N-ary Topsis

N$-$ary TOPSIS is multi-criteria decision making method, which is built to take into consideration also preference over positive or negative ideal solution. It was developed in 2020. Original TOPSIS (Technique for Order Performance by Similarity to Ideal Solution) is based on the idea of forming two ideal solutions (best possible case and worst possible case) and comparing the current alternative to these two. Several researchers have pointed out that one downside of the method is that in original TOPSIS ranking index is irrespective of the weights of the decision-maker assigns to these two separations. In N$-$ary TOPSIS preference is executed by posing weaker or stricter conditions to norm operators in creation of ideal vectors.

Justification for replacing minimum and maximum in positive and negative ideal solution computation is quite intuitive. The standard fuzzy intersection (minimum) is the weakest fuzzy intersection and hence producing largest set from among those produced by all possible fuzzy intersections (T-norms). Similarly standard fuzzy union (maximum) is the strongest fuzzy union and hence it produces the smallest set among the sets produced by all possible fuzzy unions (T-conorms). Now by changing the norm operator we can also address relative importantess type of a problem in TOPSIS at the point where we are creating Positive and Negative ideal solutions. This is obviously done separately for positive ideal solution and negative ideal solution.

N-ary norm operators

Following Klir Yuan, we can define the T-norm as follows:

Definition An aggregation operator $T:[0,1]^{2}\rightarrow [0,1]$ is called a T-norm if it is commutative, associative, monotonic, and satisfies the boundary conditions. That is, for all $x,y,z\in [0,1]$ we have that

• $T(x,y)=T(y,x)$(commutativity)
• $T(x,T(y,z))=T(T(x,y),z)$ (associativity)
• $T(x,y)\leq T(x,z)$ whenever $y\leq z$ (monotonicity)
• $T(x,1)=x$ (boundary condition)

These are minimum requirements for a norm operator to be a T-norm. Besides this often one can introduce further axioms to have even stricter norm operators. For example subidempotency and continuity.

Definition A T-norm is said to be an Archimedean t-norm if it is also continuous and $T(x,x)<x,\forall x\in (0,1)$.

Due to the associativity of T-norms, it is possible to extend the operation to the $n-$ary case, $n\geq 2$. E.g. for $n=3$ the T-norm can be computed from $T(x_{1},x_{2},x_{3})=T(T(x_{1},x_{2}),x_{3})$. For example with algebraic product ($T(x_{1},x_{2})=x_{1}x_{2}$) we would get $T(x_{1},x_{2},x_{3})=x_{1}x_{2}x_{3}$. Klement et al. gave following definition for $n-$ary case.

Definition Let T be a T-norm and $(x_{1},x_{2},...,x_{n})\in [0,1]^{n}$ be any n-ary tuple, we define $T(x_{1},x_{2},...,x_{n})$ as; $T(x_{1},x_{2},...,x_{n})=T(T(x_{1},x_{2},...,x_{n-1}),x_{n})$.

Similarly, for a norm operator to be a T-conorm following axioms needs to be satisfied.

Definition An aggregation operator $T_{co}:[0,1]^{2}\rightarrow [0,1]$ is called a T-conorm if it is commutative, associative, monotonic, and satisfies the boundary conditions. That is, for all $x,y,z\in [0,1]$ we have that

• $T_{co}(x,y)=T_{co}(y,x)$(commutativity)
• $T_{co}(x,T_{co}(y,z))=T_{co}(T_{co}(x,y),z)$ (associativity)
• $T_{co}(x,y)\leq T_{co}(x,z)$ whenever $y\leq z$ (monotonicity)
• $T_{co}(x,0)=x$ (boundary condition)

Notice that difference between T-norm and T-conorm is in the boundary condition. Again these axioms are minimum requirements for norm operator to be a T-conorm and further stricter requirements can be imposed. For example superidempotency and continuity.

Definition A T-conorm is said to be an Archimedean T-conorm if it is also continuous and $T(x,x)>x,\forall x\in [0,1]$.

Triangular conorms can be also extended to $n-$ary conorms (see e.g. Klement et al) due to their associativity. Klement et al. defined $n-$ary T-conorms as follows.

Definition Let $T_{co}$ be a T-conorm and $(x_{1},x_{2},...,x_{n})\in [0,1]^{n}$ be any n-ary tuple, then $T_{co}(x_{1},x_{2},...,x_{n})$ is given by, $T_{co}(x_{1},x_{2},...,x_{n})=T_{co}(T_{co}(x_{1},x_{2},...,x_{n-1}),x_{n})$

Method

To apply Nary TOPSIS we require a specification of the decision matrix for a set of alternatives over a set of criteria. Let $T$ and $T_{co}$ denote $n-$ary $T-$norm and $T-$conorm.

Given a set of alternatives $A=\{a_{i}|i=1,2,\cdots ,m\}$,

a set of criteria $C=\{c_{j}|j=1,2,\cdots ,n\}$ and

a set of weights $W=\{w_{j}|j=1,2,\cdots ,n\}$, $w_{j}>0$, $\sum _{j=1}^{n}w_{j}=1$, where $w_{j}$ denotes the weight of the criteria $c_{j}$.

Let $X=\{x_{ij|i=1,2,\cdots ,m,j=1,2,\cdots ,n}\}$ denote the decision matrix where $x_{ij}$ is the performance measure of the alternative $a_{i}$ with respect to the criteria $c_{j}$.

Given the decision matrix, the $n-$ary norm based TOPSIS involves following steps.

• Normalize the decision matrix into unit interval.

$z_{ij}={\frac {x_{ij}+\min _{i}(x_{ij})}{\max _{i}(x_{ij})-\min _{i}(x_{ij})}},$ $i=1,2,\cdots ,m,j=1,2,\cdots ,n$

• Let $z_{ij}$ denote normalized decision matrix. Compute the weighted normalized decision matrix. The weighted normalized value $v_{ij}$ is calculated as

$v_{ij}=w_{j}z_{ij},i=1,2,\cdots ,m,j=1,2,\cdots ,n$

• Determine the positive ideal solution (PIS) and the negative ideal solution (NIS) using chosen $n-$ary T-norm and T-conorm.

$PIS=\{v_{1}^{+},v_{2}^{+},\cdots ,v_{n}^{+}\}$

$=\{T_{\forall i}(v_{ij})|j\in J_{1},Tco_{\forall i}(v_{ij})|j\in J_{2}\}$

where $J_{1}$ denotes benefit criteria and $J_{2}$ cost criteria.

$NIS=\{v_{1}^{-},v_{2}^{-},\cdots ,v_{n}^{-}\}$

$=\{Tco_{\forall i}(v_{ij})|j\in J_{1},T_{\forall i}(v_{ij})|j\in J_{2}\}$

Here, $J_{1}$ is the set of benefit criteria, and $J_{2}$ is the set of cost criteria.

• Calculate the separation measures using the n-dimensional Euclidean distance. The separation measures $D_{i}^{+}$ and $D_{i}^{-}$ of an alternative $a_{i}$ from the PIS and NIS are

$D_{i}^{+}={\sqrt {\sum _{j=1}^{n}(v_{ij}-v_{j}^{+})^{2}}},i=1,2,\cdots ,m$

$D_{i}^{-}={\sqrt {\sum _{j=1}^{n}(v_{ij}-v_{j}^{-})^{2}}},i=1,2,\cdots ,m$

• Calculate relative closeness (RC) of the alternative $a_{i}$.

$RC_{i}={\frac {D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}}}$

$0\leq RC_{i}\leq 1$, $i=1,2,\cdots ,m$

• Arrange the ranking indexes in a descending order to obtain the best alternative.

Implementation

Method has been implemented in matlab.

References

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