On Triangular Multisets and Triangular Fuzzy Multisets

Abstract

The basic set operations between fuzzy sets are defined using the min and max functions; however, later on, new operators were introduced that used other functions, which, nevertheless, had similar properties to functions min and max. The resulting fuzzy set theories are more suitable for the description and processing of specific data sets. Crisp and fuzzy multisets have found numerous applications but still the basic operations are based on functions min and max. It is straightforward to replace these functions in the fuzzy part of fuzzy multisets; however, it is not as easy but is feasible to do the same with the multisets and the “crisp” part of fuzzy multisets. The new mathematical structures are called triangular multisets and triangular fuzzy multisets, respectively. The aim is to facilitate the processing of certain data sets so they can be used in multi-criteria decision making and computing.

1. Introduction

In set theory an element either belongs to or does not belong to a set. Suppose we allow multiple copies of an element to be part of a “set”, then new structures emerge. These structures are known in the literature as multisets [1]. Typically, a multiset A is defined by a function $A:D\to \mathbb{N}$, where D is a domain from which the elements of A are drawn. Thus $A\left(d\right)=n$ means that the element d belongs n times to the multiset A. The various operations between multisets (e.g., their union and their intersection) are natural extensions of the corresponding operations between sets.

Lotfi Askar Zadeh [2] introduced fuzzy sets in order to provide a mathematical tool to describe vagueness (see [3]). Something is vague when it is not straightforward to say to which class of objects it belongs. Assume we encounter a person whose height is $1.70\mathrm{m}$, is this person tall? Regardless of the sex of the person, we can classify him/her as either tall or short. A fuzzy set A is characterized by a function $A:D\to [0,1]$, where $[0,1]$ is the unit interval, and $A\left(d\right)=i$ means that d belongs to A with a degree that is equal to i.

Just like fuzzy sets are an extension of (classical) sets able to describe vagueness, fuzzy multisets are an extension of multisets able to accommodate vagueness [4]. Since fuzzy multisets are far too general, this author introduced multi-fuzzy sets [5]. In particular, a multi-fuzzy set A is identified by a function $A:D\to \mathbb{N}\times [0,1]$. Thus $A\left(d\right)=(m,r)$ means that the element d belongs m times with membership degree r.

Multisets have been used to define a few models of computation. The most notable examples are the GAMMA model [6,7], the chemical abstract machine [8], and P systems [9]. Similarly, there are models of computation whose core elements are fuzzy multisets [5,10] (but see also the 5th chapter of [11]).

The basic set theoretic operations between fuzzy sets are defined using functions min, max. This was not an arbitrary choice but a necessity dictated by the fact that the unit interval is a lattice. Soon after the introduction of fuzzy sets, other functions were proposed to define the basic set operations of fuzzy sets. These functions, which have properties similar to min, are known as t-norms, whereas those that have properties similar to max are known as t-conorms [12]. The resulting fuzzy sets are quite useful since they can better model various cases and the literature is full of usage examples. However, this storm did not really affected multisets and fuzzy multisets and here we plan to remedy this situation. In particular, the purpose of this work is to introduce an extension, not a generalization, of both multisets and fuzzy multisets, where the union and the intersection between two such structures would be defined using functions other than min and max.

Broadly speaking, multi-criteria decision making (MCDM) is about decision making where many people are involved and where many criteria are taken into consideration in order to make a decision [13]. Since multisets, in particular, and fuzzy multisets, in general, are about “decisions with multiplicities”, it seems quite natural to use them in MCDM. Indeed, some authors have used them in MCDM [14,15] but here we are going to demonstrate how the introduction of t-norms and t-conorms can affect MCDM.

Structure of the Paper

In what follows, there is an outline of the theory of discrete t-norms and t-conorms. Then, triangular multisets and triangular fuzzy multisets are introduced. Next, there is a discussion about the use of these new structures in MCDM.

2. Discrete Triangular Norms and Conorms

Typically, t-norms and t-conorms are functions whose domain and codomain are the sets $[0,1]\times [0,1]$ and $[0,1]$, respectively (e.g., see [16] for an overview). More specifically:

Definition 1.

A t-norm is a binary operation $\ast :[0,1]\times [0,1]\to [0,1]$ that has the following properties for all $a,b,c\in [0,1]$:

1. 

Boundary condition $a\ast 1=a$and$a\ast 0=0$.

2. 

Monotonicity $b\le c$implies$a\ast b\le a\ast c$.

3. 

Commutativity $a\ast b=b\ast a$.

4 

Associativity $a\ast (b\ast c)=(a\ast b)\ast c$.

Here ≤ is the usual ordering operator.

The dual of a t-norm is a t-conorm:

Definition 2.

A t-conorm is a binary operation $★:[0,1]\times [0,1]\to [0,1]$ that has the following properties for all $a,b,c\in [0,1]$:

1. 

Boundary condition $a★0=a$and$a★1=1$.

2. 

Monotonicity $b\le c$implies$a★b\le a★c$.

3. 

Commutativity $a★b=b★a$.

4. 

Associativity $a★(b★c)=(a★b)★c$.

More generally, if we replace the unit interval with any complete lattice, then the previous definitions do make sense. Nevertheless, if we choose not to use the unit interval but a totally ordered set $(L,\le ,\alpha ,\beta )$, where $\alpha$ and $\beta$ are the minimum and the maximum elements of L, such that $\alpha \ne \beta$, then we actually define discrete t-norms and t-conorms (see [17,18] for an overview and [19] for a discussion about the usefulness of these functions in the definition of a version of fuzzy numbers).

Assume that ${\mathbb{N}}_{+\infty }=\{0,1,2,\dots ,\infty \}$, such that $x<+\infty$ and $x+(+\infty )=(+\infty )+x=+\infty$, for all $x\in \mathbb{N}$. Assume also that ${\mathbb{N}}_m=\{0,1,2,\dots ,m\}$, for some nonnegative integer m. Then, it is easy to prove the following:

Proposition 1.

$({\mathbb{N}}_{+\infty },<,0,+\infty )$ and $({\mathbb{N}}_m,<,0,m)$ are totally ordered sets.

Example 1.

Function $gcd(m,n)$ (i.e., the greatest common divisor of m and n) is a triangular norm on $({\mathbb{N}}_{+\infty },|)$, where $d|n$ if there is a natural number q such that $n=d·q$. Clearly, gcd has the following properties:

1. 

$gcd(m,n)=gcd(n,m)$;

2. 

$gcd\left(gcd(m,n),k\right)=gcd\left(m,gcd(n,k)\right)$;

3. 

If$m|m^{\prime }$and$n|n^{\prime }$, then$gcd(m,n)|gcd(m^{\prime },n^{\prime })$; and

4. 

$gcd(m,+\infty )=m$.

Example 2.

Function ${\mathrm{lcm}}_{\mathrm{ext}}(m,n)$ (i.e., the extended least common multiple of m and n) is defined as follows:

$${\mathrm{lcm}}_{\mathrm{ext}}(m,n)=\left\{\begin{array}{cc}\mathrm{lcm}(m,n),\hfill & \mathit{if}m>0\mathit{and}n>0\hfill \\ n,\hfill & \mathit{if}m=0\mathit{and}n>0\hfill \\ +\infty ,\hfill & \mathit{if}m=+\infty \hfill \end{array}\right.$$

Since $\mathrm{lcm}(m,n)=\mathrm{lcm}(n,m)$, there is no need to specify some cases in the previous definition. Now, it is easy to see that ${\mathrm{lcm}}_{\mathrm{ext}}(m,n)$ is a triangular conorm on $({\mathbb{N}}_{+\infty },<)$.

3. Triangular Multisets and Triangular Fuzzy Multisets

In this section we introduce t-multisets and t-fuzzy multisets; that is, multisets and fuzzy multisets that are defined on some totally ordered set. As expected, the definition of a triangular multiset is a variant of the standard definition of a multiset.

Definition 3.

Let D be a set called the universe. Then, a triangular multiset or just a t-multiset A over D is identified by a function $A:D\to {\mathbb{N}}_{+\infty }$, which is called its membership function. Similarly, a k-triangular multiset or just k-t-multiset B over D is identified by a function $B:D\to {\mathbb{N}}_k$.

Obviously, $A\left(d\right)=m$, where $d\in D$ and $m\in {\mathbb{N}}_{+\infty }$ mean that d occurs m times in the triangular multiset. Similarly, $B\left(d\right)=n$ means that d occurs n times in the k-triangular multiset. For reasons of simplicity, in what follows we will discuss only t-multisets.

The intersection and the union of two t-multisets can be defined using any pair of a discrete t-norm and a discrete t-conorm, respectively, as shown below:

Definition 4.

Assume that $A,B:D\to {\mathbb{N}}_{+\infty }$ are two triangular multisets. Then, their union and their intersection are the triangular multisets $E,F:D\to {\mathbb{N}}_{+\infty }$, such that for all $a\in D$

$$\begin{array}{cc}\hfill E\left(a\right)& =A\left(a\right)★B\left(a\right),\\ \hfill F\left(a\right)& =A\left(a\right)\ast B\left(a\right)\end{array}$$

where ★ is a discrete triangular conorm and ∗ is a discrete triangular norm.

The sum of two triangular multisets is defined as follows.

Definition 5.

Suppose that $A,B:D\to {\mathbb{N}}_{+\infty }$ are two triangular multisets. Then, their sum, denoted $A\uplus B$, is the triangular multiset $C:D\to {\mathbb{N}}_{+\infty }$, such for all $a\in A$:

$$C\left(d\right)=\left\{\begin{array}{cc}+\infty ,\hfill & \mathit{if}A\left(d\right)=+\infty \mathit{or}B\left(d\right)=+\infty ,\hfill \\ A\left(d\right)+B\left(d\right),\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

The proof of the following is easy and is omitted.

Proposition 2.

The sum of triangular multisets has the following properties:

1. 

Commutative: $A\uplus B=B\uplus B$;

2. 

Associative: $(A\uplus B)\uplus C=A\uplus (B\uplus C)$;

3. 

There exists a multiset, the null multiset ∅, such that $A\uplus \varnothing =A$.

Fuzzy multisets were introduced by [4]. He started with some universe D, and then defined fuzzy multisets to be functions with domain the set D and codomain the set of all multisets over the set $[0,1]$. Thus, a fuzzy multiset A is a function $A:D\to \left(\right[0,1]\to \mathbb{N})$. Alternatively, this function is equivalent to function $A:D\times [0,1]\to \mathbb{N}$. In other words, a fuzzy multiset A is a multiset over the “universe” $D\times [0,1]$. However, this means that an element $x\in D$ may have different membership degrees. In addition, each membership degree is associated with a different multiplicity degree. This author has argued in [5] that this is far too general and in some cases unrealistic. Instead, here we use multi-fuzzy sets as they were presented in the introduction. Let us now define a triangular multi-fuzzy set.

Definition 6.

Suppose that D is a set that we call the universe. Then, a triangular multi-fuzzy set A over D is identified by a function $A:D\to {\mathbb{N}}_{+\infty }\times [0,1]$. For any $d\in D$, $A\left(d\right)=(n,i)$ means that element d belongs n times to A and its membership degree is equal to i.

Given a fuzzy multiset A, we can define the following two functions: the multiplicity function $A_m:D\to {\mathbb{N}}_{+\infty }$ and the membership function $A_{\mu }:D\to [0,1]$. Obviously, if $A\left(d\right)=(n,i)$, then $A_m\left(d\right)=n$ and $A_{\mu }\left(d\right)=i$.

The basic set operations between triangular fuzzy multisets are defined in the following definition.

Definition 7.

Let $A,B:D\to {\mathbb{N}}_{+\infty }\times [0,1]$ be two triangular fuzzy multisets and suppose we use the pair $({\ast }_D,{★}_D)$ as a discrete t-norm and t-conorm, respectively, and the pair $(\ast ,★)$ as ordinary t-norm and t-conorm, respectively, then we define their sum, union, and intersection as follows:

$$\begin{array}{cc}\hfill (A\uplus B)\left(d\right)& =\left(A_m\left(d\right)+B_m\left(d\right),A_{\mu }\left(d\right)★B_{\mu }\left(d\right)\right)\hfill \\ \hfill (A\cup B)\left(d\right)& =\left(A_m\left(d\right){★}_D{B}_m\left(d\right),A_{\mu }\left(d\right)★B_{\mu }\left(d\right)\right)\hfill \\ \hfill (A\cap B)\left(d\right)& =\left(A_m\left(d\right){\ast }_D{B}_m\left(d\right),A_{\mu }\left(d\right)\ast B_{\mu }\left(d\right)\right)\hfill \end{array}$$

4. Using Triangular Multi-Fuzzy Sets in MCDM

An MCDM problem can be easily expressed in a matrix format [13].

Definition 8.

Assume that $A=\left\{A_i\right|1\le i\le n\}$ is a (finite) set of decision alternatives and that $G=\left\{g_j\right|1\le j\le m\}$ is a (finite) set of goals according to which the desirability of an action is judged. Determine the optimal alternative $A^{\ast }$ with the highest degree of desirability with respect to all relevant goals $g_j$.

$$\begin{array}{cc}\hfill & \begin{array}{ccccc}{C}_1/w_1\hfill & C_2/w_2\hfill & C_3/w_3\hfill & \dots \hfill & C_n/w_n\hfill \end{array}\hfill \\ \begin{array}{c}{A}_1\\ A_2\\ ⋮\\ A_m\end{array}\hfill & \left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & a_{2n}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{m1}& a_{m2}& a_{m3}& \dots & a_{mn}\end{array}\right]\hfill \end{array}$$

(1)

Here $C_i$ is a criterion and $w_i$ is its relative weight.

The previous approach to MCDM is purely qualitative and completely ignores the quantitative aspects of any such problem. To demonstrate this “flaw”, consider a country that wants to upgrade its military equipment. In all cases, it matters what kind of weapons one buys (e.g., warplanes, warships) and the quantity also matters. For example, if a country can purchase 100 F-15EX warplanes and 20 F-35 warplanes, then clearly it will choose to purchase F-15EX warplanes, despite the fact that an F-35 warplane is better than an F-15EX warplane. More generally, when we have $A_i$, where $i=1,\dots ,n$ decision alternatives, and the various criteria $C_j$, where $j=1,\dots ,m$, are the various criteria involved in the decision process and their relative weights are $w_j$, $a_{ij}$ should be a positive integer that will encode the quantitative aspect of this criterion. Naturally, this scheme can be easily modeled by a multi-fuzzy set. However, in order to make the idea more general and more flexible, we should use t-multi-fuzzy sets, since we may need to perform some operations between different sets of data and we want to be able to choose the tools that will give us the best possible outcome.

In [15], the authors present a case where a merchant wants to buy some goods for his shop. In this example, the authors do not clearly show how the quantitative aspect of their problem affects the final decision. Thus the merchant has the chance to buy different goods with different qualities and different quantities and she must take these facts into consideration before making her final decision. Note that it is not immediately obvious that a product that is available at low price and in high quantities is of bad quality. For example, there are new companies that try to promote their products and sell them at really low prices.

5. Conclusions

We have introduced the notion of triangular multisets and triangular multi-fuzzy sets as an obvious generalization of multisets and multi-fuzzy sets. These mathematical structures are more flexible than their original counterparts mainly because one can select how to define the basic operations between sets. This means that data can be processed in way that is closer to their true nature. After all, this is the reason why the corresponding extensions of fuzzy sets are so useful. To illustrate the usefulness of these new structures, we have briefly discussed their properties and their possible applications in MCDM. In addition, there are models of computation that process some sort of fuzzy multisets and thus these structures could replace them. Whether the resulting systems have interesting properties remains to be seen.