Conjugate depth

The Yager's OWA (ordered weighted averaging) operators have been widely used to aggregate the crisp values in decision making schemes (such as multi-criteria decision making, multi-expert decisin making, multi-criteria multi-expert decision making). It is widely accepted that fuzzy sets are more suitable for representing preferences of criteria in decision making. But fuzzy sets are not crisp values, how can we aggregate fuzzy sets in OWA mechanism?

The type-1 OWA operators have been proposed for this purpose. So the type-1 OWA operators provides us with a new 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.

First, there are two definitions for type-1 OWA operators, one is based on Zadeh's Extension Principle, the other is based on $\alpha$-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, $\mathrm{\Phi }$,

$\mathrm{\Phi }:F(X)\times \cdots \times F(X)⟶F(X)$

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

such that

${\mu }_Y(y)={\sup }_{\sum _{k=1}^n{\overline{w}}_i{a}_{\sigma (i)}=y}\left(\begin{array}{l}{\mu }_{W^1}(w_1)\wedge \cdots \wedge {\mu }_{W^n}(w_n)\wedge {\mu }_{A^1}(a_1)\wedge \cdots \wedge {\mu }_{A^n}(a_n)\end{array}\right)$

where ${\overline{w}}_i=\frac{w_i}{\sum _{i=1}^n{w}_i}$,and $\sigma :\{1,\cdots ,n\}⟶\{1,\cdots ,n\}$ is a permutation function such that $a_{\sigma (i)}\ge a_{\sigma (i+1)},\mathrm{\forall }i=1,\cdots ,n-1$, i.e., $a_{\sigma (i)}$ is the $i$th highest element in the set $\left\{a_1,\cdots ,a_n\right\}$.

Definition 2.

The definition below is based on 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 given as

${\mathrm{\Phi }}_{\alpha }\left(A_{\alpha }^1,\dots ,A_{\alpha }^n\right)=\left\{\frac{\sum _{i=1}^n{w}_i{a}_{\sigma (i)}}{\sum _{i=1}^n{w}_i}|w_i\in W_{\alpha }^i,a_i\in A_{\alpha }^i,i=1,\dots ,n\right\}$

where $W_{\alpha }^i=\{w|{\mu }_{W_i}(w)\ge \alpha \},A_{\alpha }^i=\{x|{\mu }_{A_i}(x)\ge \alpha \}$, and $\sigma :\{1,\cdots ,n\}\to \{1,\cdots ,n\}$ is a permutation function such that $a_{\sigma (i)}\ge a_{\sigma (i+1)},\mathrm{\forall }i=1,\cdots ,n-1$, 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, these series of $\alpha$-level type-1 OWA operators are 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 ${\mathrm{\Phi }}_{\alpha }\left(A_{\alpha }^1,\cdots ,A_{\alpha }^n\right)$. Then, the resulting aggregation fuzzy set is constructed with the membership function as follows:

${\mu }_G(x)=\vee {\mathrm{\backslash limits}}_{\alpha :x\in {\mathrm{\Phi }}_{\alpha }{\left(A_{\alpha }^1,\cdots ,A_{\alpha }^n\right)}_{\alpha }}\alpha$

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

${\mathrm{\Phi }}_{\alpha }{\left(A_{\alpha }^1,\cdots ,A_{\alpha }^n\right)}_{-}=\min {\mathrm{\backslash limits}}_{\begin{array}{l}{W}_{\alpha -}^i\le w_i\le W_{\alpha +}^i{A}_{\alpha -}^i\le a_i\le A_{\alpha +}^i\end{array}}\sum _{i=1}^n{w}_i{a}_{\sigma (i)}/\sum _{i=1}^n{w}_i$

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

${\mathrm{\Phi }}_{\alpha }{\left(A_{\alpha }^1,\cdots ,A_{\alpha }^n\right)}_{+}=\max {\mathrm{\backslash limits}}_{\begin{array}{l}{W}_{\alpha -}^i\le w_i\le W_{\alpha +}^i{A}_{\alpha -}^i\le a_i\le A_{\alpha +}^i\end{array}}\sum _{i=1}^n{w}_i{a}_{\sigma (i)}/\sum _{i=1}^n{w}_i$

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

• 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 ${\rho }_{\alpha +}^{i_0}\ge A_{\alpha +}^{\sigma (i_0)}$, 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 ${\rho }_{\alpha -}^{i_0}\ge A_{\alpha -}^{\sigma (i_0)}$, 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 $\left[{\rho }_{\alpha -}^{i_0^{\ast }},{\rho }_{\alpha +}^{i_0^{\ast }}\right]$:

${\mu }_G(x)=\vee {\mathrm{\backslash limits}}_{\alpha :x\in \left[{\rho }_{\alpha -}^{i_0^{\ast }},{\rho }_{\alpha +}^{i_0^{\ast }}\right]}\alpha$

Special cases of Type-1 OWA operators

• 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.

References

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