1. Introduction
There are many decisionmaking issues in physical sciences, applied sciences, social sciences, and life sciences often containing datasets having uncertain and vague information. Fuzzy set theory [
1] and rough set theory [
2] are classical mathematical tools to characterize uncertain and inaccurate data, but as indicated in [
3,
4], each of these theories lacks theoretical parametric tools. Molodtsov [
3] initiated the concept of the soft set as a new mathematical tool for handling vague and uncertain information. Molodtsov [
3] efficiently implemented soft set theory in multiple directions, for example operations research, game theory, Riemann integral, and probability. Currently, the research on soft set theory is proceeding rapidly and has achieved many fruitful results. Some fundamental algebraic properties of soft sets were proposed by Maji et al. [
5]. Feng et al. [
6] proposed new hybrid models by combining fuzzy sets, rough sets, and soft sets. Maji and Roy [
7] presented a technique to solve decisionmaking problembased fuzzy soft sets. Xiao et al. [
8] developed a forecasting method based the fuzzy soft set model.
Henceforth, much research on parameter reduction has been completed, and several results have been derived [
9,
10,
11,
12,
13,
14]. Research based on soft set theory to solve decisionmaking problems derives from the concept of parameter reduction. The reduction of parameters in soft set theory is designed to remove redundant parameters while preserving the original decision choices. Maji et al. [
15] first solved soft set decisionmaking problems using rough setbased reduction [
16]. To improve decisionmaking problems in [
15], Chen et al. [
17] and Kong et al. [
18] respectively proposed parameterization reduction and normal parameter reduction of soft sets. Ma et al. [
19] proposed a new efficient algorithm of normal parameter reduction to improve [
15,
16,
17]. Roy and Maji [
7] proposed a new method for dealing with decisionmaking problems with fuzzy soft sets. The method deals with a comparison table derived from a fuzzy soft set in the sense of parameters to make a decision. Kong et al. [
20] indicated that the Roy and Maji technique [
7] was inaccurate, and they proposed a modified approach to solve this issue. They described the effectiveness of the Roy and Maji technique [
7] and demonstrated its boundaries. Ma et al. [
21] proposed extensive parameter reduction methods for intervalvalued fuzzy soft sets. Decisionmaking research for the reduction of fuzzy soft sets has been given considerable attention. Using the idea of the level soft set, Feng et al. [
22] gave the idea of the parameter reduction of fuzzy soft sets and proposed an adaptable method to decisionmaking based on fuzzy soft sets. Moreover, Feng et al. [
23] presented another intuition about decisionmakingbased intervalvalued fuzzy soft sets. Jiang et al. [
24] proposed a reduction method of intuitionistic fuzzy soft sets for decisionmaking using the level soft sets of intuitionistic fuzzy sets. The theory of fuzzy systems has rich applications in different areas, including engineering [
25,
26,
27]. Zhang [
28,
29] first proposed the idea of bipolar fuzzy sets (Yin Yang bipolar fuzzy sets) in the space
$\{\forall \phantom{\rule{3.33333pt}{0ex}}(x,y\left)\phantom{\rule{3.33333pt}{0ex}}\right\phantom{\rule{3.33333pt}{0ex}}(x,y)\in [1,0]\times [0,1]\}$ as an extension of fuzzy sets. In the case of bipolar fuzzy sets, membership degree range is enlarged from the interval
$[0,1]$–
$[1,1]$. The idea behind this description is related to the existence of bipolar information. For example, profit and loss, feedback and feedforward, competition and cooperation, etc., are usually two aspects of decisionmaking. In Chinese medicine, Yin and Yang are the two sides. Yin is the negative side of a system, and Yang is the positive side of a system. Bipolar fuzzy set theory has many applications in different fields, including pattern recognition and machine learning. Saleem et al. [
30] presented a new hybrid model, namely bipolar fuzzy soft sets, by combining bipolar fuzzy sets with soft sets. Motivated by these concerns, in this paper, we present four ways to reduce parameters in bipolar fuzzy soft sets by developing another bipolar fuzzy soft set theoretical approach to solve decisionmaking problems. In particular, we solve the decisionmaking problem in [
30] by our proposed decisionmaking algorithmbased bipolar fuzzy soft sets. We propose an algorithm of each reduction technique. Furthermore, we compare these reduction methods and discuss their pros and cons in detail. We also present a reallife application to show the validity of our proposed reduction algorithms. For other terminologies not mentioned in the paper, the readers are referred to [
31,
32,
33,
34,
35,
36,
37,
38,
39].
The rest of this paper is structured as follows.
Section 2 introduces the basic definitions and develops a new technique for decisionmakingbased bipolar fuzzy soft sets.
Section 3 defines four kinds of parameter reductions of bipolar fuzzy soft sets and presents their reduction algorithms, which are illustrated by corresponding examples. A comparison among the reduction algorithms is presented in
Section 4.
Section 5 is devoted to solving a reallife decisionmaking application. In the end, the conclusions of this paper are provided in
Section 6. Throughout this paper, the following notations given in
Table 1 will be used.
2. Another Bipolar Fuzzy Soft Sets Approach to DecisionMaking Problems
Saleem et al. [
30] presented an efficient approach to solve practical decisionmaking problems based on bipolar fuzzy soft sets. In this section, we first review the definitions of bipolar fuzzy sets and bipolar fuzzy soft sets, and then, we introduce a novel approach based on bipolar fuzzy soft sets, which can effectively solve decisionmaking problems, followed by an algorithm. Moreover, we use our proposed algorithm to solve the decisionmaking application presented by Saleem et al. [
30] and observe that the optimal decisions obtained by both methods are the same.
Definition 1. [29,40] Let O be a nonempty universe of objects. A bipolar fuzzy set B in O is defined as:where ${\lambda}^{+}:O\to [0,1]$ and ${\lambda}^{}:O\to [1,0]$ are mappings. The positive membership degree ${\lambda}^{+}\left(o\right)$ denotes the satisfaction degree of an object o for the property corresponding to a bipolar fuzzy set B, and the negative membership degree ${\lambda}^{}\left(o\right)$ denotes the satisfaction degree of an object o for some implicit counterproperty corresponding to a bipolar fuzzy set B. Definition 2. [30] Let O be a nonempty universe of objects and R a universe of parameters related to objects in O. A pair $(G,R)$ is called a BFSS over universe O, where G is a mapping from R into $B{F}^{O}$. It is defined as follows: Assume that
$O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ is a universe of objects and
$R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ is a universe of parameters related to objects in
O. Then, a BFSS
$(G,R)$ can also be presented by tabular arrangement, as shown in
Table 2.
Definition 3. Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ be a universe of objects and $R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ a universal set of parameters associated with objects in O. For a BFSS $(G,R)$, ${\lambda}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and ${\lambda}_{{r}_{j}}^{}\left({o}_{i}\right)$ are the membership and nonmembership degrees of each element ${o}_{i}$ to $[0,1]$ and $[1,0]$, respectively. We define the score of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for each ${r}_{j},\phantom{\rule{3.33333pt}{0ex}}(j=1,2,\dots ,m)$ as: Definition 4. Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ be a universe of objects and $R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ a universal set of parameters associated with objects in O. For a BFSS $(G,R)$, the score of the membership degrees for ${r}_{j}$$\left({b}_{{r}_{j}}\left({o}_{i}\right)\right)$ is given by:where ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ are the scores of positive and negative membership degrees for each ${r}_{j}$, respectively. Definition 5. Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ be a universe of objects, and $R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ a universal set of parameters associated with objects in O. For a BFSS $(G,R)$, the final score for each object ${o}_{i}$ denoted by ${S}_{i}$ is defined as follows: We present a new decisionmaking technique based on BFSSs as follows:
Example 1. Reconsider Example 8 in [30]. Let $O=\{{o}_{1},{o}_{2},{o}_{3},{o}_{4}\}$ be the set of four cars, $R=\{{r}_{1}=\mathrm{costly},{r}_{2}=\mathrm{beautiful},{r}_{3}=\mathrm{fuel}\phantom{\rule{3.33333pt}{0ex}}\mathrm{efficient},{r}_{4}=\mathrm{modern}\phantom{\rule{3.33333pt}{0ex}}\mathrm{technology},{r}_{5}=\mathrm{luxurious}\}$ a collection of parameters, and $Q=\{{r}_{1},{r}_{2},{r}_{5}\}\subseteq R$. Then, a BFSS $(G,Q)$ is given by Table 3. We proceed to applying Algorithm 1 to $(G,Q)$. By using (2) and (3), the scores of positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i=1,2,3,4$ and $j=1,2,5$ are given by: Now, by using Definition 4, the tabular arrangement for the score of the membership degrees of BFSS $(G,Q)$ is given by Table 6. By Definition 5, the final score of each car ${o}_{i}$ is given by Table 7. By way of illustration, Clearly, ${S}_{3}=0.7$ is the maximum score for the object ${o}_{3}$. Thus, ${o}_{3}$ is the decision object, which coincides with the decision obtained in [30]. Algorithm 1 Selection of an object based on BFSSs. 
Input O, a universal set having n objects. R, a universe of parameters with m elements. $(G,R)$, a BFSS, which is given by Definition 2. Find the score of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees, where $i=1,2,\dots ,n$ and $j=1,2,\dots ,m.$ Calculate the score of the membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ where $i=1,2,\dots ,n$ and $j=1,2,\dots ,m,$ by Definition 4. Evaluate the final score ${S}_{i}$ for each object ${o}_{i}$, $i=1,2,\dots ,n$ by Definition 5. Compute all indices l for which ${S}_{l}={max}_{i=1,2,\dots ,n}{S}_{i}$. Output The decision will be any ${o}_{l}$ corresponding to the list of indices obtained in Step 5.

Example 2. Let $O=\{{o}_{1},{o}_{2},{o}_{3},{o}_{4},{o}_{5}\}$ be a collection of five objects under consideration and $R\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\{{r}_{1},{r}_{2},{r}_{3},{r}_{4}\}$ a collection of parameters related to the objects in O. Then, a BFSS $(G,R)$ is given by Table 8. By using (2) and (3), the score of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i=1,2,\dots ,5$ and $j=1,2,3,4$ are given by Table 9 and Table 10, respectively. Now, by using Definition 4, the score of membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ for $i=1,2,\dots ,5$ and $j=1,2,3,4$ of BFSS $(G,R)$ is given by Table 11. Using $\left(5\right)$, the final score of each object ${o}_{i}$ is given by Table 12. Clearly, ${S}_{4}=5.6$ is the maximum score for the object ${o}_{4}$, which coincides with the decision object obtained using the decisionmaking algorithm in [30]. From the above analysis, it can be easily perceived that our proposed decisionmaking approach based on BFSSs is efficient and reliable. However, in a realistic perspective, it contains redundant parameters for decisionmaking. To overcome this issue, the parameter reduction of BFSS is proposed. A parameter reduction is a technique in which the set of parameters is reduced to obtain a minimal subset that gives the same decision as the whole set.
3. Four Types of Parameter Reductions of BFSSs
1. OCBPR:
We first define OCBPR and then provide an algorithmic approach to obtain it, which is illustrated via an example.
Definition 6. Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ be a universe of objects and $R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ a universal set of parameters associated with objects in O. For a BFSS $(G,R)$, denote ${M}_{R}$ as the family of objects in O, which takes the maximum value of ${S}_{i}$. For each $B\subseteq R$, if ${M}_{RB}={M}_{R}$, then B is said to be dispensable in R, else B is called indispensable in the set R. The parameter set R is called independent if every $B\subset R$ is indispensable in R, else R is dependent. A subset P of R is said to be an OCBPR of R if the following axioms hold.
 1.
P is independent (that is, P is the smallest subset of R that keeps the optimal decision object invariant).
 2.
${M}_{P}={M}_{R}$.
Based on Definition 6, we propose an OCBPR algorithm that deletes redundant parameters while keeping the optimal decision object unchanged.
Example 3. Let $O=\{{o}_{1},{o}_{2},{o}_{3},{o}_{4}\}$ be the set of four cars and $R=\{{r}_{1}=\mathrm{costly},{r}_{2}=\mathrm{beautiful},{r}_{3}=\mathrm{fuel}\mathrm{efficient},{r}_{4}=\mathrm{modern}\phantom{\rule{3.33333pt}{0ex}}\mathrm{technology},{r}_{5}=\mathrm{luxurious}\}$ a collection of parameters. Reconsider a BFSS $(G,Q)$ as in Example 8 [30], where $Q=\{{r}_{1},{r}_{2},{r}_{5}\}\subseteq R$. We proceed to applying Algorithm 2 to the BFSS $(G,Q)$. From Table 7, we compute that for $B=\{{r}_{2},{r}_{5}\}$, we obtain ${M}_{QB}={M}_{B}$. Hence, $\left\{{r}_{1}\right\}$ is an OCBPR of BFSS $(G,Q)$ given by Table 13. From Table 13, it can be easily observed that ${o}_{3}$ is the optimal decision object after reduction. Clearly, the subset $\left\{{r}_{1}\right\}\subset Q$ is minimal, which keeps the optimal decision object unchanged. Algorithm 2 OCBPR. 
Input O, a universal set having n objects. R, a universe of parameters with m elements. $(G,R)$, a BFSS, which is given by Definition 2. Calculate the score of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1,2,\dots ,n$ and $j=1,2,\dots ,m.$ Calculate the score of membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ for $i=1,2,\dots ,n$ and $j=1,2,\dots ,m$ by using $\left(4\right)$. Evaluate the final score ${S}_{i}$ for each object ${o}_{i}$, $i=1,2,\dots ,n$ by Definition 5. Compute all $B=\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}},\dots ,{r}_{p}^{{}^{\prime}}\}\subset R$ that satisfy the following condition:
Output The set $RB$ is referred as an OCBPR of BFSS $(G,R)$, if there does not exist such $B\subset R$ that they satisfy ( 6), then there is no OCBPR of BFSS $(G,R)$.

2. IRDCBPR:
There are several real situations in which our main task is to compute the rank of optimal and suboptimal choices. The suboptimal choices are not considered by the OCBPR method because OCBPR only studies the optimal choice. To overcome this drawback, we define IRDCBPR and present an algorithmic approach that keeps the rank of optimal and suboptimal choices unchanged after deleting the irrelevant parameters.
Definition 7. Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ be a universe of objects, $R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ a universe of parameters, and $P\subset R$. For a BFSS $(G,R)$, an indiscernibility relation is given by:where ${S}_{P}\left({o}_{i}\right)={\displaystyle \sum _{{r}_{s}\in P}}{b}_{{r}_{s}}\left({o}_{i}\right)$. For an arbitrary BFSS $(G,R)$ over $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$, the decision partition is given by:where for each subclass ${\{{o}_{u},{o}_{u+1},\dots ,{o}_{u+v}\}}_{{S}_{i}},\phantom{\rule{3.33333pt}{0ex}}{S}_{R}\left({o}_{u}\right)={S}_{R}\left({o}_{u+1}\right)=\dots ={S}_{R}\left({o}_{u+v}\right)={S}_{i},$ and ${S}_{1}\ge {S}_{2}\ge \dots \ge {S}_{z},$ that is, there are z subclasses. Actually, objects are ranked with respect to the score value of ${S}_{i}$, where $i=1,2,\dots ,m$. Definition 8. Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ be a universe of objects and $R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ a universal set of parameters associated with objects in O, and let $(G,R)$ be a BFSS. For each $B\subseteq R$, if ${D}_{RB}={D}_{R}$, then B is said to be dispensable in R, else B is referred to as an indispensable set in the set R. The parameter set R is said to be independent if each $B\subset R$ is indispensable in R, else R is dependent. A subset P of R is said to be an IRDCBPR of R if the following axioms hold.
 1.
P is independent (that is, P is the minimal subset of R that keeps the rank of optimal and suboptimal decision choices unchanged).
 2.
${D}_{P}={D}_{R}$.
Based on Definition 8, we propose an IRDCBPR algorithm (see Algorithm 3) that deletes irrelevant parameters while keeping the rank of optimal and suboptimal decision choice objects unchanged.
Algorithm 3 IRDCBPR. 
Input O, a universal set having n objects. R, a universe of parameters with m elements. $(G,R)$, a BFSS, which is given by Definition 2. Calculate the score of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i=1,2,\dots ,n$ and $j=1,2,\dots ,m.$ Calculate the score of membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ for $i=1,2,\dots ,n$ and $j=1,2,\dots ,m$ by using $\left(4\right)$. Evaluate the final score ${S}_{i}$ for each object ${o}_{i}$, $i=1,2,\dots ,n$ by using $\left(5\right)$. Compute all $B=\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}},\dots ,{r}_{p}^{{}^{\prime}}\}\subset R$ that satisfy the following condition:
Output The set $RB$ is referred to as an IRDCBPR of BFSS $(G,R)$, if there does not exist $B\subset R$ that satisfy ( 7), then there is no IRDCBPR of BFSS $(G,R)$.

Example 4. Let $O=\{{o}_{1},{o}_{2},{o}_{3},{o}_{4}\}$ be a universal set of four objects and $R=\{{r}_{1},{r}_{2},{r}_{3}\}$ a set of parameters related to the objects in O. Then, a BFSS $(G,R)$ is given by Table 14. By using (2) and (3), the scores of positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i=1,2,3,4$ and $j=1,2,3$ are given by Table 15 and Table 16, respectively. Now, by using Definition 4, the tabular arrangement for the score of membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ where $i=1,2,3,4$ and $j=1,2,3$ of $(G,R)$ is given by Table 17. From $\left(5\right)$, the final score of each object ${o}_{i},\phantom{\rule{3.33333pt}{0ex}}(i=1,2,3,4)$ is given by Table 18. Clearly, ${S}_{4}=3.7$ is the maximum score for the object ${o}_{4}$. Thus, ${o}_{4}$ is the optimal decision object, which coincides with the decision obtained using the algorithm in [30]. From Table 16, it can readily be computed: Using Algorithm 3, we can proceed further by examining the subsets of R. Thus, for $B=\{{r}_{1},{r}_{2}\}$, we have ${D}_{RB}=\{{\left\{{o}_{4}\right\}}_{1.5},{\left\{{o}_{3}\right\}}_{0.7},{\left\{{o}_{2}\right\}}_{0.3},{\left\{{o}_{1}\right\}}_{2.5}\}$ with ${D}_{RB}={D}_{R}$. Note that after reduction, the rank and partition of objects are not changed. Hence, $\left\{{r}_{3}\right\}$ (not all) is the IRDCBPR of BFSS $(G,R)$ given by Table 19. Clearly, $\left\{{r}_{3}\right\}\subset R$ is minimal that keeps the rank of decision choices unchanged.
3. NPR:
The parameter reduction techniques such as OCBPR and IRDCBPR are not always workable in many practical applications. Therefore, we provide the normal parameter reduction of BFSSs, which studies the issues of added parameters and suboptimal choice. We present a definition of NPR and provide an algorithmic method to obtain it, which are illustrated via an example.
Definition 9. Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ be a universe of objects and $R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ a universal set of parameters associated with objects in O. For a BFSS $(G,R)$, B is called dispensable if there exist $B=\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}},\dots ,{r}_{p}^{{}^{\prime}}\}\subset R$ that satisfy the following expression. Else, B is indispensable. A subset $P\subset R$ is called NPR of R, if the following axioms hold.
 1.
P is indispensable.
 2.
$\sum _{{r}_{j}\in RP}}{b}_{{r}_{j}}\left({o}_{1}\right)={\displaystyle \sum _{{r}_{j}\in RP}}{b}_{{r}_{j}}\left({o}_{2}\right)=\dots ={\displaystyle \sum _{{r}_{j}\in RP}}{b}_{{r}_{j}}\left({o}_{n}\right).$
Based on Definition 9, we propose the NPR algorithm (Algorithm 4) as follows.
Algorithm 4 NPR. 
Input O, a universal set having n objects. R, a universe of parameters with m elements. $(G,R)$, a BFSS, which is given by Definition 2. Calculate the score of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i=1,2,\dots ,n$ and $j=1,2,\dots ,m.$ Calculate the score of membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ for $i=1,2,\dots ,n$ and $j=1,2,\dots ,m$ by Definition 4. Evaluate the final score ${S}_{i}$ for each object ${o}_{i}$, $i=1,2,\dots ,n$ by Definition 5. For each $B\subset R$ with the cardinality $\leftR\right1$, we check if it verifies the following condition:
Output If any of these subsets verifies the condition ( 8), then we select any of their complements in R as an optimal NPR. Otherwise, for each $B\subset R$ with cardinality $\leftR\right2$, we check if it verifies the condition ( 8), then we select $RB$ as an optimal NPR, and so on; if there does not exist such $B\subset R$ that satisfy ( 8), then there is no NPR of BFSS $(G,R)$.

Example 5. Let $O=\{{o}_{1},{o}_{2},{o}_{3},{o}_{4},{o}_{5},{o}_{6}\}$ be the set of six objects and $R=\{{r}_{1},{r}_{2},{r}_{3},{r}_{4},{r}_{5}\}$ a set of parameters. Then, a BFSS $(G,R)$ is defined by Table 20. By using (2) and (3), the scores of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i=1,2,\dots ,6$ and $j=1,2,\dots ,5$ are described by Table 21 and Table 22, respectively. Now, by using Definition 9, the tabular arrangement for the score of membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ where $i=1,2,\dots ,6$ and $j=1,2,\dots ,5$ of $(G,R)$ is given by Table 23. From $\left(5\right)$, the final score of each object ${o}_{i}$ is given by Table 24. Clearly, ${S}_{6}=6.2$ is the maximum score for the object ${o}_{6}$. Thus, ${o}_{6}$ is the optimum decision object. From Table 24, it can be easily observed that $B=\{{r}_{3},{r}_{5}\}$, satisfying: Thus, $\{{r}_{1},{r}_{2},{r}_{4}\}$ (not all) is the NPR of BFSS $(G,R)$ given by Table 25. Clearly, NPR method maintains the invariable rank of decision choices, as well as takes into account immutable differences between the decision choice objects. Thus, if we add new parameters in the set of parameters, there is no need to compute new reduction again. The issue of added parameters is discussed by examples in Section 4. 4. ANPR:
NPR is an outstanding technique for the reduction of parameters. It is very difficult to compute NPR taking into account that BFSS provides bipolar information to explain membership degrees. To improve this method, we propose a new reduction method, namely ANPR, which is a compromise between IRDCBPR and NPR.
Definition 10. Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{n}\}$ be a universe of objects and $R=\{{r}_{1},{r}_{2},\dots ,{r}_{m}\}$ a universal set of parameters associated with objects in O. For a BFSS $(G,R)$, given an arbitrary error value α, if there exists $B=\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}},\dots ,{r}_{p}^{{}^{\prime}}\}\subset R$ such that:inside the range of α and ${D}_{RB}={D}_{R}$, then B is dispensable, else B is indispensable. The subset $P\subset R$ is called an ANPR of BFSS $(G,R)$, when the following three axioms hold.  1.
P is indispensable.
 2.
$\sum _{{r}_{j}\in RP}}{b}_{{r}_{j}}\left({o}_{1}\right)\approx {\displaystyle \sum _{{r}_{j}\in RP}}{b}_{{r}_{j}}\left({o}_{2}\right)\approx \dots \approx {\displaystyle \sum _{{r}_{j}\in RP}}{b}_{{r}_{j}}\left({o}_{n}\right)$ inside the range of α.
 3.
${D}_{P}={D}_{R}$.
We are ready to propose the ANPR algorithm (Algorithm 5 below):
Algorithm 5 ANPR. 
Input O, a universal set having n objects. R, a universe of parameters with m elements. $\alpha $, an error value. $(G,R)$, a BFSS, which is given by Definition 2. Calculate the score of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i=1,2,\dots ,n$ and $j=1,2,\dots ,m.$ Calculate the score of the membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ for $i=1,2,\dots ,n$ and $j=1,2,\dots ,m$ by Definition 4. Evaluate the final score ${S}_{i}$ for each object ${o}_{i}$, $i=1,2,\dots ,n$ by Definition 5. For each $B\subset R$ with the cardinality $\leftR\right1$, we check if it verifies the following expressions:
inside range of $\alpha $ and:
Output If any of these subsets verifies the conditions ( 9) and ( 10), then we select any of their complements in R as an optimal ANPR. Otherwise, for each $B\subset R$ with cardinality $\leftR\right2$, we check if it verifies the conditions ( 9) and ( 10), then we select any of their complements in R as an optimal ANPR, and so on; if there does not exist such $B\subset R$ that satisfy Conditions ( 9) and ( 10), then there is no ANPR of BFSS $(G,R)$.

As mentioned earlier, ANPR is a compromise between IRDCBPR and NPR. Note that if there is no limitation from $\alpha $ (that is, without $\alpha $), ANPR is IRDCBPR, and when $\alpha =0$, ANPR is NPR. It can be easily observed that the reduction set by ANPR relies on the outcomes of IRDCBPR and the provided range $\alpha $, because the ANPR algorithm is relying on IRDCBPR. In other words, reduction sets through ANPR are computed based on the reduction sets through IRDCBPR. If the computing difference between the highest and lowest sum of scores of reduced parameters is lower than $\alpha $, the set of reduction is referred to as parameter reduction through ANPR, else it is not referred to as the parameter reduction through ANPR. Note that the ANPR method preserves the rank of decision choices.
Example 6. Let $O=\{{o}_{1},{o}_{2},{o}_{3},{o}_{4},{o}_{5},{o}_{6}\}$ be a universal set of six objects and $R=\{{r}_{1},{r}_{2},{r}_{3},{r}_{4},{r}_{5}\}$ a set of parameters Then, a BFSS $(G,R)$ is defined by Table 26. By using (2) and (3), the scores of the positive ${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative ${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for $i=1,2,\dots ,6$ and $j=1,2,\dots ,5$ are given by Table 27 and Table 28, respectively. Now, by using Definition 4, the tabular arrangement for the score of membership degrees ${b}_{{r}_{j}}\left({o}_{i}\right)$ where $i=1,2,\dots ,6$ and $j=1,2,\dots ,5$ of BFSS $(G,R)$ is given by Table 29. From $\left(5\right)$, the final score of each object ${o}_{i}$ is given by Table 30. Clearly, ${S}_{5}=3.0$ is the maximum score for the object ${o}_{5}$. Thus, ${o}_{5}$ is an optimal decision object. Given an error value $\alpha =0.9$, using Table 30, we can easily compute that $B=\{{r}_{2},{r}_{5}\}$, $\sum _{{r}_{j}\in B}}{b}_{{r}_{j}}\left({o}_{1}\right)=0.1,\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{{r}_{j}\in B}}{b}_{{r}_{j}}\left({o}_{2}\right)=0.7,\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{{r}_{j}\in B}}{b}_{{r}_{j}}\left({o}_{3}\right)=0.1,\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{{r}_{j}\in B}}{b}_{{r}_{j}}\left({o}_{4}\right)=0.1,\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{{r}_{j}\in B}}{b}_{{r}_{j}}\left({o}_{5}\right)=0.1,\phantom{\rule{3.33333pt}{0ex}}{\displaystyle \sum _{{r}_{j}\in B}}{b}_{{r}_{j}}\left({o}_{6}\right)=0.1,$ satisfying: From Table 30, ${D}_{R}=\{{\left\{{o}_{5}\right\}}_{3.0},{\left\{{o}_{4}\right\}}_{0.6},{\left\{{o}_{2}\right\}}_{0.2},{\left\{{o}_{3}\right\}}_{1.2},{\left\{{o}_{6}\right\}}_{1.6},{\left\{{o}_{1}\right\}}_{2.0}\}$. Also, ${D}_{RB}=\{{\left\{{o}_{5}\right\}}_{3.1},{\left\{{o}_{4}\right\}}_{0.7},{\left\{{o}_{2}\right\}}_{0.9},{\left\{{o}_{3}\right\}}_{1.1},{\left\{{o}_{6}\right\}}_{1.5},{\left\{{o}_{1}\right\}}_{1.9}\}$, satisfying ${D}_{RB}={D}_{R}$. Hence, $\{{r}_{1},{r}_{3},{r}_{4}\}$ is an ANPR of BFSS $(G,R)$ given in Table 31. 4. Comparison
This section compares our proposed parameter reduction algorithms regarding the EDCR, applicability, exact degree of reduction, reduction result, multiuse of the reduction set, and applied situation.
1. Comparison of EDCR and applicability:
Assume that a coefficient q represents the ratio of correctlycomputed parameter reduction in different datasets. In other words, q represents the applicability of our proposed reduction techniques in practical applications and will be interpreted as EDCR. OCBPR only preserves the optimal decision object. Therefore, a parameter reduction is easy to compute with OCBPR. For example, $\left\{{r}_{1}\right\}$ is the OCBPR in Example 3; $\left\{{r}_{1}\right\}$ and $\left\{{r}_{3}\right\}$ are the OCBPR in Example 4; $\left\{{r}_{1}\right\},\left\{{r}_{2}\right\},\left\{{r}_{3}\right\}$ and $\left\{{r}_{4}\right\}$ are the OCBPR in Example 5; $\left\{{r}_{4}\right\}$ is the OCBPR in Example 6. Hence, $q=100\%$.
IRDCBPR is designed to delete the irrelevant parameters by preserving the partitioning and rank of objects. Obviously, parameter reduction using IRDCBPR is more difficult than OCBPR. For instance, $\left\{{r}_{3}\right\}$ is the IRDCBPR in Example 4, and $\{{r}_{1},{r}_{2},{r}_{4}\}$ and $\{{r}_{2},{r}_{3},{r}_{4}\}$ are the IRDCBPR in Example 5. We can observe that there is no IRDCBPR in Examples 3 and 6. Thus, $q=2/4=50\%$.
NPR maintains both invariable rank and unchangeable differences between decision choices. Using the NPR algorithm is the most difficult to obtain parameter reduction as compared to other proposed reduction methods. We can see that $\{{r}_{1},{r}_{2},{r}_{4}\}$ and $\{{r}_{2},{r}_{3},{r}_{4}\}$ are the NPRs in Example 5. Unfortunately, there is no NPR in Examples 3, 4, and 6. Thus, $q=1/3=33.3\%.$
ANPR is a compromise between IRDCBPR and NPR. Without $\alpha $, ANPR is IRDCBPR, and when $\alpha =0$, ANPR is NPR. Thus, the EDCR of ANPR depends on $\alpha $. Therefore, $q=1/3=33.3\%.$
2. Comparison of the exact degree of reduction and reduction results:
The exact degree of parameter reduction considers the precision of parameter reduction and its impact on the postreduction decision object. OCBPR only keeps the optimal decision object unchanged after reduction (that is, the rank of decision choices may be changed after reduction). Therefore, the exact degree of reduction is lower. IRDCBPR reduces redundant parameters by preserving the partitioning and rank of objects. Therefore, the exact degree of reduction is higher as compared to OCBPR. NPR preserves both rank and unchangeable differences between decision choices. Therefore, its exact degree of reduction is highest.
3. Comparison of the multiple use of the parameter reduction and applied situation:
The multiple use of parameter reduction means that the reduction sets can be reused when the expert demands suboptimal parameters and when he/she adds some new parameters.
(i) Comparison of the multiple use of the parameter reduction and applied situation of OCBPR:
OCBPR usually has a wider range of applications. As we know, it only provides the optimal option. After selecting the best choice, if the data of the optimal object are deleted from the dataset, then, for the next decision, we need to make a new reduction again, which wastes much time on the parameter reduction. Furthermore, the added parameter set has not been considered. If new parameters are added to the parameter set, a new reduction is required. We explain these issues by the following example.
Example 7. From Example 2, clearly, ${o}_{4}=5.6$ is the best option in Table 12. An OCBPR of $(G,R)$ is $\left\{{r}_{2}\right\}$, which is given by Table 32. When the object ${o}_{4}$ is deleted from Table 12, the suboptimal choice object is ${o}_{1}$. From Table 32, it can be easily observed that the suboptimal choice is ${o}_{3}$. It is clear that the suboptimal choice has changed. Let $A=\{{a}_{1},{a}_{2}\}$ be the set of added parameters for the BFSS $(G,R)$ in Example 2, given by: For the parameters ${a}_{1}$ and ${a}_{2}$, the score of membership degrees ${b}_{{a}_{j}}\left({o}_{i}\right)$ where $i=1,2,3,4.5$ and $j=1,2$ of BFSS $(G,R)$ is given by Table 33. By combining Table 12 and Table 33, we can observe that ${o}_{1}$ is the optimal decision object from Table 34, while by combining Table 32 and Table 33, ${o}_{4}$ is the best option from Table 35. Clearly, these two optimal options are different. Thus, OCBPR has a lower degree of the multiple use of parameter reduction. (ii) Comparison of multiuse of parameter reduction and applied situation of IRDCBPR:
IRDCBPR maintains the rank of suboptimal decision choices. However, the issue of added parameters is not solved by the IRDCBPR method. We give the following example to explain this idea.
Example 8. Let $A=\{{a}_{1}^{{}^{\prime}},{a}_{2}^{{}^{\prime}}\}$ be the set of added parameters for the BFSS $(G,R)$ in Example 4, given by: For the parameters ${a}_{1}^{{}^{\prime}}$ and ${a}_{2}^{{}^{\prime}}$, the score of membership degrees is given by Table 36. Combine Table 18 and Table 19 with Table 36. From Table 37, we see that ${D}_{R+\{{a}_{1}^{{}^{\prime}},{a}_{2}^{{}^{\prime}}\}}=\{{\left\{{o}_{3}\right\}}_{2.5},{\left\{{o}_{4}\right\}}_{2.1},{\left\{{o}_{2}\right\}}_{0.3},{\left\{{o}_{1}\right\}}_{3.1}\}$. Similarly, using Table 38, we get ${D}_{\left\{{r}_{3}\right\}+\{{a}_{1}^{{}^{\prime}},{a}_{2}^{{}^{\prime}}\}}=\{{\left\{{o}_{2}\right\}}_{0.3},{\{{o}_{1},{o}_{3},{o}_{4}\}}_{0.1}\}$. Clearly, the ranks of choice objects in Table 37 and Table 38 are different. From Table 18, we observe that ${D}_{R}=\{{\left\{{o}_{4}\right\}}_{3.7},{\left\{{o}_{3}\right\}}_{3.3},{\left\{{o}_{2}\right\}}_{0.3},{\left\{{o}_{1}\right\}}_{5.5}\}$. IRDCBPR of the BFSS $(G,R)$ is $\left\{{r}_{3}\right\}$. We can compute that ${D}_{\left\{{r}_{3}\right\}}=\{{\left\{{o}_{4}\right\}}_{1.5},{\left\{{o}_{3}\right\}}_{0.7},{\left\{{o}_{2}\right\}}_{0.3},{\left\{{o}_{1}\right\}}_{2.5}\}$. Thus, IRDCBPR preserves the partition and rank of the objects after parameter reduction. From the above analysis, we observe that the issue of suboptimal choice can be solved by the IRDCBPR method, while the issue of added parameters cannot be solved by the IRDCBPR technique. (iii) Comparison of the multiple use of parameter reduction and applied situation of NPR:
The problems of the suboptimal choice and rank of decision choice objects can be solved using NPR. The following example addresses this issue.
Example 9. Let $\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}}\}$ be the set of added parameters for the BFSS $(G,R)$ in Example 5, which are given by: For the parameters ${r}_{1}^{{}^{\prime}}$ and ${r}_{2}^{{}^{\prime}}$, the score of membership degrees is given by Table 39. Combine Table 24 (final score table of BFSS $(G,R)$) and Table 25 (NPR for the BFSS $(G,R)$) with Table 39 (added parameters’ score table). From Table 40 and Table 41, we can easily compute that ${D}_{R+\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}}\}}=\{{\left\{{o}_{3}\right\}}_{5.2},{\left\{{o}_{6}\right\}}_{2.0},{\left\{{o}_{1}\right\}}_{1.0},{\left\{{o}_{4}\right\}}_{0.2},{\left\{{o}_{5}\right\}}_{2.2},{\left\{{o}_{2}\right\}}_{4.0}\}$ and ${D}_{\{{r}_{1},{r}_{2},{r}_{4}\}+\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}}\}}=\{{\left\{{o}_{3}\right\}}_{5.2},{\left\{{o}_{6}\right\}}_{2.0},{\left\{{o}_{1}\right\}}_{1.0},{\left\{{o}_{4}\right\}}_{0.2},{\left\{{o}_{5}\right\}}_{2.2},{\left\{{o}_{2}\right\}}_{4.0}\}$, respectively. Hence, the ranks of decision choices are the same. Thus, NPR has the highest degree of the multiple use of reduction sets. (iv) Comparison of the multiple use of parameter reduction and applied situation of ANPR
No doubt, NPR is a suitable approach for parameter reduction, but it is very hard to compute the NPR because BFSS provides bipolar information to describe membership degrees. To reduce this computational difficulty, ANPR is given as a compromise between IRDCBPR and NPR.
Example 10. By combining Table 30 (final score table for the BFSS $(G,R)$ in Example 6) and Table 31 (ANPR of BFSS $(G,R)$) with Table 39 (added parameters’ score table), we get Table 42 and Table 43, respectively. From
Table 42 and
Table 43, we find that
${D}_{R+\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}}\}}=\{{\left\{{o}_{4}\right\}}_{3.6},{\left\{{o}_{2}\right\}}_{1.6},{\left\{{o}_{3}\right\}}_{0},{\left\{{o}_{1}\right\}}_{0.2},{\left\{{o}_{5}\right\}}_{0.6},{\left\{{o}_{6}\right\}}_{5.7}\}$ and
${D}_{\{{r}_{1},{r}_{2},{r}_{4}\}+\{{r}_{1}^{{}^{\prime}},{r}_{2}^{{}^{\prime}}\}}=\{{\left\{{o}_{4}\right\}}_{3.7},{\left\{{o}_{2}\right\}}_{0.9},{\left\{{o}_{3}\right\}}_{0.1},{\left\{{o}_{1}\right\}}_{0.1},{\left\{{o}_{5}\right\}}_{0.5},{\left\{{o}_{6}\right\}}_{5.7}\}$, respectively. Hence, the ranks of decision choices are the same, but there is a little difference among decision choices within the range of
$\alpha =0.9$. Thus, ANPR has the highest degree of the multiple use of reduction sets.
5. Application
To demonstrate our proposed techniques, they were applied to a practical application.
Let $O=\{{o}_{1},{o}_{2},\dots ,{o}_{12}\}$ be a set of twelve investment avenues, where:
‘${o}_{1}$’ represents “Bank Deposits”,
‘${o}_{2}$’ represents “Insurance”,
‘${o}_{3}$’ represents “Foreign or Overseas Mutual Fund”,
‘${o}_{4}$’ represents “Bonds Offered by the Government and Corporates”,
‘${o}_{5}$’ represents “Equity Mutual Funds”,
‘${o}_{6}$’ represents “Precious Objects”,
‘${o}_{7}$’ represents “Postal Savings”,
‘${o}_{8}$’ represents “Shares and Stocks”,
‘${o}_{9}$’ represents “Employee Provident Fund”,
‘${o}_{10}$’ represents “Company Deposits”,
‘${o}_{11}$’ represents “Real Estate”,
‘${o}_{12}$’ represents “Money Market Instruments”,
and $R=\{{r}_{1},{r}_{2},\dots ,{r}_{10}\}$ be a collection of parameters associated with the objects in O (${r}_{i}$’s are basically factors influencing investment decision), where:
‘${r}_{1}$’ denotes “Safety of Funds”,
‘${r}_{2}$’ denotes “Liquidity of Funds”,
‘${r}_{3}$’ denotes “State Policy”,
‘${r}_{4}$’ denotes “Maximum Profit in Minimum Period”,
‘${r}_{5}$’ denotes “Stable Return”,
‘${r}_{6}$’ denotes “Easy Accessibility”,
‘${r}_{7}$’ denotes “Tax Concession”,
‘${r}_{8}$’ denotes “Minimum Risk of Possession”,
‘${r}_{9}$’ denotes “Political Climate”,
‘${r}_{10}$’ denotes “Level of Income”.
An investor
Z wants to invest in a most suitable investment avenue from the abovementioned investment avenues. The information between the investment avenues and influenced factors is given in the form of a BFSS
$(G,R)$, which is given by
Table 44.
By using (
2) and (
3), the score of the positive
${b}_{{r}_{j}}^{+}\left({o}_{i}\right)$ and negative
${b}_{{r}_{j}}^{}\left({o}_{i}\right)$ membership degrees for
$i=1,2,\dots ,12$ and
$j=1,2,\dots ,10$ are described by
Table 45 and
Table 46, respectively.
Now, by using Definition 9, the tabular arrangement for the score of membership degrees
${b}_{{r}_{j}}\left({o}_{i}\right)$ where
$i=1,2,\dots ,12$ and
$j=1,2,\dots ,10$ of BFSS
$(G,R)$ is given by
Table 47.
From
$\left(5\right)$, the final score of each object
${o}_{i},\phantom{\rule{3.33333pt}{0ex}}i=1,2,\dots ,12$ is given by
Table 48.
Clearly, ${S}_{11}=25.9$ is the maximum score for the object ${o}_{11}$. Thus, the investment avenue, ${o}_{11}$, namely real estate, is the best choice for the investor Z. Our proposed reduction algorithms were executed by the investment avenue dataset. Consequently, the parameter reduction sets were readily computed by OCBPR, and the minimal reduction was ${r}_{1}$ (not all) that kept the optimal decision invariant. Regrettably, we obtained no parameter reduction through IRDCBPR, ANPR, and NPR. This means that OCBPR can be applied in many reallife decisionmaking situations as compared to IRDCBPR, ANPR, and NPR.
6. Conclusions
Parameter reduction is one of the main issues in soft set modelization and its hybrid models, including fuzzy soft set theory. Parameter reduction preserves the decision by removing the irrelevant parameters. In this paper, a novel approach for decisionmaking based on BFSSs was introduced, and some decisionmaking problems were solved by this newlyproposed approach to prove its validity, including a decisionmaking problem presented in [
30]. It was also observed that the results were the same by applying this novel decisionmaking approach. Using this concept, four novel definitions of parameter reductions, namely, OCBPR, IRDCBPR, NPR, and ANPR, of BFSSs were presented and illustrated through examples. Due to the existence of bipolar information in many realworld problems, the newlyproposed decisionmaking method based on BFSSs and parameter reductions of BFSSs were very efficient approaches to solve such problems, when compared to some existing methods, including fuzzy soft sets [
32] and their parameter reduction [
33]. An algorithm for each parameter reduction approach was developed. Moreover, our proposed reduction methods were compared with respect to the theoretical and experimental points of view as displayed in
Table 49. Finally, an application was studied to show the feasibility of our proposed reduction algorithms. In the future, we expect to extend our research work to (1) parameter reduction of the Pythagorean fuzzy soft sets, (2) parameter reduction of the Pythagorean fuzzy bipolar soft sets, and (3) parameter reduction of
mpolar fuzzy soft sets.