Type-1 OWA operators

The Yager's OWA (ordered weighted averaging) operators are used to aggregate the crisp values in decision making schemes (such as multi-criteria decision making, multi-expert decision making and multi-criteria/multi-expert decision making). It is widely accepted that Fuzzy sets are more suitable for representing preferences of criteria in decision making.

The type-1 OWA operators have been proposed for this purpose. The type-1 OWA operators provides a technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets.

The two definitions for type-1 OWA operators are based on Zadeh's Extension Principle and $\alpha$-cuts of fuzzy sets. The two definitions lead to equivalent results.

Definitions

Definition 1

Let $F(X)$ be the set of fuzzy sets with domain of discourse $X$, a type-1 OWA operator is defined as follows:

Given n linguistic weights $\left\{{W^{i}}\right\}_{i=1}^{n}$ in the form of fuzzy sets defined on the domain of discourse $U=[0,1]$, a type-1 OWA operator is a mapping, $\Phi$,

$\Phi \colon F(X)\times \cdots \times F(X)\longrightarrow F(X)$

$(A^{1},\cdots ,A^{n})\mapsto Y$

such that

$\mu _{Y}(y)=\displaystyle \sup _{\displaystyle \sum _{k=1}^{n}{\bar {w}}_{i}a_{\sigma (i)}=y}\left({\begin{array}{*{1}l}\mu _{W^{1}}(w_{1})\wedge \cdots \wedge \mu _{W^{n}}(w_{n})\wedge \mu _{A^{1}}(a_{1})\wedge \cdots \wedge \mu _{A^{n}}(a_{n})\end{array}}\right)$

where ${\bar {w}}_{i}={\frac {w_{i}}{\sum _{i=1}^{n}{w_{i}}}}$,and $\sigma \colon \{1,\cdots ,n\}\longrightarrow \{1,\cdots ,n\}$ is a permutation function such that $a_{\sigma (i)}\geq a_{\sigma (i+1)},\ \forall i=1,\cdots ,n-1$, i.e., $a_{\sigma (i)}$ is the $i$th highest element in the set $\left\{{a_{1},\cdots ,a_{n}}\right\}$.

Definition 2

Using the alpha-cuts of fuzzy sets:

Given the n linguistic weights $\left\{{W^{i}}\right\}_{i=1}^{n}$ in the form of fuzzy sets defined on the domain of discourse $U=[0,\;\;1]$, then for each $\alpha \in [0,\;1]$, an $\alpha$-level type-1 OWA operator with $\alpha$-level sets $\left\{{W_{\alpha }^{i}}\right\}_{i=1}^{n}$ to aggregate the $\alpha$-cuts of fuzzy sets $\left\{{A^{i}}\right\}_{i=1}^{n}$ is:

$\Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)=\left\{{{\frac {\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}}}{\sum \limits _{i=1}^{n}{w_{i}}}}\left|{w_{i}\in W_{\alpha }^{i},\;a_{i}}\right.\in A_{\alpha }^{i},\;i=1,\ldots ,n}\right\}$

where $W_{\alpha }^{i}=\{w|\mu _{W_{i}}(w)\geq \alpha \},A_{\alpha }^{i}=\{x|\mu _{A_{i}}(x)\geq \alpha \}$, and $\sigma :\{\;1,\cdots ,n\;\}\to \{\;1,\cdots ,n\;\}$ is a permutation function such that $a_{\sigma (i)}\geq a_{\sigma (i+1)},\;\forall \;i=1,\cdots ,n-1$, i.e., $a_{\sigma (i)}$ is the $i$th largest element in the set $\left\{{a_{1},\cdots ,a_{n}}\right\}$.

Representation theorem of Type-1 OWA operators

Given the n linguistic weights $\left\{{W^{i}}\right\}_{i=1}^{n}$ in the form of fuzzy sets defined on the domain of discourse $U=[0,\;\;1]$, and the fuzzy sets $A^{1},\cdots ,A^{n}$, then we have that

$Y=G$

where $Y$ is the aggregation result obtained by Definition 1, and $G$ is the result obtained by in Definition 2.

Programming problems for Type-1 OWA operators

According to the Representation Theorem of Type-1 OWA Operators, a general type-1 OWA operator can be decomposed into a series of $\alpha$-level type-1 OWA operators. In practice, this series of $\alpha$-level type-1 OWA operators is used to construct the resulting aggregation fuzzy set. So we only need to compute the left end-points and right end-points of the intervals $\Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)$. Then, the resulting aggregation fuzzy set is constructed with the membership function as follows:

$\mu _{G}(x)=\operatorname {\bigvee } \limits _{\alpha :x\in \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{\alpha }}\alpha$

For the left end-points, we need to solve the following programming problem:

$\Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{-}=\operatorname {\min } \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}$

while for the right end-points, we need to solve the following programming problem:

$\Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{+}=\operatorname {\max } \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}$

A fast method has been presented to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently, for details, please see the paper.

Alpha-level approach to Type-1 OWA operation

Three-step process:

• Step 1—To set up the $\alpha$- level resolution in [0, 1].
• Step 2—For each $\alpha \in [0,1]$,

• Step 2.1—To calculate $\rho _{\alpha +}^{i_{0}^{\ast }}$

1. Let $i_{0}=1$;
2. If $\rho _{\alpha +}^{i_{0}}\geq A_{\alpha +}^{\sigma (i_{0})}$, stop, $\rho _{\alpha +}^{i_{0}}$ is the solution; otherwise go to Step 2.1-3.
3. $i_{0}\leftarrow i_{0}+1$, go to Step 2.1-2.

• Step 2.2 To calculate$\rho _{\alpha -}^{i_{0}^{\ast }}$

1. Let $i_{0}=1$;
2. If $\rho _{\alpha -}^{i_{0}}\geq A_{\alpha -}^{\sigma (i_{0})}$, stop, $\rho _{\alpha -}^{i_{0}}$ is the solution; otherwise go to Step 2.2-3.
3. $i_{0}\leftarrow i_{0}+1$, go to step Step 2.2-2.

• Step 3—To construct the aggregation resulting fuzzy set $G$ based on all the available intervals $\left[{\rho _{\alpha -}^{i_{0}^{\ast }},\;\rho _{\alpha +}^{i_{0}^{\ast }}}\right]$:

$\mu _{G}(x)=\operatorname {\bigvee } \limits _{\alpha :x\in \left[{\rho _{\alpha -}^{i_{0}^{\ast }},\;\rho _{\alpha +}^{i_{0}^{\ast }}}\right]}\alpha$

Special cases

• Any OWA operators, like maximum, minimum, mean operators;
• Join operators of (type-1) fuzzy sets, i.e., fuzzy maximum operators;
• Meet operators of (type-1) fuzzy sets, i.e., fuzzy minimum operators;
• Join-like operators of (type-1) fuzzy sets;
• Meet-like operators of (type-1) fuzzy sets.

Generalizations

Type-2 OWA operators have been suggested to aggregate the type-2 fuzzy sets for soft decision making.