N-ary Topsis

Nformula_1ary 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 Nformula_1ary 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 formula_3 is called a T-norm if it is commutative, associative, monotonic, and satisfies the boundary conditions. That is, for all formula_4 we have that
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 formula_10.
Due to the associativity of T-norms, it is possible to extend the operation to the formula_11ary case, formula_12. E.g. for formula_13 the T-norm can be computed from formula_14. For example with algebraic product (formula_15) we would get formula_16. Klement et al. gave following definition for formula_11ary case.
Definition
Let T be a T-norm and formula_18 be any n-ary tuple, we define formula_19 as;
formula_20.
Similarly, for a norm operator to be a T-conorm following axioms needs to be satisfied.
Definition
An aggregation operator formula_21 is called a T-conorm if it is commutative, associative, monotonic, and satisfies the boundary conditions. That is, for all formula_4 we have that
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 formula_28.
Triangular conorms can be also extended to formula_11ary conorms (see e.g. Klement et al) due to their associativity. Klement et al. defined formula_11ary T-conorms as follows.
Definition
Let formula_31 be a T-conorm and formula_18 be any n-ary tuple, then formula_33 is given by,
formula_34
Method.
To apply Nary TOPSIS we require a specification of the decision matrix for a set of alternatives over a set of criteria. Let formula_35 and formula_31 denote formula_11ary formula_38norm and formula_38conorm. 
Given a set of alternatives formula_40, 
a set of criteria formula_41 and 
a set of weights formula_42, formula_43, formula_44, where formula_45 denotes the weight of the criteria formula_46. 
Let formula_47 denote the decision matrix where formula_48 is the performance measure of the alternative formula_49 with respect to the criteria formula_46. 
Given the decision matrix, the formula_11ary norm based TOPSIS involves following steps.
formula_52 formula_53
formula_56
formula_58
formula_59
where formula_60 denotes benefit criteria and formula_61 cost criteria.
formula_62
formula_63
Here, formula_60 is the set of benefit criteria, and formula_61 is the set of cost criteria.
formula_69
 formula_70
formula_72
formula_73, formula_74
Implementation.
Method has been implemented in matlab.