### 统计代写|运筹学作业代写operational research代考|Econ308

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|运筹学作业代写operational research代考|Neutrosophical Plant Hybridization in Decision-Making Problems

This section presents some of the basic properties of neutrosophic sets and operations on neutrosophic sets which are used for further study.

Definition 1 [15] Let $X$ be a non-empty set. A neutrosophic set $A$ having the form $A=\left{\left(x, \mu_{A}(x), \sigma_{A}(x), \gamma_{A}(x)\right): x \in X\right}$, where $\mu_{A}(x), \sigma_{A}(x)$ and $\left.\gamma_{A}(x) \in\right] 0^{-}, 1^{+}[$ represent the degree of membership (namely, $\mu_{A}(x)$ ), the degree of indeterminacy (namely, $\sigma_{A}(x)$ ), and the degree of nonmembership (namely, $\gamma_{A}(x)$ ), respectively, for each $x \in X$ to the set $A$ such that $0^{-} \leq \mu_{A}(x)+\sigma_{A}(x)+\gamma_{A}(x) \leq 3^{+}$for all $x \in X$. For a non-empty set $X, N(X)$ denotes the collection of all neutrosophic sets of $X$.
Definition 2 [16] The following statements are true for neutrosophic sets $A$ and $B$ on $X$ :
i. $\mu_{A}(x) \leq \mu_{B}(x), \sigma_{A}(x) \leq \sigma_{B}(x)$, and $\gamma_{A}(x) \geq \gamma_{B}(x)$ for all $x \in X$ if and only if $A \subseteq B$
ii. $A \subseteq B$ and $B \subseteq A$ if and only if $A=B$.
iii. $A \cap B=\left{\left(x, \min \left{\mu_{A}(x), \mu_{B}(x)\right}, \min \left{\sigma_{A}(x), \sigma_{B}(x)\right}, \max \left{\gamma_{A}(x), \gamma_{B}(x)\right}\right): x \in X\right}$.
iv. $A \cup B=\left{\left(x, \max \left{\mu_{A}(x), \mu_{B}(x)\right}, \max \left{\sigma_{A}(x), \sigma_{B}(x)\right}, \min \left{\gamma_{A}(x), \gamma_{B}(x)\right}\right): x \in X\right}$.

More generally, the intersection and the union of a collection of neutrosophic sets $\left{A_{i}\right}_{i \in \Lambda}$ are defined by $\bigcap_{i \in \Lambda} A_{i}=\left{\left(x, \inf {i \in \Lambda}\left{\mu{A_{i}}(x)\right}, \inf {i \in \Lambda}\left{\sigma{A_{i}}(x)\right}\right.\right.$, $\left.\left.\sup {i \in \Lambda}\left{\gamma{A_{i}}(x)\right}\right): x \in X\right}$ and $\bigcup_{i \in \Lambda} A_{i}=\left{\left(x, \sup {i \in \Lambda}\left{\mu{A_{i}}(x)\right}, \sup {i \in \Lambda} \sigma{A_{i}}{(x)}\right.\right.$, $\left.\left.\inf {i \in A}\left{\gamma A{i}(x)\right}\right): x \subset X\right}$.

Corollary 1 [16] The following statements are true for the neutrosophic sets $A, B$, $C$, and $D$ on $X$ :
i. $A \cap C \subseteq B \cap D$ and $A \cup C \subseteq B \cup D$, if $A \subseteq B$ and $C \subseteq D$.
ii. $A \subseteq B \cap C$, if $A \subseteq B$ and $A \subseteq C . A \cup B \subseteq C$, if $A \subseteq C$ and $B \subseteq C$.
iii. $A \subseteq C$, if $A \subseteq B$ and $B \subseteq C$.

## 统计代写|运筹学作业代写operational research代考|Neutrosophic Methodologies in Multi-criteria Decision-Making Problems

This section systematically develops a new methodological approach in multicriteria decision making (MCDM) problems with single-valued neutrosophic information for neutrosophic sets structure. The following methodological approach gives the necessary steps to select the best alternatives in each division and the best alternative overall regions in the MCDM situations.

Step 1: Problem Selection Consider the multi-criteria decision-making problem shown in Table 1, with $\mathrm{m}$ alternatives $A_{1}, A_{2}, \ldots, A_{m}$ and $p$ attributes $B_{1}, B_{2}$, $\ldots, B_{p}$ for $n$ regions $D_{1}, D_{2}, \ldots, D_{n}$ to identify the best alternatives of each region and the best alternative of the entire regions.

Step 2: Problem Division Divide the selected problem into $n$ subproblems for $n$ regions.
Step 3: Direct Direction
Step 3(a): Subproblem Selection Select first subproblem for the corresponding region.

Step 3(b): Neutrosophic Operations For $j=1,2, \ldots, m$, find $C_{j 1}=\left{\left(a_{j 1}\right){1},\left(a{j 2}\right){1}\right.$, $\left.\ldots,\left(a{j p}\right){1}\right}$ and $D{j 1}=\left{\left(a_{j 1}\right){1} \cup\left(a{j 2}\right){1},\left(a{j 1}\right){1} \cup\left(a{j 3}\right){1}, \ldots,\left(a{j 1}\right){1} \cup\left(a{j p}\right){1},\left(a{j 2}\right){1}\right.$ $\left.\cup\left(a{j 3}\right){1}, \ldots,\left(a{j 2}\right){1} \cup\left(a{j p}\right){1}, \ldots,\left(a{j(p-1)}\right){1} \cup\left(a{j p}\right){1}\right} \ldots,\left(a{j 2}\right){1} \cup\left(a{j p}\right){1}, \ldots$, $\left.\left(a{j(p-1)}\right){1} \cup\left(a{j p}\right){1}\right}$ such that $\left(a{j k}\right){1} \cup\left(a{j l}\right){1} \notin C{j 1}$.

Step 3(c): Finding Single-Valued Neutrosophic Score Functions For $j=1,2$, $\ldots, m$, find single-valued neutrosophic score functions of $C_{j 1}$ and $D_{j 1}$ that are defined as follows: $S V N S F\left(C_{j 1}\right)=1 / 3 p\left[\sum_{i=1}^{p}\left[2+\mu_{j i}-\sigma_{j i}-\gamma_{j i}\right]\right]$, and $\operatorname{SVNSF}\left(D_{j 1}\right)=1 / 3 q\left[\sum_{i=1}^{q}\left[2+\mu_{j i}-\sigma_{j i}-\gamma_{j i}\right]\right]$, where $q$ is the number of elements of $D_{j 1} . S V N S F\left(A_{j}\right)=\left{S V N S F\left(C_{j 1}\right)\right.$, if $S V N S F\left(D_{j 1}\right)=0$. Otherwise, 1/2[ SVNSF $\left.\left(C_{j 1}\right)+\operatorname{SVNSF}\left(D_{j 1}\right)\right}$.
Step 3(d): Arrangement For $j=1,2, \ldots, m$, arrange all the single-valued neutrosophic score function’s values for the alternatives $A_{1}, A_{2}, \ldots, A_{m}$ in ascending order.

Step 3(e): Repetition Repeat steps 3(a)-3(d) for each subproblem of the corresponding region.

## 统计代写|运筹学作业代写operational research代考|Neutrosophical Plant Hybridization in Decision-Making Problems

$\mathrm{A}=\mathrm{~ I l e f t { l e f t ( X , ~ \ m u _ { A } ( X ) , ~ I}$
$\left.\gamma_{A}(x) \in\right] 0^{-}, 1^{+}\left[\right.$表示隶属度 (即， $\mu_{A}(x)$ )，不确定度 (即， $\sigma_{A}(x)$ )，以及非会员的程度（即， $\gamma_{A}(x)$ ) 分别 对每个 $x \in X$ 到集合 $A$ 这样 $0^{-} \leq \mu_{A}(x)+\sigma_{A}(x)+\gamma_{A}(x) \leq 3^{+}$对所有人 $x \in X$. 对于非空集 $X, N(X)$ 表 示所有中智集的集合 $X$.

-。 $\mu_{A}(x) \leq \mu_{B}(x), \sigma_{A}(x) \leq \sigma_{B}(x)$ ，和 $\gamma_{A}(x) \geq \gamma_{B}(x)$ 对所有人 $x \in X$ 当且仅当 $A \subseteq B$ ii. $A \subseteq B$ 和 $B \subseteq A$ 当且仅当 $A=B$.
iii.
A \cap B=\left } { \backslash \text { left } ( x , \backslash m i n \backslash \text { left } \backslash
iv.

\bigcap_{i \in \Lambda} A_{i} $}=\backslash$ eft ${\backslash e f t(x, \backslash i n f{i \backslash \mathrm{~ i n ~ \ L a m b d a }}$ 和
$\mathrm{~ \ b i g c u p _ { i ~ \ i n ~ \ L a m b d a } ~ A _ { i } = \ l e f t {}$

• $A \cap C \subseteq B \cap D$ 和 $A \cup C \subseteq B \cup D$ ， 如果 $A \subseteq B$ 和 $C \subseteq D$.
ii. $A \subseteq B \cap C$ ，如果 $A \subseteq B$ 和 $A \subseteq C . A \cup B \subseteq C$ ，如果 $A \subseteq C$ 和 $B \subseteq C$.
iii. $A \subseteq C$ ，如果 $A \subseteq B$ 和 $B \subseteq C$.

## 统计代写|运筹学作业代写operational research代考|Neutrosophic Methodologies in Multi-criteria Decision-Making Problems

$\mathrm{~ C _ { j ~ 1 } = V l e f t { \ l e f t ( a _ { j ~ 1 } \ r i g h t ) ~ { 1 } , V}$
\begin{tabular}{|c|c|} \end{tabular} $\mathrm{~ , ~ V e f t . V e f t ( a { j ( p – 1 ) } \ r i g h t ) { 1 } \ c u p}$

$\operatorname{SVNSF}\left(C_{j 1}\right)=1 / 3 p\left[\sum_{i=1}^{p}\left[2+\mu_{j i}-\sigma_{j i}-\gamma_{j i}\right]\right]$ ，和
$\operatorname{SVNSF}\left(D_{j 1}\right)=1 / 3 q\left[\sum_{i=1}^{q}\left[2+\mu_{j i}-\sigma_{j i}-\gamma_{j i}\right]\right]$ ，在哪里 $q$ 是元素的数量
$\mathrm{~ D _ { j ~ 1 } ~ 。 S ~ V ~ N ~ S ~ F V e f t ( A _ { j } \ r i g h t ) = V l e f t { S ~ V ~ N ~ S ~ F V l e f t ( C _ { j ~ 1 }}$

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