "PSW_X".

Furthermore, the values of PSW of k different individual are given as follows: ${PSW}_{1} = (p_{1}^{1}, p_{2}^{1}, . . . p_{s (E)}^{1})$ ${PSW}_{2} = (p_{1}^{2}, p_{2}^{2}, . . . p_{s (E)}^{2})$ $\dots$ ${PSW}_{k} = (p_{1}^{k}, p_{2}^{k}, . . . p_{s (E)}^{k})$

Remark 3.3. Note that the significance weights defined for each parameter have to fall into the closed interval [0, 1], i.e. $p_{i}^{X} \in [0, 1]$ . If the significance weights are not within the desired range, the normalization process for each parameter should be applied as follows: ${\tilde{p}}_{i}^{X} = \frac{p_{i}^{X} - P_{X}^{-}}{P_{X}^{+} - P_{X}^{-}}$ (3.4) where $\max_{1 \leq i \leq s (E)} (p_{i}^{X}) = P_{X}^{+}$ and $\min_{1 \leq i \leq s (E)} (p_{i}^{X}) = P_{X}^{-}$ .

Assume that, a decision group has K decision-makers, then we get a common PSW_NET value for all these values as follows: ${PSW}_{NET} = (p_{1}^{NET}, p_{2}^{NET}, . . ., p_{s (E)}^{NET})$ (3.5) $p_{i}^{NET} = \frac{1}{K} \sum_{m = 1}^{K} {\tilde{p}}_{i}^{m}$ (3.6)

Definition 3.4. In order to obtain the "personal object code" which is denoted by ${POC}_{X}^{k}$ , we multiply the vector value "OC" for each object in the initial universe and the "PSW" vector value of an individual. Note that, these vector values are scalars. Eventually, we get ${POC}_{X}^{k} = (c_{1}^{X, k}, c_{2}^{X, k}, . . ., c_{s (E)}^{X, k})$ (3.7) where $c_{i}^{X, k} = f (u_{k})_{i} \times p_{i}^{X}$ for all 1 ≤ i ≤ s (E) and 1 ≤ k ≤ n.

Remark 3.5. If a person specifies a significance weight for each element in the parameter set, a POC code of this person can be obtained.

Definition 3.6. We define the PIPOC (positive-ideal POC) and NIPOC (negative-ideal POC) vectors as follows $PIPOC = (1, 1, . . ., 1)$ $NIPOC = (0, 0, . . ., 0)$ where $PIPOC, NIPOC \in ℝ^{s (E)}$ .

Definition 3.7. We define the function $d : ℝ^{s (E)} \times ℝ^{s (E)} \to ℝ$ to compute the metric distance of the ${POC}_{X}^{k}$ value from the PIPOC and NIPOC vectors. For this purpose, we use the metrics defined as follows: $d ({POC}_{X}^{k}, PIPOC) = \sum_{i = 1}^{s (E)} | c_{i}^{X, k} - 1 |$ (3.8) $d_{p} ({POC}_{X}^{k}, PIPOC) = [\sum_{i = 1}^{s (E)} | c_{i}^{X, k} - 1 |^{p}]^{\frac{1}{p}}$ (3.9)

$d_{\infty} ({POC}_{X}^{k}, PIPOC) = \max {| c_{i}^{X, k} - 1 | : 1 \leq i \leq s (E)}$ (3.10) $d ({POC}_{X}^{k}, NIPOC) = \sum_{i = 1}^{s (E)} | c_{i}^{X, k} - 0 |$ (3.11)

$d_{p} ({POC}_{X}^{k}, NIPOC) = [\sum_{i = 1}^{s (E)} | c_{i}^{X, k} - 0 |^{p}]^{\frac{1}{p}}$ (3.12) $d_{\infty} ({POC}_{X}^{k}, NIPOC) = \max {| c_{i}^{X, k} - 0 | : 1 \leq i \leq s (E)}$ (3.13)

for 1 ≤ k ≤ n.

In the meantime, we define the parametric distance, which is abbreviated by "PD", between objects using the ${POC}_{X}^{k}$ values.

Definition 3.8. (Binary parametric distance) The function $d : ℝ^{s (E)} \times ℝ^{s (E)} \to ℝ$ is defined to obtain the "binary parametric distance (BPD)" metric which measures the distance of the ${POC}_{X}^{t}$ value from the ${POC}_{X}^{k}$ . Consequently, the relevant metrics are defined as follows: $d ({POC}_{X}^{k}, {POC}_{X}^{t}) = \sum_{i = 1}^{s (E)} | c_{i}^{X, k} - c_{i}^{X, t} |$ (3.14) $d_{p} ({POC}_{X}^{k}, {POC}_{X}^{t}) = [\sum_{i = 1}^{s (E)} | c_{i}^{X, k} - c_{i}^{X, t} |^{p}]^{\frac{1}{p}}$ (3.15) $d_{\infty} ({POC}_{X}^{k}, {POC}_{X}^{t}) = \max {| c_{i}^{X, k} - c_{i}^{X, t} | : 1 \leq i \leq s (E)}$ (3.16)

for 1 ≤ k, t ≤ n.

Finally, we define the necessary metrics to measure the triple parametric distance among the POC values.

Definition 3.9. (Triple parametric distance) Let $d : ℝ^{s (E)} \times ℝ^{s (E)} \times ℝ^{s (E)} \to ℝ$ be a function to define the D-metric which measures the "triple parametric distance (TPD)" of ${POC}_{X}^{k}, {POC}_{X}^{t}$ and ${POC}_{X}^{l}$ values. In this regard, we use the D-metrics defined as follows: $\begin{matrix} D ({POC}_{X}^{k}, {POC}_{X}^{t}, {POC}_{X}^{l}) = \sum_{i = 1}^{s (E)} | c_{i}^{X, k} - c_{i}^{X, t} | \\ + \sum_{i = 1}^{s (E)} | c_{i}^{X, k} - c_{i}^{X, l} | + \sum_{i = 1}^{s (E)} | c_{i}^{X, t} - c_{i}^{X, l} | \end{matrix}$ (3.17) $\begin{matrix} D_{\infty} ({POC}_{X}^{k}, {POC}_{X}^{t}, {POC}_{X}^{l}) = \max {| c_{i}^{X, k} - c_{i}^{X, t} |, \\ | c_{i}^{X, k} - c_{i}^{X, l} |, | c_{i}^{X, t} - c_{i}^{X, l} | : 1 \leq i \leq s (E)} \end{matrix}$ (3.18) for 1 ≤ k, t, l ≤ n.

4 Multi-attribute decision making applications

In this section, we present the algorithms which are based on the soft set theory. We begin with an algorithm which can be used to solve the problem of selecting the best object in the given universe set.

Algorithm 1.

Step 1: Choose a soft set on E for V ⊆ U and determine how many elements are contained in the parameter set.

Step 2: Find the "OC" value of each object using the selected soft set.

Step 3: Select a particular element, e.g. Mr.X, and calculate its "PSW_X" value.

Step 4: Obtain the " ${POC}_{X}^{k}$ " value by scalar multiplication of "OC" and "PSW_X" values.

Step 5: Find the metric distance of the " ${POC}_{X}^{k}$ " values against the PIPOC or NIPOC values.

Step 6: Select the object with the smallest distance to the PIPOC value in the decision-making phase. Equivalently, select the object with the greatest distance to the NIPOC value.

With the procedure given in Algorithm 1, one can obtain the best option for a given decision making problem. Now, we present an example which illustrates the application of this algorithm.

Example 4.1. Assume that a food company aims to hire a chef due to the increase in the number of customers in a catering company. For this purpose, the food company publishes a job advertisement and arranges a recruitment interview. Every chef applying for the job advertisement is asked to show his/her skills. A jury consisted of three people evaluates the dishes made by the chefs and tries to select the most suitable candidate.

We first define the universe as U = {u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀} and the parameter set as E = {e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀} and the subset V of the universe as V = {u₂, u₃, u₆, u₈, u₉} ⊆ U. For i = 1, 2, . . . , 10, the parameters e_i stand for "experience", "food presentation", "foreign language knowledge", "effective speech", "working discipline", "hat trick", "higher education", "young age", "marriage status" and "presentation skills", respectively.

Step 1: If A = {e₂, e₃, e₅, e₇, e₈, e₉, e₁₀} ⊆ E, F (e₂) = {u₂, u₆, u₉}, F (e₃) = {u₃, u₈, u₉}, F (e₅) = {u₃, u₆, u₈}, F (e₇) = {u₂, u₃}, F (e₈) = {u₆}, F (e₉) = {u₂, u₉} and F (e₁₀) = {u₃, u₈}, then the soft set (F, A) is given by

$(F, A) = {(e_{2}, {u_{2}, u_{6}, u_{9}}), (e_{3}, {u_{3}, u_{8}, u_{9}}), (e_{5}, {u_{3}, u_{6}, u_{8}}), (e_{7}, {u_{2}, u_{3}}),$ $(e_{8}, {u_{6}}), (e_{9}, {u_{2}, u_{9}}), (e_{1} 0, {u_{3}, u_{8}})}$ .

Step 2: We calculate "OC" values for each object in the selected soft set.

Table 3
The values of "OC" for all u_k ∈ V

Objects f (u_k) ₁ f (u_k) ₂ f (u_k) ₃ f (u_k) ₄ f (u_k) ₅ f (u_k) ₆ f (u_k) ₇

u ₂ 1 0 0 1 0 1 0

u ₃ 0 1 1 1 0 0 1

u ₆ 1 0 1 0 1 0 0

u ₈ 0 1 1 0 0 0 1

u ₉ 1 1 0 0 0 1 0

Step 3: For each parameter, three decision-makers express the most appropriate PSW values which are given in the following: ${PSW}_{1} = (5.5, 4.8, 3.8, 7.5, 3.4, 2.3, 6.6)$ ${PSW}_{2} = (0.65, 0.75, 0.43, 0.52, 0.21, 1, 0.88)$ ${PSW}_{3} = (0.4, 0.8, 0.4, 0.92, 1, 0.56, 0.28)$

We recall that the components in all PSW_X values must lie in the range [0, 1]. However, the values for PSW₁ are out of the intended interval. Therefore, PSW₁ values have to be normalized with a normalization procedure. For the first component, the process, which is given below, should be applied to each component:

${\tilde{p}}_{1}^{1} = \frac{p_{1}^{1} - P_{1}^{-}}{P_{1}^{+} - P_{1}^{-}} = \frac{5.5 - 2.3}{7.5 - 2.3} = 0.62$

where $\max_{1 \leq i \leq s (A)} (p_{i}^{1}) = P_{1}^{+} = 7.5$ and $\min_{1 \leq i \leq s (A)} (p_{i}^{1}) = P_{1}^{-} = 2.3$ . Thus, the normalized values of PSW₁ are calculated as follows: ${PSW}_{1} = (0.62, 0.48, 0.28, 1, 0.21, 0, 0.827)$

To achieve the best result by making a common evaluation, we calculate the PSW_NET value as: ${PSW}_{NET} = (0.56, 0.68, 0.372, 0.813, 0.473, 0.52, 0.662)$

For example, the first component is calculated as follows: $p_{1}^{NET} = \frac{1}{3} \sum_{m = 1}^{3} {\tilde{p}}_{1}^{m} = \frac{0.62 + 0.65 + 0.4}{3} = 0.56$

Step 4: To calculate the " ${POC}_{NET}^{k}$ " code of each object, we obtain the scalar multiplication of "OC" and "PSW_NET" values. The following table shows the calculated values: (for k = 2, 3, 6, 8, 9)

Table 4

The values of ${POC}_{NET}^{k}$ for all u_k ∈ V

Objects	$c_{1}^{NET, k}$	$c_{2}^{NET, k}$	$c_{3}^{NET, k}$	$c_{4}^{NET, k}$	$c_{5}^{NET, k}$	$c_{6}^{NET, k}$	$c_{7}^{NET, k}$
u ₂	0.56	0	0	0.813	0	0.52	0
u ₃	0	0.68	0.372	0.813	0	0	0.662
u ₆	0.56	0	0.372	0	0.473	0	0
u ₈	0	0.68	0.372	0	0	0	0.662
u ₉	0.56	0.7	0	0	0	0.52	0

To elaborate the calculation of these values, we present the case for the ${POC}_{NET}^{3}$ value. Note that, the others can be obtained in a similar manner. The calculation of the ${POC}_{NET}^{3}$ value is done with the following procedure: $c_{1}^{NET, 3} = f (u_{3})_{1} \times p_{1}^{NET} = 0 \times 0.56 = 0$ $c_{2}^{NET, 3} = f (u_{3})_{2} \times p_{2}^{NET} = 1 \times 0.68 = 0.68$ $c_{3}^{NET, 3} = f (u_{3})_{3} \times p_{3}^{NET} = 0 \times 0.372 = 0.372$ $c_{4}^{NET, 3} = f (u_{3})_{4} \times p_{4}^{NET} = 1 \times 0.813 = 0.813$ $c_{5}^{NET, 3} = f (u_{3})_{5} \times p_{5}^{NET} = 0 \times 0.473 = 0$ $c_{6}^{NET, 3} = f (u_{3})_{6} \times p_{6}^{NET} = 0 \times 0.52 = 0$ $c_{7}^{NET, 3} = f (u_{3})_{7} \times p_{7}^{NET} = 1 \times 0.662 = 0.662$

Step 5: We calculate how much the POC values deviated from the ideal values. The result is given in the following:

Table 5

The values of $d ({POC}_{NET}^{k}, PIPOC)$ and $d ({POC}_{NET}^{k}, NIPOC)$ for k = 2, 3, 6, 8, 9

Objects	$d ({POC}_{NET}^{k}, PIPOC)$	Objects	$d ({POC}_{NET}^{k}, NIPOC)$
u ₂	5.107	u ₂	1.893
u ₃	4.473	u ₃	2.527
u ₆	5.595	u ₆	1.405
u ₈	5.286	u ₈	1.714
u ₉	5.24	u ₉	1.76

For example, using the ${POC}_{NET}^{2}$ value obtained for u₂, we calculate the deviation as follows:

$d ({POC}_{NET}^{2}, PIPOC) = \sum_{i = 1}^{7} | c_{i}^{NET, 2} - 1 | = | 0 - 1 | + | 0.68 - 1 | + | 0.372 - 1 | + | 0.81 - 1 | +$ $| 0 - 1 | + | 0 - 1 | + | 0.662 - 1 | = 4.473$ $d ({POC}_{NET}^{2}, NIPOC) = \sum_{i = 1}^{7} | c_{i}^{NET, 2} - 0 | = | 0 - 0 | + | 0.68 - 0 | + | 0.372 - 0 | + | 0.81 - 0 | +$ $| 0 - 0 | + | 0 - 0 | + | 0.662 - 0 | = 2.527$

Step 6: Finally, we evaluate the candidates according to the amount of deviation calculated on Step 5.

According to the results obtained using the PIPOS value, the smallest value is attained by the candidate u₂. In addition, according to the results obtained using the NIPOS value, the highest value is the candidate u₃. It is therefore appropriate to choose the candidate u₃.

Now we give an algorithm which can be used to solve the problem of selecting the two most compatible objects in the given universe set.

Algorithm 2.

The first four steps given in Algorithm 1 are used again.

Step 5: Assume that the number of objects is n. Therefore, the dimension of the ${POC}_{NET}^{k}$ vector is n. So, one gets as much value as the binomial choices binomn2. Select the appropriate metric. For instance, we used the metric $d ({POC}_{X}^{k}, {POC}_{X}^{t}) = \sum_{i = 1}^{s (E)} | c_{i}^{X, k} - c_{i}^{X, t} |$ in this study. However, any other metric given in the previous section can be used at this stage.

Step 6: Compare the results of the binary parametric distance of two objects. If the smallest value is $d ({POC}_{X}^{k}, {POC}_{X}^{t})$ , then we deduce that the objects k and t are the most compatible ones. So, we suggest choosing these two objects.

Example 4.2. Consider the decision making problem given in Example 4.1. However, instead of choosing the best candidate, the food company might choose two candidates that can work in the most harmonious manner.

Initially, we apply the first four steps given in Algorithm 1. Further, we use Algorithm 4 to solve the remainder of the problem.

Step 5: After the calculations addressed by Algorithm 2, we obtain the results given below:

Table 6

The values of binary parametric distance for all u_k ∈ V

BPD	u ₂	u ₃	u ₆	u ₈	u ₉
u ₂	0	2.794	2.178	3.607	1.493
u ₃	2.794	0	3.188	0.813	2.927
u ₆	2.178	3.188	0	2.375	2.045
u ₈	3.607	0.813	2.375	0	2.114
u ₉	1.493	2.927	2.045	2.114	0

The calculation of the binary parametric distance between u₃ and u₆ is conducted as follows: $d ({POC}_{NET}^{3}, {POC}_{NET}^{6}) = \sum_{i = 1}^{7} | c_{i}^{NET, 3} - c_{i}^{NET, 6} | = | 0 - 0.56 | + | 0.68 - 0 | + | 0.372 - 0.372 | +$ $+ | 0.813 - 0 | + | 0 - 0.473 | + | 0 - 0 | + | 0.662 - 0 | = 3.188$

Step 6: According to the results, the smallest value is achieved by the metric $d ({POC}_{NET}^{3}, {POC}_{NET}^{8}) = 0.813$ . Thus, the food company should hire the candidates u₃ and u₈, if they want to select the most compatible pair.

Finally, we give an algorithm which can be used to solve the problem of selecting the three most compatible objects in the given universe.

Algorithm 3.

The first four steps of this algorithm are identical with the ones given in Algorithm 1.

Step 5: Assume that the number of objects is n. Therefore, the dimension of the ${POC}_{NET}^{k}$ vector is n. So, one gets as much value as the binomial choices binomn3. Select the appropriate metric. For instance, we used the D-metric $D ({POC}_{X}^{k}, {POC}_{X}^{t}, {POC}_{X}^{l}) = \sum_{i = 1}^{s (E)} | c_{i}^{X, k} - c_{i}^{X, t} | + \sum_{i = 1}^{s (E)} | c_{i}^{X, k} - c_{i}^{X, l} | + \sum_{i = 1}^{s (E)} | c_{i}^{X, t} - c_{i}^{X, l} |$ in this study. Note again that, any other D-metric given in the previous section can be used at this stage.

Step 6: Compare the results of the triple parametric distance among the objects. If the smallest value is $D ({POC}_{X}^{k}, {POC}_{X}^{t}, {POC}_{X}^{l})$ , then it is implied that the objects k, t and l are the most compatible ones. So, we suggest choosing these three objects.

Example 4.3. We again consider the decision making problem given in Example 4.1. However, this time instead of choosing the best candidate, we assume that the food company wants to choose three candidates who are compatible to work in the most harmonious manner.

After applying the first four steps given in Algorithm 1, we proceed with the remaining steps of Algorithm 4 to further solve the problem.

Step 5: We calculate the values given in the following table by using Algorithm 3:

Table 7

The values of triple parametric distance for all u_k ∈ V

TPD	u ₂	u ₃	u ₆	u ₈	u ₉	TPD
u ₂	1.493	7.214	5.716	7.214	1.493	u ₉
u ₃	2.794	2.794	8.16	7.214	7.214	u ₂
u ₆	8.16	3.188	3.188	6.376	8.16	u ₃
u ₈	8.16	6.376	2.375	2.375	6.534	u ₆
u ₉	7.214	5.854	6.534	2.114	2.114	u ₈

In order to obtain the triple parametric distance between u₂, u₃ and u₆, we do the following calculations: $D ({POC}_{NET}^{2}, {POC}_{NET}^{3}, {POC}_{NET}^{6}) =$ $\sum_{i = 1}^{7} | c_{i}^{NET, 2} - c_{i}^{NET, 3} | + \sum_{i = 1}^{7} | c_{i}^{NET, 2} - c_{i}^{NET, 6} | +$ $\sum_{i = 1}^{7} | c_{i}^{NET, 3} - c_{i}^{NET, 6} | = 2.794 + 2.178 + 3.188 = 8.16$ L

Step 6: The results indicate that the smallest value is achieved by the metric $D ({POC}_{NET}^{2}, {POC}_{NET}^{6}, {POC}_{NET}^{9}) = 5.716 .$ As a result, it is recommended that the food company should hire the candidates u₂, u₆ and u₉, if they aim to select the most compatible triplet.

5 Conclusion

In this paper, we have introduced a novel approach to address uncertainty in decision-making problems by leveraging soft set theory. Our contributions include the development of algorithms for selecting the best candidate, best pair, and best triplet, along with practical examples illustrating their application. Our primary objective has been to identify alternatives that form a compatible group of objects for optimal outcomes, distinguishing our approach from those solely focused on selecting the single best alternative. However, it’s crucial to acknowledge that our method is not without limitations. Its effectiveness hinges on the precise modeling of uncertainty using soft set theory, which may not be universally applicable, and its performance could be influenced by the complexity of decision scenarios and data availability.

Looking ahead, several promising research directions emerge. Firstly, further research should investigate more advanced algorithms capable of handling complex decision scenarios with higher dimensions and intricate constraints. Additionally, exploring the integration of our approach with other decision support frameworks to create hybrid methods could leverage the strengths of multiple theories. Empirical studies and real-world applications across various decision-making contexts are essential to assess the practicality and effectiveness of our approach. Developing methods to quantify uncertainty in decision problems will enable more informed choices, and creating user-friendly software tools or decision support systems will make the proposed approach accessible to a broader audience. By addressing these limitations and pursuing these research directions, we can refine and expand the application of our approach in decision-making, providing decision-makers with robust tools for handling uncertainty and selecting optimal alternatives.

Footnotes

Acknowledgments

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. GRANT4356].

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of interest

The authors declare no competing interests.

References

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Bellman

R.E.

and Zadeh

L.A.

, Decision-making in a fuzzy environment, Management Sci 17 (1970), 141–164.

Celik

and Yamak

, Fuzzy soft set theory applied to medical diagnosis using fuzzy arithmetic operations, Journal of Inequalities and Applications 82 (2013), 1–9.

Chen-Tung

, Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets and Systems 114 (2000), 1–9.

Delgado

, Verdegay

J.L.

and Vila

M.A.

, Linguistic decision-making models, Int J Intelligent System 7 (1992), 479–492.

Dhage

B.C.

, Generalized metric spaces and topological structure II, Pure Appl. Math. Sci. 40 (1994), 37–412.

Feng

, Soft rough sets applied to multicriteria group decision making, Ann Fuzzy Math Inform 2 (2013), 69–80.

Herrera

, Herrera-Viedma

and Verdegay

J.L.

, A model of consensus in group decision making under linguistic assessments, Fuzzy Sets and Systems 78 (1996), 73–87.

Hsu

H.M.

and Chen

C.T.

, Fuzzy credibility relation method for multiple criteria decision-making problems, Inform. Sci. 96 (1997), 79–91.

10.

Demirtaş

and Dalkılıç

, An application in the diagnosis of prostate cancer with the help of bipolar soft rough sets, on Mathematics and Mathematics Education (ICMME 2019), KONYA, 283, 2019.

11.

Demirtaş

, Hussaın

and Dalkılış

, New approaches of inverse soft rough sets and their applications in a decision making problem, Journal of Applied Mathematics and Informatics 38(3–4) (2020), 335–349.

12.

Dalkılıç

, An Application of VFPFSS’s in Decision Making Problems, Journal of Polytechnic, https://doi.org/10.2339/politeknik.758474, 2020.

13.

Dalkılıç

and Demirtas

, VFP-Soft Sets and Its Application on Decision Making Problems, Journal of Polytechnic, https://doi.org/10.2339/politeknik.685634, 2020.

14.

Khalil

A.M.

, Li

S.G.

, Lin

, Li

H.X.

and Ma

S.G.

, A new expert system in prediction of lung cancer disease based on fuzzy soft sets, Soft Computing (2020), 1–29.

15.

Demirtaş

and Dalkılıç

, Decompositions of Soft α-continuity and Soft A-continuity, Journal of New Theory (31) (2020), 86–94.

16.

Dalkılıç

and Demirtaş

, Bipolar soft filter, Journal of Universal Mathematics 3(1) (2020), 21–27.

17.

Hwang

C.L.

, Yoon

, Multiple Attributes Decision Making Methods and Applications, Springer, Berlin Heidelberg, 1981.

18.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Comput. Math. Appl. 45 (2003), 555–562.

19.

Molodtsov

, Soft set theory first results, Comput. Math. Appl. 37 (1999), 19–31.

20.

Naidu

S.V.R.

, Rao

K.P.R.

and Srinivasa

, Rao, On the topology of D-metric spaces and the generation of D-metric spaces from metric spaces, Int. J. Math. Sci. 51 (2004), 2719–2740.

21.

Pawlak

, Rough sets, Int. J. Inf. Comp. Sci. 11 (1982), 341–356.

22.

Yuksel

, Tozlu

and Tuğba Han

, An Application of Multicriteria Group Decision Making by Soft Covering Based Rough Sets, Filomat 29 (2015), 209–219.

23.

Zadeh

L.A.

, Fuzzy sets, Inform Control 8 (1965), 338–353.

24.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning, Inform Sci. 8 (1975), 199–249(I), 301-357(II).

25.

Selvachandran

and Peng

, A modified TOPSIS method based on vague parameterized vague soft sets and its application to supplier selection problems, Neural Computing and Applications 10 (2019), 5901–5916.

26.

Al-Quran

, Al-Sharqi

, Ullah

, Romdhini

M.U.

, Balti

and Alomair

, Bipolar fuzzy hypersoft set and its application in decision making, International Journal of Neutrosophic Science 20 (2023), 65–77.

27.

Khan

M.J.

, Kumam

, Ashraf

and Kumam

, Generalized picture fuzzy soft sets and their application in decision support systems, Symmetry 11(3) (2019), 415.

28.

Jan

, Mahmood

, Zedam

and Ali

, Multi-valued picture fuzzy soft sets and their applications in group decision-making problems, Soft Computing 24(24) (2020), 18857–18879.

29.

Mahmood

, A novel approach towards bipolar soft sets and their applications, Journal of Mathematics (2020).

30.

Al-Quran

and Hassan

, Neutrosophic vague soft multiset for decision under uncertainty, Songklanakarin Journal of Science and Technology 40 (2018), 290–305.

31.

Mukherjee

and Das

, Neutrosophic bipolar vague soft set and its application to decision making problems, Neutrosophic Sets and Systems 32(1) (2020), 2020.

32.

Kamal

N.L.A.M.

, Abdullah

, Alkhazaleh

and Karaaslan

, Multi-valued interval neutrosophic soft set: Formulation and Theory, Neutrosophic Sets and Systems 30(1) (2019), 12.

33.

Al-Sharqi

, Al-Quran

, Ahmad

A.G.

and Broumi

, Interval-valued complex neutrosophic soft set and its applications in decision-making, Neutrosophic Sets and Systems 40 (2021), 149–168.

34.

Abu Qamar

and Hassan

, An approach toward a Q-neutrosophic soft set and its application in decision making, Symmetry 11(2) (2019), 139.

35.

Gündoğdu

F.K.

and Kahraman

, A novel fuzzy TOPSIS method using emerging interval-valued spherical fuzzy sets, Engineering Applications of Artificial Intelligence 85 (2019), 307–323.

36.

Rashidi

and Cullinane

, A comparison of fuzzy DEA and fuzzy TOPSIS in sustainable supplier selection: Implications for sourcing strategy, Expert Systems with Applications 121 (2019), 266–281.

37.

Zhang

, Zhan

and Yao

, Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: an application to biomedical problems, Information Sciences 517 (2020), 315–339.

38.

Yazdi

, Korhan

and Daneshvar

, Application of fuzzy fault tree analysis based on modified fuzzy AHP and fuzzy TOPSIS for fire and explosion in the process industry, International Journal of Occupational Safety and Ergonomics 26(2) (2020), 319–335.

Objects	f (u_k) ₁	f (u_k) ₂	f (u_k) ₃	f (u_k) ₄	f (u_k) ₅	f (u_k) ₆	f (u_k) ₇
u ₂	1	0	0	1	0	1	0
u ₃	0	1	1	1	0	0	1
u ₆	1	0	1	0	1	0	0
u ₈	0	1	1	0	0	0	1
u ₉	1	1	0	0	0	1	0

Mathematical analysis of parameters belonging to the universe in the soft set theory with new distance measures

Abstract

Introduction:

Methods:

Results and Conclusions:

Keywords

1 Introduction

Table 1 Evaluation results for each gear Sequence First Gear Second Gear A,D 10 2 B,D 10 4 C,D 10 3

3 Proposed methodology

Table 3 The values of "OC" for all u k ∈ V Objects f (u k ) 1 f (u k ) 2 f (u k ) 3 f (u k ) 4 f (u k ) 5 f (u k ) 6 f (u k ) 7 u 2 1 0 0 1 0 1 0 u 3 0 1 1 1 0 0 1 u 6 1 0 1 0 1 0 0 u 8 0 1 1 0 0 0 1 u 9 1 1 0 0 0 1 0

Footnotes

Acknowledgments

Data availability

Conflict of interest

References

Table 1
Evaluation results for each gear

Sequence First Gear Second Gear

A,D 10 2

B,D 10 4

C,D 10 3

Table 3
The values of "OC" for all u_k ∈ V

Objects f (u_k) ₁ f (u_k) ₂ f (u_k) ₃ f (u_k) ₄ f (u_k) ₅ f (u_k) ₆ f (u_k) ₇

u ₂ 1 0 0 1 0 1 0

u ₃ 0 1 1 1 0 0 1

u ₆ 1 0 1 0 1 0 0

u ₈ 0 1 1 0 0 0 1

u ₉ 1 1 0 0 0 1 0