Efficient Algorithms for Data Processing under Type-3 (and Higher) Fuzzy Uncertainty

Abstract

It is known that, to more adequately describe expert knowledge, it is necessary to go from the traditional (type-1) fuzzy techniques to higher-order ones: type-2, probably type-3 and even higher. Until recently, only type-1 and type-2 fuzzy sets were used in practical applications. However, lately, it turned out that type-3 fuzzy sets are also useful in some applications. Because of this practical importance, it is necessary to design efficient algorithms for data processing under such type-3 (and higher-order) fuzzy uncertainty. In this paper, we show how we can combine known efficient algorithms for processing type-1 and type-2 uncertainty to come up with a new algorithm for the type-3 case.

1. Outline

Usual data processing algorithms treat data points as if they were exact. In practice, data come with uncertainty. When data come from experts who describe their knowledge by using imprecise (“fuzzy”) words from natural language, a natural way to describe the corresponding uncertainty is to use fuzzy techniques. To get a more accurate representation of expert uncertainty, it is necessary to use higher-order fuzzy techniques, i.e., go from the usual $[0,1]$-based type-1 techniques to type-2, type-3, and maybe even higher types. In this paper, we describe efficient algorithms for data processing under such higher-order fuzzy uncertainty.

The structure of this paper is as follows. In Section 2, we recall the need for data processing. In Section 3, we recall the need for fuzzy techniques and for higher-order fuzzy techniques. In Section 4, Section 5 and Section 6, we recall how data can be processed under type-1, interval type-2, and general type-2 fuzzy uncertainty. Finally, in Section 7, we use these known results to come up with new efficient algorithms for data processing under type-3 and higher-order fuzzy uncertainty. Section 8 contains conclusions and plans for future work.

2. Why Data Processing

One of the main objectives of science is to describe the current state of the world and to predict its future state. One of the main objectives of engineering is to design new buildings, gadgets, and/or new algorithms to make this future better. To describe the state of the world—and to describe the engineered objects—we need to list the numerical values of the quantities that characterize different natural and artificial objects.

Some quantities we can simply measure: we can directly measure the temperature outside, we can directly measure the distance between the two nearby buildings, etc. However, many quantities we cannot measure directly: e.g., we cannot directly measure the distance to a faraway star or the amount of oil in a given oilfield. Furthermore, it is definitely not possible to directly measure the future state—e.g., future temperature. To estimate such a difficult-to-measure quantity y, a natural idea is to find easier-to-measure-or-estimate quantities $x_1,x_2,\dots$ that are related to the desired quantity y by a known dependence $y=f(x_1,x_2,\dots )$. Then, we can measure or estimate the quantities $x_i$, and use the results $\tilde{y}$ of measurement or estimation to estimate y as $\tilde{y}=f({\tilde{x}}_1,{\tilde{x}}_2,\dots )$.

Computing this estimate, i.e., applying the algorithm $f(x_1,x_2,\dots )$ to the results of measurements and/or expert estimations is what is usually called data processing—see, e.g., [1].

3. Need for Fuzzy Uncertainty and Need for Higher-Order Fuzzy Uncertainty

3.1. Need for Fuzzy Uncertainty

Often, estimates for $x_i$ come from experts, and experts rarely provide exact values. Expert knowledge is usually formulated by using imprecise (“fuzzy”) words from natural language. An experienced driver explaining their driving strategy will not say that in a certain situation, you need to show down by exactly 5.0 km/h, he/she will probably say “slow down a little bit”, or “slow down by about 5 km/h”.

We want to use this imprecise knowledge in computer-related data processing. The challenge is that computers were designed to process numbers, not words from natural language. So, we need to transform expert statements into computer-understandable numerical form. For this purpose, Lotfi Zadeh invented fuzzy techniques (see, e.g., [2,3,4,5,6,7]), where each imprecise term like “small” is described by assigning, to each possible value x of the corresponding quantity, the degree $m\left(x\right)$—from the interval $[0,1]$—to which, according to the expert, this value is small. The resulting function $m\left(x\right)$ is known as the membership function or, alternatively, as the fuzzy set. This original idea is also called type-1 fuzzy techniques.

Let us describe this idea in precise terms.

Definition 1

([2,3,4,5,6,7]). Let U be a set. By a fuzzy subset of U, or, for short, a fuzzy set, we mean a function $m:U\to [0,1]$.

Usually, only normalized fuzzy sets are considered, i.e., fuzzy sets for which $m\left(x_0\right)$ for some $x_0\in U$.

Definition 2

([2,3,4,5,6]). A fuzzy set is called normalized if $m\left(x_0\right)=1$ for some $x_0$.

3.2. Fuzzy Numbers

For most terms, the membership function first (non-strictly) increases from 0 and then (non-strictly) decreases to 0. Such membership functions are known as fuzzy numbers.

Definition 3

([2,3,4,5,6,7]). A fuzzy set $m:\mathrm{I}\mathrm{R}\to [0,1]$ is called a fuzzy number if it satisfies the following two conditions:

• We have $m\left(x\right)\to 0$ when $x\to -\infty$ and when $x\to +\infty$.
• There exists a number $x_0$ such that $m\left(x\right)$ is (non-strictly) increasing for $x\le x_0$ and (non-strictly) decreasing for $x\ge x_0$.

It should be mentioned that sometimes, an additional requirement is added to this definition: that there exists an interval $[\underline{x},\overline{x}]$ such that $m\left(x\right)=0$ for all values x outside this interval.

3.3. “And”- and “Or”-Operations (T-Norms and T-Conorms)

Expert rules often involve logical connectives like “and” and “or”. For example, a rule can say that if a car in front of you is close and it slows down a little bit, then you should break a little bit. Strictly speaking, in this case, we need to find out, for each pair consisting of a distance value and a change-in-velocity value, the degree to which, for this pair, the condition “a car in front of you is close and it slows down a little bit” is satisfied. In this case, we may be able to do it, but, e.g., in medicine, we have rules with 5 or 6 different conditions. Even if we only try 10 values for each of the 5–6 variables, this still means asking ${10}^5$ to ${10}^6$ questions to an expert—this is not feasible. In such situations, to estimate the degree of confidence in a composite statement $A\&B$ or $A\vee B$, the only information we have is the expert’s degrees of confidence a and b in the original statements A and B.

The algorithm $f_{\&}(a,b)$ that estimates the degree of confidence in $A\&B$ based on this information is known as an “and”-operation or, for historical reason, a t-norm.

Definition 4

([2,3,4,5,6]). An “and”-operation (t-norm) is a function $f_{\&}:[0,1]\times [0,1]\to [0,1]$ that satisfies the following properties for all a, b, $a^{\prime }$, $b^{\prime }$, and c:

• $f_{\&}(a,b)=f_{\&}(b,a)$ (commutativity);
• $f_{\&}(a,f_{\&}(b,c))=f_{\&}(f_{\&}(a,b),c)$ (associativity);
• if $a\le a^{\prime }$ and $b\le b^{\prime }$, then $f_{\&}(a,b)\le f_{\&}(a^{\prime },b^{\prime })$ (monotonicity);
• $f_{\&}(0,a)=0$ and $f_{\&}(1,a)=a$.

Similarly, the algorithm $f_{\vee }(a,b)$ that estimates the degree of confidence in $A\vee B$ based on this information is known as an “or”-operation or, for historical reason, a t-conorm.

Definition 5

([2,3,4,5,6]). An“or”-operation (t-conorm) is a function $f_{\vee }:[0,1]\times [0,1]\to [0,1]$ that satisfies the following properties for all a, b, $a^{\prime }$, $b^{\prime }$, and c:

• $f_{\vee }(a,b)=f_{\vee }(b,a)$ (commutativity);
• $f_{\vee }(a,f_{\vee }(b,c))=f_{\vee }(f_{\vee }(a,b),c)$ (associativity);
• if $a\le a^{\prime }$ and $b\le b^{\prime }$, then $f_{\vee }(a,b)\le f_{\vee }(a^{\prime },b^{\prime })$ (monotonicity);
• $f_{\vee }(0,a)=a$ and $f_{\vee }(1,a)=1$.

The simplest, and frequently used, “and”- and “or”-operations are $f_{\&}(a,b)=min(a,b)$ and $f_{\vee }(a,b)=max(a,b)$.

3.4. Operations on Fuzzy Sets

For usual sets, the intersection $S_1\cap S_2$ of two sets is the set of all of all elements that belong to the first set $S_1$and that belong to the second set $S_2$. Similarly, the union $S_1\cup S_2$ of two sets is the set of all of all elements that belong to the first set $S_1$or that belong to the second set $S_2$. Thus, once we have selected “and”- and “or”-operations, we can define intersection and union of fuzzy sets $m_1\left(x\right)$ and $m_2\left(x\right)$ as, correspondingly, $m_{\cap }\left(x\right)=f_{\&}(m_1\left(x\right),m_2\left(x\right))$ and $m_{\cup }\left(x\right)=f_{\vee }(m_1\left(x\right),m_2\left(x\right))$. In particular, for the usual choice of $f_{\&}(a,b)=min(a,b)$ and $f_{\vee }(a,b)=max(a,b)$, we arrive at the following definitions:

Definition 6

([2,3,4,5,6,7]). Let U be a set and let $m_1:U\to [0,1]$ and $m_2:U\to [0,1]$ be fuzzy sets; then:

• by the intersection $m_{\cap }=m_1\cap m_2$ of these fuzzy sets, we mean the set $m_{\cap }\left(x\right)=min(m_1\left(x\right),$$m_2\left(x\right))$;
• by the union $m_{\cup }=m_1\cup m_2$ of these fuzzy sets, we mean the set $m_{\cup }\left(x\right)=max(m_1\left(x\right),m_2\left(x\right))$.

3.5. Data Processing under Fuzzy Uncertainty

Since fuzzy techniques are practically useful, it is desirable to develop efficient algorithms for data processing under such uncertainty:

• We know that the quantity-of-interest y is a function $y=f(x_1,x_2,\dots )$ of several auxiliary quantities $x_1,x_2,\dots$
• We also know, for each i, the membership function $m_i\left(x_i\right)$ that describes, for each real number $x_i$, the degree to which this number is a possible value of the i-th input.

Based on this information, we want to describe, for each real number y, the degree $m\left(y\right)$ to which this number is a possible value of the quantity of interest.

To determine this degree, let us take into account that a value y is possible if $y=f(x_1,x_2,\dots )$ for some possible values $x_i$. We know the degree $m_i\left(x_i\right)$ to which each value $x_i$ is possible. We can therefore use the min “and”-operation to describe, for each tuple $(x_1,x_2,\dots )$ for which $y=f(x_1,x_2,\dots )$, the degree to which all its values are possible—i.e.,. $x_1$ is possible and$x_2$ is possible, etc.—as $min(m_1\left(x_1\right),m_2\left(x_2\right),\dots )$.

The value y if possible if either the first tuple $(x_1,x_2,\dots )$ for which $y=f(x_1,x_2,\dots )$ is possible, or the second such tuple is possible, etc. We can therefore us the max “or”-operation to estimate the degree to which y is possible as

$$m\left(y\right)=sup\{min(m_1\left(x_1\right),m_2\left(x_2\right),\dots ):y=f(x_1,x_2,\dots )\}.$$

(1)

This formula was first described by Zadeh himself and is therefore known as Zadeh’s extension principle.

Definition 7

([2,3,4,5,6]). Let $U_1,U_2,\dots ,U$ be sets, let $m_i:U_i\to [0,1]$ be fuzzy sets, and let $f:U_1\times U_2\times \dots \to U$ be a function. By the result $m=f(m_1,m_2,\dots )$ of applying the function f to fuzzy sets $m_i$ we mean a fuzzy set $m:U\to [0,1]$ defined by the Equation (1).

3.6. Need for Type-2 Fuzzy Technique

The challenge with type-1 fuzzy technique is that similarly to the fact that an expert cannot name the exact value of the quantity, the same expert cannot produce the exact degree $m\left(x\right)$. At best, the expert can provide an interval of possible values of this degree—e.g., $[0.6,0.7]$—or even a fuzzy statement like “the degree is close to 0.6”. So, a natural idea is to allow the degree $m\left(x\right)$ to be an interval—which leads to interval-valued fuzzy sets—or even a fuzzy number corresponding to a statement like “the degree is close to 0.6”—this leads to so-called type-2 fuzzy sets. In general, an interval $[\underline{x},\overline{x}]$ can be viewed as a fuzzy set—the degree of confidence is 1 for all the values inside this interval and 0 for all the values outside this interval. Thus, interval-values fuzzy sets are particular cases of type-2 fuzzy sets. Type-2 fuzzy sets—both interval-valued and general—turned out to be useful in many applications, see, e.g., [4,8,9,10,11,12].

Definition 8

([4,5]). Let U be a set, and let I denote the set of all subintervals $[\underline{m},\overline{m}]\subseteq [0,1]$ of the interval $[0,1]$. By an interval-valued fuzzy subset of U, or, for short, an interval-valued fuzzy set, we mean a function $m:U\to I$.

In the interval-valued case, for each x, the expert-generated degree of confidence that x has the desired property (e.g., is small) is an interval $\mathbf{m}\left(x\right)=[\underline{m}\left(x\right),\overline{m}\left(x\right)]$. In the general type-2 fuzzy case, we have the following definition.

Definition 9

([4,5]). Let U be a set, and let $F\left(\right[0,1\left]\right)$ denote the set of all fuzzy subsets of the interval $[0,1]$. By a type-2 fuzzy subset of U, or, for short, a type-2 fuzzy set, we mean a function $m:U\to F\left(\right[0,1\left]\right)$.

In the general type-2 case, for each x and for each number t from the interval $[0,1]$, the expert provides a degree to which this number t is a degree of confidence that x has the desired property (like “small”). We will denote this degree by $m(x,t)$.

3.7. Operations on Interval-Valued and General Type-2 Fuzzy Sets

To describe union and intersection of interval-valued and general type-2 fuzzy sets, it is natural to use formulas similar to formulas from Definition 6. To make sense of these formulas, we need to describe what is the meaning of $min(m_1,m_2)$ and $max(m_1,m_2)$ for the case when $m_i$ are both fuzzy sets—but that meaning is already provided by Definition 7, for the case when $U_1=U_2=U=[0,1]$ and $f(a,b)=min(a,b)$ or $f(a,b)=max(a,b)$. Thus, we arrive at the following definitions:

Definition 10

([4,5]). Let U be a set and let $m_1:U\to F\left([0,1]\right)$ and $m_2:U\to F\left([0,1]\right)$ be type-2 fuzzy sets; then:

• by the intersection $m_{\cap }=m_1\cap m_2$ of these type-2 fuzzy sets, we mean the type-2 fuzzy set $m_{\cap }\left(x\right)=min(m_1\left(x\right),m_2\left(x\right))$, where, for each x, the result $min(m_1\left(x\right),m_2\left(x\right))$ of applying the function $f(a,b)=min(a,b)$ to fuzzy sets $m_1\left(x\right)$ and $m_2\left(x\right)$ is defined by Definition 7.
• by the union $m_{\cup }=m_1\cup m_2$ of these type-2 fuzzy sets, we mean the set $m_{\cup }\left(x\right)=max(m_1\left(x\right),$$m_2\left(x\right))$, where, for each x, the result $max(m_1\left(x\right),m_2\left(x\right))$ of applying the function $f(a,b)=max(a,b)$ to fuzzy sets $m_1\left(x\right)$ and $m_2\left(x\right)$ is defined by Definition 7.

One can show that for interval-valued fuzzy sets, when $m_i\left(x\right)=[{\underline{m}}_i\left(x\right),{\overline{m}}_i\left(x\right)]$, the resulting interval-valued membership functions $m_{\cap }\left(x\right)$ and $m_{\cup }\left(x\right)$ have the following form:

$$m_{\cap }\left(x\right)=[min({\underline{m}}_1\left(x\right),{\underline{m}}_2\left(x\right)),min({\overline{m}}_1\left(x\right),{\overline{m}}_2\left(x\right))];$$

$$m_{\cup }\left(x\right)=[max({\underline{m}}_1\left(x\right),{\underline{m}}_2\left(x\right)),max({\overline{m}}_1\left(x\right),{\overline{m}}_2\left(x\right))].$$

3.8. Data Processing under Type-2 Fuzzy Uncertainty

Since, as we have mentioned, type-2 fuzzy techniques are practically useful, it is desirable to develop efficient algorithms for data processing under such uncertainty:

• We know that the quantity-of-interest y is a function $y=f(x_1,x_2,\dots )$ of several auxiliary quantities $x_1,x_2,\dots$.
• We also know, for each i, the membership function $m_i\left(x_i\right)$ that describes, for each real number $x_i$, the (fuzzy-valued) degree to which this number is a possible value of the i-th input.

Based on this information, we want to describe, for each real number y, the (fuzzy-valued) degree $m\left(y\right)$ to which this number is a possible value of the quantity of interest.

To describe the result of applying a function $f(x_1,x_2,\dots )$ to type-2 fuzzy sets, it is natural to use the same Equation (1) as for the usual (type-1) fuzzy sets. To make sense of this formula, we need to describe what is the meaning of its right-hand side when the values $m_i\left(x_i\right)$ are themselves fuzzy sets—but that meaning is already provided by Definition 7. Thus, we arrive at the following definition.

Definition 11

([4]). Let $U_1,U_2,\dots ,U$ be sets, let $m_i:U_i\to F\left([0,1]\right)$ be type-2 fuzzy sets, and let $f:U_1\times U_2\times \dots \to U$ be a function. By the result $m=f(m_1,m_2,\dots )$ of applying the function f to type-2 fuzzy sets $m_i$ we mean a fuzzy set $m:U\to F\left(\right[0,1\left]\right)$ defined by the Equation (1), in which the right-hand side is understood according to Definition 7.

3.9. Need for Type-3 and Higher-Order Fuzzy Techniques

Similarly to the fact that an expert cannot describe their degree of confidence—that x is small—by a single number, the same expert cannot describe their degree of confidence that t is a degree of confidence that x is small by a single number. At best, the expert can provide either an interval $[\underline{m}(x,t),\overline{m}(x,t)]$ or a fuzzy number that describes this degree of confidence. The fuzzy case is known as type-3 fuzzy technique, and the interval-valued case is known as interval type-3.

Definition 12

([13]). Let U be a set, and let $F_2\left([0,1]\right)$ denote the set of all type-2 fuzzy subsets of the interval $[0,1]$. By a type-3 fuzzy subset of U, or, for short, a type-3 fuzzy set, we mean a function $m:U\to F_2\left([0,1]\right)$.

In the general type-3 case, for each value s from the interval $[0,1]$, we provide a degree—denoted by $m(x,t,s)$—that s is degree of confidence in the statement “t is a degree of confidence that x has the desired property”.

3.10. Operations on Type-3 Fuzzy Sets

To describe union and intersection of type-3 fuzzy sets, it is natural to use formulas similar to formulas from Definition 6. To make sense of these formulas, we need to describe what is the meaning of $min(m_1,m_2)$ and $max(m_1,m_2)$ for the case when $m_i$ are both type-2 fuzzy sets—but that meaning is already provided by Definition 11, for the case when $U_1=U_2=U=[0,1]$ and $f(a,b)=min(a,b)$ or $f(a,b)=max(a,b)$. Thus, we arrive at the following definitions:

Definition 13

([13]). Let U be a set and let $m_1:U\to F_2\left([0,1]\right)$ and $m_2:U\to F_2\left([0,1]\right)$ be type-3 fuzzy sets; then:

• by the intersection $m_{\cap }=m_1\cap m_2$ of these type-3 fuzzy sets, we mean the type-3 fuzzy set $m_{\cap }\left(x\right)=min(m_1\left(x\right),m_2\left(x\right))$, where, for each x, the result $min(m_1\left(x\right),m_2\left(x\right))$ of applying the function $f(a,b)=min(a,b)$ to type-2 fuzzy sets $m_1\left(x\right)$ and $m_2\left(x\right)$ is defined by Definition 11.
• by the union $m_{\cup }=m_1\cup m_2$ of these type-3 fuzzy sets, we mean the type-3 fuzzy set $m_{\cup }\left(x\right)=max(m_1\left(x\right),m_2\left(x\right))$, where, for each x, the result $max(m_1\left(x\right),m_2\left(x\right))$ of applying the function $f(a,b)=max(a,b)$ to type-2 fuzzy sets $m_1\left(x\right)$ and $m_2\left(x\right)$ is defined by Definition 11.

3.11. Is This Worth Considering?

At first glance, the difference between type-2 and type-3 is so subtle and complicated that one can doubt whether it is necessary to use type-3 in practical applications. Actually, people doubted that type-2 would be practically useful—and, as we have mentioned, it turned out that it is often useful. Similarly, it turned out that type-3 techniques are also useful in many practical cases; see, e.g., [13,14] and references therein. Examples of successful use of type-3 fuzzy techniques range from improving the quality of automatic tuning of a television image [15] to more accurate stock market predictions [16].

It should be mentioned that current applications of type-3 fuzzy techniques only use interval-valued type-3 fuzzy sets, i.e., function $m:U\to F_2\left([0,1]\right)$ for which, for every m, the degree $m\left(x\right)$ is an interval-valued fuzzy set. This limitation is caused largely by the fact that processing general type-3 fuzzy sets has been, so far, computationally complicated. This paper’s new efficient algorithm for processing type-3 fuzzy data will help make general type-3 more feasible and will, thus, hopefully, will lead to useful applications of general type-3 fuzzy sets.

3.12. Data Processing under Type-3 Fuzzy Uncertainty

Since, as we have mentioned, type-3 fuzzy techniques are practically useful, it is desirable to develop efficient algorithms for data processing under such uncertainty.

• We know that the quantity-of-interest y is a function $y=f(x_1,x_2,\dots )$ of several auxiliary quantities $x_1,x_2,\cdots$.
• We also know, for each i, the membership function $m_i\left(x_i\right)$ that describes, for each real number $x_i$, the (type-2-fuzzy-valued) degree to which this number is a possible value of the i-th input.

Based on this information, we want to describe, for each real number y, the (type-2-fuzzy-valued) degree $m\left(y\right)$ to which this number is a possible value of the quantity of interest.

To describe the result of applying a function $f(x_1,x_2,\dots )$ to type-3 fuzzy sets, it is natural to use the same Equation (1) as for type-1 and type-2 fuzzy sets. To make sense of this formula, we need to describe what is the meaning of its right-hand side when the values $m_i\left(x_i\right)$ are themselves type-2 fuzzy sets—but that meaning is already provided by Definition 11. Thus, we arrive at the following definition.

Definition 14.

Let $U_1,U_2,\dots ,U$ be sets, let $m_i:U_i\to F_2\left([0,1]\right)$ be type-3 fuzzy sets, and let $f:U_1\times U_2\times \dots \to U$ be a function. By the result $m=f(m_1,m_2,\dots )$ of applying the function f to type-3 fuzzy sets $m_i$ we mean a fuzzy set $m:U\to F_2\left([0,1]\right)$ defined by Equation (1), in which the right-hand side is understood according to Definition 11.

3.13. What about Higher Order Types?

Clearly, an expert cannot provide the exact degree $m(x,t,s)$, so a natural idea is to allow an expert to provide interval-valued of fuzzy degrees—which leads to type-4, where for each real number r from the interval $[0,1]$, we ask the expert to describe their degree of confidence $m(x,t,s,r)$ that r is a proper value of $m(x,t,s)$.

The expert cannot describe the precise value of $m(x,t,s,r)$, so this value can also be fuzzy—we get type-5, etc. We can have the following inductive definitions, that describe, for every natural number $L>3$, type-L fuzzy sets and operations on them in terms of fuzzy sets of type $(L-1)$.

Definition 15.

Let U be a set, and let $F_{L-1}\left([0,1]\right)$ denote the set of all type-$(L-1)$ fuzzy subsets of the interval $[0,1]$. By a type-L fuzzy subset of U, or, for short, a type-L fuzzy set, we mean a function $m:U\to F_{L-1}\left([0,1]\right)$.

Definition 16.

Let U be a set and let $m_1:U\to F_{L-1}\left([0,1]\right)$ and $m_2:U\to F_{L-1}\left([0,1]\right)$ be type-L fuzzy sets; then:

• by the intersection $m_{\cap }=m_1\cap m_2$ of these type-L fuzzy sets, we mean the type-L fuzzy set $m_{\cap }\left(x\right)=min(m_1\left(x\right),m_2\left(x\right))$, where, for each x, the result $min(m_1\left(x\right),m_2\left(x\right))$ of applying the function $f(a,b)=min(a,b)$ to type-$(L-1)$ fuzzy sets $m_1\left(x\right)$ and $m_2\left(x\right)$ is defined by Definition 14 (for $L=4$) or Definition 17 (for other L).
• by the union $m_{\cup }=m_1\cup m_2$ of these type-L fuzzy sets, we mean the type-L fuzzy set $m_{\cup }\left(x\right)=max(m_1\left(x\right),m_2\left(x\right))$, where, for each x, the result $max(m_1\left(x\right),m_2\left(x\right))$ of applying the function $f(a,b)=max(a,b)$ to type-$(L-1)$ fuzzy sets $m_1\left(x\right)$ and $m_2\left(x\right)$ is defined by Definition 14 (for $L=4$) or Definition 17 (for other L).

Definition 17.

Let $U_1,U_2,\dots ,U$ be sets, let $m_i:U_i\to F_{L-1}\left([0,1]\right)$ be type-L fuzzy sets, and let $f:U_1\times U_2\times \dots \to U$ be a function. By the result $m=f(m_1,m_2,\dots )$ of applying the function f to type-L fuzzy sets $m_i$ we mean a fuzzy set $m:U\to F_{L-1}\left([0,1]\right)$ defined by the Equation (1), in which the right-hand side is understood according to Definition 14 (for $L=4$) or according to this same definition (for other L).

3.14. Need for Data Processing under Such Uncertainty

Since type-1, type-2, and type-3 fuzzy techniques are practically useful, it is desirable to develop efficient algorithms for data processing under such uncertainty. Efficient algorithms for type-1 and type-2 are known—we describe them in the following sections. Efficient algorithms for type-3 case are described in the last section of this paper.

We do not know yet whether type-4, type-5, etc., will be practically useful, but the fact that type-2 and type-3 turned out to be useful makes us think that it is quite probable that higher-order fuzzy sets will be useful. So it makes sense to think of efficient algorithms for these cases too, and this is what we will do in the same last section.

4. Effective Algorithms for Data Processing under Type-1 Fuzzy Uncertainty: Reminder

4.1. How to Actually Perform Data Processing: Analysis of the Problem

Straightforward computation of the Equation (1) requires solving a complex constraint optimization problem—which is, in general, time-consuming. It is known, however, that there are more efficient ways to compute $m\left(y\right)$. These ways are related to the notion of α-cuts of a fuzzy sets, which are defined, for each $\alpha \in (0,1]$, as $\{x:m(x)\ge \alpha \}$. For fuzzy numbers, each $\alpha$-cut is an interval; we will denote it by $\mathbf{x}\left(\alpha \right)=[\underline{m}\left(\alpha \right),\overline{m}\left(\alpha \right)]$.

For $\alpha =0$, we can use a slightly different formulation of the $\alpha$-cut: it the closure $\mathbf{x}\left(0\right)=\overline{\{x:m(x)>0\}}$ of the set $\{x:m(x)>0\}$.

Definition 18

([2,3,4,5,6]). Let U be a set, let $m:U\to [0,1]$ be a fuzzy set, and let $\alpha \in [0,1]$ be a real number. Then, by the$\alpha$-cut of m, we mean the following set:

• when $\alpha >0$, we take $\{x:m(x)\ge \alpha \}$;
• when $\alpha =0$, we take $\overline{\{x:m(x)>0\}}$.

In the following text, for simplicity, we will only list the simpler formula which is valid for $\alpha >0$, but, of course, for $\alpha =0$, we have to use the more complex formula.

Once we know all the $\alpha$-cuts, we can reconstruct the membership function as $m\left(x\right)=sup\{\alpha :x\in \mathbf{x}(\alpha \left)\right\}$. In particular, if we know $\alpha$-cuts for $\alpha =0,0.1,0.2,\dots ,1.0$, then we can reconstruct $m\left(x\right)$ with accuracy 0.1—which is usually sufficient, since experts rarely produce their degree of confidence with higher accuracy. So, to find $m\left(y\right)$, it is sufficient to find the $\alpha$-cuts $\mathbf{y}\left(\alpha \right)$ for the corresponding 11 values $\alpha$.

Because of the possibility to easily move from the usual representation of the membership function $m\left(x\right)$ and its $\alpha$-cut representation, sometimes the membership function is stored by listing the corresponding $\alpha$-cuts.

To find the $\alpha$-cuts corresponding to the desired quantity y, we can take into account that the value $m\left(y\right)$ as described by the Equation (1) is larger than or equal to $\alpha$ if and only if for one of the tuples $(x_1,x_2,\dots )$ for which $y=f(x_1,x_2,\dots ,)$, we have $min(m_1\left(x_1\right),m_2\left(x_2\right),\dots )\ge \alpha$. This inequality, in its turn, is equivalent to requiring that $m_i\left(x_i\right)\ge \alpha$ for all i. Thus, the $\alpha$-cut for y is equal to the range of the function $y=f(x_1,x_2,\dots )$ when each $x_i$ is in the corresponding $\alpha$-cut:

$$\mathbf{y}\left(\alpha \right)=f({\mathbf{x}}_1\left(\alpha \right),{\mathbf{x}}_2\left(\alpha \right),\dots ),$$

(2)

where for each sets $X_1,X_2,\dots$, the range $f(X_1,X_2,\dots )$ is defined as

$$f(X_1,X_2,\dots )\stackrel{\mathrm{def}}{=}\{f(x_1,x_2,\dots ):x_1\in X_1,x_2\in X_2,\dots \}.$$

(3)

The problem of computing the range of a function when each input is in a known interval is known as the problem of interval computations; there are efficient general algorithms for estimating this range, see, e.g., [17,18,19,20].

4.2. Comment

In some important cases, interval computation is easy, no general complex algorithms are needed. For example, if the function $f(x_1,x_2,\dots )$ is (non-strictly) increasing in each of its variables, then the smallest value of this function on intervals $X_i=[{\underline{x}}_i,{\overline{x}}_i]$ is attained when each input $x_i$ is the smallest, i.e., when $x_i={\underline{x}}_i$ for all i. Similarly, the largest value of this function on intervals $X_i=[{\underline{x}}_i,{\overline{x}}_i]$ is attained when each input $x_i$ is the largest, i.e., when $x_i={\overline{x}}_i$ for all i. Thus,

$$f([{\underline{x}}_1,{\overline{x}}_1],[{\underline{x}}_2,{\overline{x}}_2],\dots )=[f({\underline{x}}_1,{\underline{x}}_2,\dots ),f({\overline{x}}_1,{\overline{x}}_2,\dots )].$$

4.3. Resulting Algorithm

• First, if the information about the inputs $x_i$ is stored in the form of the usual membership functions $m_i\left(x_i\right)$, we compute, for each i and for each value $\alpha \in \{0,0.1,\dots ,1.0\}$, the corresponding $\alpha$-cut

  $${\mathbf{x}}_i\left(\alpha \right)=\{x_i:m_i\left(x_i\right)\ge \alpha \}.$$
  (Recall that for $\alpha =0$, we will have to use a slightly more complex formula.)
• Then, for each value $\alpha$ from the above list, we use an interval computation algorithm to compute the range $\mathbf{y}\left(\alpha \right)=f({\mathbf{x}}_1\left(\alpha \right),{\mathbf{x}}_2\left(\alpha \right),\dots ).$ These ranges form the $\alpha$-cut representation of the desired membership function $m\left(y\right)$.
• Finally, if we want to represent this membership function in the usual form, we compute $m\left(y\right)=max\{\alpha :y\in \mathbf{y}(\alpha \left)\right\}$.

4.4. How Many Computation Steps Do We Need

These computations need to be repeated for all $\alpha$. So, if we use 11 values $\alpha =0,0.1,\dots ,1.0$, then, to find the result of data processing under type-1 fuzzy uncertainty, we need to apply an interval computations algorithm 11 times.

5. Data Processing under Interval-Valued Fuzzy Uncertainty: Reminder

5.1. Formulation of the Problem

In the interval-valued case, the relation between $m\left(y\right)$ and $m_i\left(x_i\right)$ is described by the same Equation (1); the main difference is that now, values $m\left(y\right)$ and $m_i\left(x_i\right)$ are not numbers but intervals.

The corresponding efficient algorithms are described in [21,22].

5.2. Interval Case: Analysis of the Problem

In the interval case, each value $m_i\left(x_i\right)$ is an interval $[{\underline{m}}_i\left(x_i\right),{\overline{m}}_i\left(x_i\right)]$. The right-hand side of the Equation (1) is a non-strictly increasing function of all the values $m_i\left(x_i\right)$. Thus, the desired range is equal to $[\underline{m}\left(y\right),\overline{m}\left(y\right)]$, where

$$\underline{m}\left(y\right)=sup\{min({\underline{m}}_1\left(x_1\right),{\underline{m}}_2\left(x_2\right),\dots ):y=f(x_1,x_2,\dots )\}\mathrm{and}$$

$$\overline{m}\left(y\right)=sup\{min({\overline{m}}_1\left(x_1\right),{\overline{m}}_2\left(x_2\right),\dots ):y=f(x_1,x_2,\dots )\}.$$

These are exactly formulas (1) for membership functions ${\underline{m}}_i\left(x_i\right)$ and ${\overline{m}}_i\left(x_i\right)$. So, to compute each of the two bounds $\underline{m}\left(y\right)$ and $\overline{m}\left(y\right)$, we can use the efficient $\alpha$-cut-based algorithm.

5.3. Interval Case: Resulting Algorithm

We are given interval-valued membership functions $[{\underline{m}}_i\left(x_i\right),{\overline{m}}_i\left(x_i\right)]$.

• Based on each of these membership functions, for each i and for each value $\alpha$ from the given list, we compute the orrepsonding $\alpha$-cuts as:

  $${\underline{\mathbf{x}}}_i\left(\alpha \right)=\{x_i:{\underline{m}}_i\left(x_i\right)\ge \alpha \}\mathrm{and} {\overline{\mathbf{x}}}_i\left(\alpha \right)=\{x_i:{\overline{m}}_i\left(x_i\right)\ge \alpha \}.$$
• We compute the $\alpha$-cuts $\underline{\mathbf{y}}\left(\alpha \right)$ and $\overline{\mathbf{y}}\left(\alpha \right)$ for the endpoints $\underline{m}\left(y\right)$ and $\overline{m}\left(y\right)$ of the interval-valued membership function $[\underline{m}\left(y\right),\overline{m}\left(y\right)]$ as follows:

  $$\underline{\mathbf{y}}\left(\alpha \right)=f({\underline{\mathbf{x}}}_1\left(\alpha \right),{\underline{\mathbf{x}}}_2\left(\alpha \right),\dots )\mathrm{and} \overline{\mathbf{y}}\left(\alpha \right)=f({\overline{\mathbf{x}}}_1\left(\alpha \right),{\overline{\mathbf{x}}}_2\left(\alpha \right),\dots ).$$
• Finally, the compute the endpoints $\underline{m}\left(y\right)$ and $\overline{m}\left(y\right)$ of the desired interval-valued membership function $[\underline{m}\left(y\right),\overline{m}\left(y\right)]$ as

  $$\underline{m}\left(y\right)=max\{\alpha :y\in \underline{\mathbf{y}}\left(\alpha \right)\}\mathrm{and} \overline{m}\left(y\right)=max\{\alpha :y\in \overline{\mathbf{y}}\left(\alpha \right)\}.$$

5.4. How Many Computation Steps Do We Need

These computations need to be repeated for all $\alpha$. So, if we use 11 values $\alpha =0,0.1,\dots ,1.0$, then, to find the result of data processing under type-2 fuzzy uncertainty, we need to apply an interval computations algorithm $2\times 11=22$ times.

6. Data Processing under General Type-2 Fuzzy Uncertainty: Reminder

6.1. Formulation of the Problem

In the general type-2 case, the relation between $m\left(y\right)$ and $m_i\left(x_i\right)$ is described by the same Equation (1); the main difference is that now, values $m\left(y\right)$ and $m_i\left(x_i\right)$ are not numbers but fuzzy sets.

The corresponding efficient algorithms are described in [21,22].

6.2. General Type-2 Case: Analysis of the Problem

In the general type-2 case, $m\left(y\right)$ and $m_i\left(x_i\right)$ are fuzzy numbers. In this case, we can use the general type-1 result that the processing of fuzzy numbers is equivalent to computing the ranges of the processing function on different $\alpha$-cuts. In this case, the data processing is described by the Equation (1).

To distinguish $\alpha$-cuts of the original membership functions for $x_i$ and y and the $\alpha$-cuts of each fuzzy number $m\left(y\right)$ and $m_i\left(x_i\right)$, we will use the letter $\beta$ for the new alpha-cuts. Thus, we get the following for each $\beta$:

$$\mathbf{m}\left(y\right)\left(\beta \right)=sup\{min({\mathbf{m}}_1\left(x_1\right)\left(\beta \right),{\mathbf{m}}_2\left(x_2\right)\left(\beta \right),\dots ):y=f(x_1,x_2,\dots )\},$$

where

$$\mathbf{m}\left(y\right)\left(\beta \right)\stackrel{\mathrm{def}}{=}\{t:m(y,t)\ge \beta \}\mathrm{and} {\mathbf{m}}_i\left(x_i\right)\left(\beta \right)\stackrel{\mathrm{def}}{=}\{t:m_i(x_i,t)\ge \beta \}.$$

For fuzzy numbers, $\beta$-cuts are intervals, and the corresponding relation (1) is increasing. Thus, the above formula means that to get the lower endpoint $\underline{m}\left(y\right)\left(\beta \right)$ of a y’s $\beta$-cut, we need to use only lower endpoints for $\beta$-cuts for $x_i$, and similarly for the upper endpoints:

$$\underline{m}\left(y\right)\left(\beta \right)=sup\{min({\underline{m}}_1\left(x_1\right)\left(\beta \right),{\underline{m}}_2\left(x_2\right)\left(\beta \right),\dots ):y=f(x_1,x_2.\dots )\}\mathrm{and}$$

$$\overline{m}\left(y\right)\left(\beta \right)=sup\{min({\overline{m}}_1\left(x_1\right)\left(\beta \right),{\overline{m}}_2\left(x_2\right)\left(\beta \right),\dots ):y=f(x_1,x_2,\dots )\}.$$

Each of these formulas is, in effect, Zadeh’s extension principle for the corresponding membership functions. Thus, there formulas can be reformulated in terms of $\alpha$-cuts of the corresponding membership functions:

$$\underline{\mathbf{y}}(\alpha ,\beta )=f({\underline{\mathbf{x}}}_1(\alpha ,\beta ),{\underline{\mathbf{x}}}_2(\alpha ,\beta ),\dots )\mathrm{and}$$

$$\overline{\mathbf{y}}(\alpha ,\beta )=f({\overline{\mathbf{x}}}_1(\alpha ,\beta ),{\overline{\mathbf{x}}}_2(\alpha ,\beta ),\dots ),$$

where

$$\underline{\mathbf{y}}(\alpha ,\beta )\stackrel{\mathrm{def}}{=}\{y:\underline{m}\left(y\right)\left(\beta \right)\ge \alpha \},{\underline{\mathbf{x}}}_i(\alpha ,\beta )\stackrel{\mathrm{def}}{=}\{x_i:{\underline{m}}_i\left(x_i\right)\left(\beta \right)\ge \alpha \},$$

$$\overline{\mathbf{y}}(\alpha ,\beta )\stackrel{\mathrm{def}}{=}\{y:\overline{m}\left(y\right)\left(\beta \right)\ge \alpha \},{\overline{\mathbf{x}}}_i(\alpha ,\beta )\stackrel{\mathrm{def}}{=}\{x_i:{\overline{m}}_i\left(x_i\right)\left(\beta \right)\ge \alpha \}.$$

Hence, we arrive at the following algorithm:

6.3. General Type-2 Case: Resulting Algorithm

We start with type-2 membership functions $m_i(x_i,t)$.

• First, for each i and for each value $\beta$ from the given list, we compute the $\beta$-cuts

  $$[{\underline{m}}_i\left(x_i\right)\left(\beta \right),{\overline{m}}_i\left(x_i\right)\left(\beta \right)]\stackrel{\mathrm{def}}{=}\{t:m_i(x_i,t)\ge \beta \}.$$
• Then, for each i and for each pair of values $(\alpha ,\beta )$ from the given list, we compute the $\alpha$-cuts

  $${\underline{\mathbf{x}}}_i(\alpha ,\beta )\stackrel{\mathrm{def}}{=}\{x_i:{\underline{m}}_i\left(x_i\right)\left(\beta \right)\ge \alpha \}\mathrm{and} {\overline{\mathbf{x}}}_i(\alpha ,\beta )\stackrel{\mathrm{def}}{=}\{x_i:{\overline{m}}_i\left(x_i\right)\left(\beta \right)\ge \alpha \}.$$
• For each $\alpha$ and $\beta$, we then use an interval computation algorithm to compute:

  $$\underline{\mathbf{y}}(\alpha ,\beta )=f({\underline{\mathbf{x}}}_1(\alpha ,\beta ),{\underline{\mathbf{x}}}_2(\alpha ,\beta ),\dots )\mathrm{and}$$

  $$\overline{\mathbf{y}}(\alpha ,\beta )=f({\overline{\mathbf{x}}}_1(\alpha ,\beta ),{\overline{\mathbf{x}}}_2(\alpha ,\beta ),\dots ).$$
• Based on these intervals, for each $\beta$, we compute

  $$\underline{m}\left(y\right)\left(\beta \right)=sup\{\alpha :y\in \underline{\mathbf{y}}(\alpha ,\beta )\}\mathrm{and} \overline{m}\left(y\right)\left(\beta \right)=sup\{\alpha :y\in \overline{\mathbf{y}}(\alpha ,\beta )\}.$$
• Finally, we compute the desired membership function

  $$m(y,t)=max\{\beta :t\in [\underline{m}\left(y\right)\left(\beta \right),\overline{m}\left(y\right)\left(\beta \right)]\}.$$

6.4. How Many Computation Steps Do We Need

These computations need to be repeated for all $\alpha$ and $\beta$. So, if for each of these two parameters, we use 11 values $\alpha ,\beta =0,0.1,\dots ,1.0$, then, to find the result of data processing under type-2 fuzzy uncertainty, we need to apply an interval computations algorithm $2\times {11}^2=242$ times.

7. Data Processing under Type-3 (and Higher Order) Fuzzy Uncertainty: A New Algorithm

7.1. Formulation of the Problem

Let us show the above type-2 algorithms can be used to come with an efficient algorithm for the type-3 case.

7.2. Type-3 Case: Analysis of the Problem

In the type-3 case, each value $m\left(y\right)$ and $m_i\left(x_i\right)$ is a type-2 fuzzy set. Thus, we have the relation (1) between these type-2 fuzzy sets. So, based on the algorithm presented in the previous section, for each pair of values $\beta$ and $\gamma$ from the interval $[0,1]$, we have:

$$\underline{\mathbf{m}}\left(y\right)(\beta ,\gamma )=$$

$$sup\{min({\underline{\mathbf{m}}}_1\left(x_1\right)(\beta ,\gamma ),{\underline{\mathbf{m}}}_2\left(x_2\right)(\beta ,\gamma ),\dots ):y=f(x_1,x_2,\dots )\}$$

(4)

and

$$\overline{\mathbf{m}}\left(y\right)(\beta ,\gamma )=$$

$$sup\{min({\overline{\mathbf{m}}}_1\left(x_1\right)(\beta ,\gamma ),{\overline{\mathbf{m}}}_2\left(x_2\right)(\beta ,\gamma ),\dots ):y=f(x_1,x_2,\dots )\},$$

(5)

where

$$\underline{\mathbf{m}}\left(y\right)(\beta ,\gamma )=[{\underline{m}}^{-}\left(y\right)(\beta ,\gamma ),{\underline{m}}^{+}\left(y\right)(\beta ,\gamma )]\stackrel{\mathrm{def}}{=}\{t:\underline{m}(y,t)\left(\gamma \right)\ge \beta \},$$

$${\underline{\mathbf{m}}}_i\left(x_i\right)(\beta ,\gamma )=[{\underline{m}}_i^{-}\left(x_i\right)(\beta ,\gamma ),{\underline{m}}_i^{+}\left(x_i\right)(\beta ,\gamma )]\stackrel{\mathrm{def}}{=}\{t:{\underline{m}}_i(x_i,t)\left(\gamma \right)\ge \beta \},$$

$$\overline{\mathbf{m}}\left(y\right)(\beta ,\gamma )=[{\overline{m}}^{-}\left(y\right)(\beta ,\gamma ),{\overline{m}}^{+}\left(y\right)(\beta ,\gamma )]\stackrel{\mathrm{def}}{=}\{t:\overline{m}(y,t)\left(\gamma \right)\ge \beta \},$$

$${\overline{\mathbf{m}}}_i\left(x_i\right)(\beta ,\gamma )=[{\overline{m}}_i^{-}\left(x_i\right)(\beta ,\gamma ),{\overline{m}}_i^{+}\left(x_i\right)(\beta ,\gamma )]\stackrel{\mathrm{def}}{=}\{t:{\overline{m}}_i(x_i,t)\left(\gamma \right)\ge \beta \},$$

and

$$[\underline{m}(y,t)\left(\gamma \right),\overline{m}(y,t)\left(\gamma \right)]\stackrel{\mathrm{def}}{=}\{s:m(y,t,s)\ge \gamma \},$$

$$[{\underline{m}}_i(x_i,t)\left(\gamma \right),{\overline{m}}_i(x_i,t)\left(\gamma \right)]\stackrel{\mathrm{def}}{=}\{s:m_i(x_i,t,s)\ge \gamma \}.$$

The corresponding transformation (1) is non-strictly increasing, thus the Equations (4) and (5) lead to similar relations between endpoints of the corresponding intervals:

$${\underline{m}}^{-}\left(y\right)(\beta ,\gamma )=$$

$$sup\{min({\underline{m}}_1^{-}\left(x_1\right)(\beta ,\gamma ),{\underline{m}}_2^{-}\left(x_2\right)(\beta ,\gamma ),\dots ):y=f(x_1,x_2,\dots )\},$$

(6)

$${\underline{m}}^{+}\left(y\right)(\beta ,\gamma )=$$

$$sup\{min({\underline{m}}_1^{+}\left(x_1\right)(\beta ,\gamma ),{\underline{m}}_2^{+}\left(x_2\right)(\beta ,\gamma ),\dots ):y=f(x_1,x_2,\dots )\},$$

(7)

$${\overline{m}}^{-}\left(y\right)(\beta ,\gamma )=$$

$$sup\{min({\overline{m}}_1^{-}\left(x_1\right)(\beta ,\gamma ),{\overline{m}}_2^{-}\left(x_2\right)(\beta ,\gamma ),\dots ):y=f(x_1,x_2,\dots )\},$$

(8)

$${\overline{m}}^{+}\left(y\right)(\beta ,\gamma )=$$

$$sup\{min({\overline{m}}_1^{+}\left(x_1\right)(\beta ,\gamma ),{\overline{m}}_2^{+}\left(x_2\right)(\beta ,\gamma ),\dots ):y=f(x_1,x_2,\dots )\}.$$

(9)

Each of the Equations (6)–(9) is, in effect, Zadeh’s extension principle for the corresponding membership functions. Thus, there formulas can be reformulated in terms of $\alpha$-cuts of the corresponding membership functions:

$${\underline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )=f({\underline{\mathbf{x}}}_1^{-}(\alpha ,\beta ,\gamma ),{\underline{\mathbf{x}}}_2^{-}(\alpha ,\beta ,\gamma ),\dots ),$$

$${\underline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )=f({\underline{\mathbf{x}}}_1^{+}(\alpha ,\beta ,\gamma ),{\underline{\mathbf{x}}}_2^{+}(\alpha ,\beta ,\gamma ),\dots ),$$

$${\overline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )=f({\overline{\mathbf{x}}}_1^{-}(\alpha ,\beta ,\gamma ),{\overline{\mathbf{x}}}_2^{-}(\alpha ,\beta ,\gamma ),\dots ),$$

$${\overline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )=f({\overline{\mathbf{x}}}_1^{+}(\alpha ,\beta ,\gamma ),{\overline{\mathbf{x}}}_2^{+}(\alpha ,\beta ,\gamma ),\dots ),$$

where

$${\underline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{y:{\underline{m}}^{-}\left(y\right)(\beta ,\gamma )\ge \alpha \},{\underline{\mathbf{x}}}_i^{-}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{x_i:{\underline{m}}_i^{-}\left(x_i\right)(\beta ,\gamma )\ge \alpha \},$$

$${\underline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{y:{\underline{m}}^{+}\left(y\right)(\beta ,\gamma )\ge \alpha \},{\underline{\mathbf{x}}}_i^{+}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{x_i:{\underline{m}}_i^{+}\left(x_i\right)(\beta ,\gamma )\ge \alpha \},$$

$${\overline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{y:{\overline{m}}^{-}\left(y\right)(\beta ,\gamma )\ge \alpha \},{\overline{\mathbf{x}}}_i^{-}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{x_i:{\overline{m}}_i^{-}\left(x_i\right)(\beta ,\gamma )\ge \alpha \},$$

$${\overline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{y:{\overline{m}}^{+}\left(y\right)(\beta ,\gamma )\ge \alpha \},{\overline{\mathbf{x}}}_i^{+}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{x_i:{\overline{m}}_i^{+}\left(x_i\right)(\beta ,\gamma )\ge \alpha \}.$$

Hence, we arrive at the following algorithm.

7.3. Type-3 Case: Resulting Algorithm

We start with type-3 membership functions $m_i(x_i,t,s)$.

• First, for every i and for all $\gamma$ from the selected list of values, we compute:

  $$[{\underline{m}}_i(x_i,t)\left(\gamma \right),{\overline{m}}_i(x_i,t)\left(\gamma \right)]\stackrel{\mathrm{def}}{=}\{s:m_i(x_i,t,s)\ge \gamma \}.$$
• Then, for each i, $\beta$, and $\gamma$, we compute:

  $$[{\underline{m}}_i^{-}\left(x_i\right)(\beta ,\gamma ),{\underline{m}}_i^{+}\left(x_i\right)(\beta ,\gamma )]\stackrel{\mathrm{def}}{=}\{t:{\underline{m}}_i(x_i,t)\left(\gamma \right)\ge \beta \}\mathrm{and}$$

  $$[{\overline{m}}_i^{-}\left(x_i\right)(\beta ,\gamma ),{\overline{m}}_i^{+}\left(x_i\right)(\beta ,\gamma )]\stackrel{\mathrm{def}}{=}\{t:{\overline{m}}_i(x_i,t)\left(\gamma \right)\ge \beta \}.$$
• Then, for each i, $\alpha$, $\beta$, and $\gamma$, we compute

  $${\underline{\mathbf{x}}}_i^{-}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{x_i:{\underline{m}}_i^{-}\left(x_i\right)(\beta ,\gamma )\ge \alpha \},$$

  $${\underline{\mathbf{x}}}_i^{+}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{x_i:{\underline{m}}_i^{+}\left(x_i\right)(\beta ,\gamma )\ge \alpha \},$$

  $${\overline{\mathbf{x}}}_i^{-}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{x_i:{\overline{m}}_i^{-}\left(x_i\right)(\beta ,\gamma )\ge \alpha \},$$

  $${\overline{\mathbf{x}}}_i^{+}(\alpha ,\beta ,\gamma )\stackrel{\mathrm{def}}{=}\{x_i:{\overline{m}}_i^{+}\left(x_i\right)(\beta ,\gamma )\ge \alpha \}.$$
• For each $\alpha$, $\beta$, and $\gamma$, we then use an interval computation algorithm to compute:

  $${\underline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )=f({\underline{\mathbf{x}}}_1^{-}(\alpha ,\beta ,\gamma ),{\underline{\mathbf{x}}}_2^{-}(\alpha ,\beta ,\gamma ),\dots ),$$

  $${\underline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )=f({\underline{\mathbf{x}}}_1^{+}(\alpha ,\beta ,\gamma ),{\underline{\mathbf{x}}}_2^{+}(\alpha ,\beta ,\gamma ),\dots ),$$

  $${\overline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )=f({\overline{\mathbf{x}}}_1^{-}(\alpha ,\beta ,\gamma ),{\overline{\mathbf{x}}}_2^{-}(\alpha ,\beta ,\gamma ),\dots ),$$

  $${\overline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )=f({\overline{\mathbf{x}}}_1^{+}(\alpha ,\beta ,\gamma ),{\overline{\mathbf{x}}}_n^{+}(\alpha ,\beta ,\gamma ),\dots ).$$
• Next, for each y, $\beta$, and $\gamma$, we compute

  $${\underline{m}}^{-}\left(y\right)(\beta ,\gamma )=max\{\alpha :y\in {\underline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )\},$$

  $${\underline{m}}^{+}\left(y\right)(\beta ,\gamma )=max\{\alpha :y\in {\underline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )\},$$

  $${\overline{m}}^{-}\left(y\right)(\beta ,\gamma )=max\{\alpha :y\in {\overline{\mathbf{y}}}^{-}(\alpha ,\beta ,\gamma )\},$$

  $${\overline{m}}^{+}\left(y\right)(\beta ,\gamma )=max\{\alpha :y\in {\overline{\mathbf{y}}}^{+}(\alpha ,\beta ,\gamma )\}.$$
• For each y, t, and $\gamma$, we compute

  $$\underline{m}(y,t)\left(\gamma \right)=max\{\beta :t\in [{\underline{m}}^{-}\left(y\right)(\beta ,\gamma ),{\underline{m}}^{+}\left(y\right)(\beta ,\gamma )]\}and$$

  $$\overline{m}(y,t)\left(\gamma \right)=max\{\beta :t\in [{\overline{m}}^{-}\left(y\right)(\beta ,\gamma ),{\overline{m}}^{+}\left(y\right)(\beta ,\gamma )]\}.$$
• Finally, for all y, t and s, we compute

  $$m(y,t,s)=max\{\gamma :s\in [\underline{m}(y,t)\left(\gamma \right),\overline{m}(y,t)\left(\gamma \right)]\}.$$

7.4. What about Higher Order Fuzzy Sets?

In this section, we showed how processing type-2 fuzzy information can be used to processing type-2 fuzzy information. This reduction was based on the fact that in the type-3 case, each value $m\left(y\right)$ and $m_i\left(x_i\right)$ is a type-2 fuzzy set. Thus, we have the relation (1) between these type-2 fuzzy sets.

Similarly, in the type-4 case, each value $m\left(y\right)$ and $m_i\left(x_i\right)$ is a type-3 fuzzy set. Thus, we have the relation (1) between these type-3 fuzzy sets—and we can use the above algorithm to process these values. Similarly, for every level L, in the type-L case, each value $m\left(y\right)$ and $m_i\left(x_i\right)$ is a type-$(L-1)$ fuzzy set. Thus, we have the relation (1) between these type-$(L-1)$ fuzzy sets. This way, we can reduce processing type-L fuzzy sets to processing type-$(L-1)$ fuzzy sets; similarly, we can reduce processing type-$(L-1)$ fuzzy sets to processing type-$(L-2)$ fuzzy sets, etc., until we get to the known algorithms for processing type-1 and type-2 fuzzy sets.

7.5. How Many Computational Steps Do We Need

The only (minor) problem with processing type-3 and higher-order fuzzy sets is that as we go to higher and higher order, the computational complexity increases. Indeed:

• For type-1, for each y, the desired information $m\left(y\right)$ consists of a single number. In this case, if we use 11 values of $\alpha$, we need to use an interval computation algorithm 11 times.
• For type-2, for each y, we need to find the values $m(y,t)$ corresponding to different values $t\in [0,1]$. If we use 11 values for t, we thus need at least 11 times more computations than in the type-1 case—and indeed, we need order of $11\times 11$ calls to an interval computation algorithm—namely, $2\times {11}^2$ calls.
• For type-3, for each y, we need to find the values $m(y,t,s)$ corresponding to different values $t,s\in [0,1]$. If we use 11 values of each of the variables t and s, we thus need at least ${11}^2$ times more computations than in the type-1 case—and indeed, we need order of ${11}^2\times 11={11}^3$ calls to an interval computation algorithm—namely, $2^2\times {11}^3$ calls.
• In general, for type-L, for each y, we need to find the values $m(y,t_1,\dots ,t_{L-1})$ corresponding to different values $t_1,\times ,t_{L-1}\in [0,1]$. If we use 11 values for each of the variables $t_i$, we thus need at least ${11}^{L-1}$ times more computations than in the type-1 case—and indeed, as one can show by induction over L, we need order of ${11}^{L-1}\times 11={11}^L$ calls to an interval computation algorithm—namely, $2^{L-1}\times {11}^L$ calls.

8. Conclusions and Future Work

8.1. Conclusions

Usual data processing algorithms treat data points as if they were exact. In practice, data come with uncertainty. When data come from experts who describe their knowledge by using imprecise (“fuzzy”) words from natural language, a natural way to describe the corresponding uncertainty is to use fuzzy techniques. To get a more accurate representation of expert uncertainty, it is necessary to use higher-order fuzzy techniques, i.e., go from the usual $[0,1]$-based type-1 techniques to type-2, or even to higher-order: type-3 etc.

In many practical applications, the use of type-2 fuzzy uncertainty leads to better results. To more efficiently handle such situations, efficient algorithms have been proposed, and used, for data processing under type-2 fuzzy uncertainty.

Recently, it has been shown that in several applications, the use of type-3 fuzzy techniques leads to further improvements. In view of these successes, it has become necessary to develop efficient algorithms for data processing under such uncertainty. In this paper, we show how to use the existing efficient type-2 algorithms to design efficient algorithms for data processing under type-3 (and, if needed, higher-order) fuzzy uncertainty.

8.2. Future Work

Now that an efficient algorithm for data processing under general type-2 fuzzy uncertainty has been designed, a natural next step is to implement it and to apply it to different practical situations—with hope that in some of these applications it will lead to better results.

It is also desirable to take into account that, in addition to fuzzy techniques, there are many other techniques for representing and processing uncertainty. Many of these techniques have been successfully combined with type-1 and even type-2 fuzzy sets to produce even more adequate results. For example, type-2 fuzzy techniques have been successfully combined with rough sets; see, e.g., [23,24,25,26,27]. In view of these successes, it is desirable to try to combine type-3 fuzzy approach with these alternative uncertainty techniques.