Developments of Semi-Type-2 Interval Approach with Mathematics and Order Relation: A New Uncertainty Tackling Technique

Abstract

This paper aims to introduce a new interval approach called the Semi-Type-2 interval to represent imprecise parameters in uncertain decision-making. The proposed work establishes arithmetic operations of Semi-Type-2 intervals with algebraic properties. Additionally, a new interval ranking is proposed in order to compare Semi-Type-2 interval numbers, and the corresponding properties of total order relations are also derived. All the definitions and properties related to Semi-Type-2 intervals are illustrated with the help of numerical examples. Numerical illustrations confirm the consistency of the framework and its effectiveness in extending classical interval mathematics. Finally, some probable applications of the Semi-Type-2 interval approach are demonstrated for future implementation.

1. Introduction

Decision-making in uncertain environments is a major challenge in many real-life problems across Operations Research and Management Science (especially in the fields where real-world problems often involve parameters that are imprecise, fluctuating, or only partially known). For example, in supply chain management, sudden shifts in demand or disruptions in transportation can make cost and delivery times unpredictable. In healthcare, treatment outcomes and patient responses vary widely, introducing uncertainty into medical planning. Financial markets face constant volatility in prices and interest rates, while in engineering design, material properties and system loads may deviate from expected values.

Similar issues arise in areas like time-series signal analysis, product design, multi-criteria decision-making, data mining, and remote-control systems, where parameters are inherently imprecise and often unpredictable. Recently, several researchers, viz., Maleki et al. [1], Ahmadi et al. [2], Ruiz and Dashti [3], Hamlehvar and Aazami [4], Deldadehasl et al. [5], and others, have reported their findings in these sectors without considering the uncertainty of parameters. In all these contexts, the presence of uncertainty can hinder the reliability of mathematical models and complicate the search for optimal solutions.

Researchers have therefore devoted substantial effort to model imprecise parameters in a way that preserves practical usefulness. Approaches such as fuzzy sets, stochastic methods, fuzzy–stochastic hybrids, and interval analysis have been widely used to address uncertainty. Among those, the interval approach has gained prominence due to its simplicity and ease of implementation. Recently, a new type of interval approach, named the Type-2 interval approach, was developed by Rahman et al. [6] as a generalization of the interval approach. The goal of the present study is to propose another generalization of the interval approach, named as Semi-Type-2 interval.

Depending upon the several generalizations of the usual interval approach, it can be categorized into the following approaches:

❖

Type-1 interval approach;

❖

Type-2 interval approach;

❖

Proposed Semi-Type-2 interval approach.

In the Type-1 interval approach, an uncertain parameter is represented by an interval with known upper and lower bounds. The research area of the Type-1 interval approach was enriched by the scholarly contributions of several researchers. Moore [7] was the pioneer of this area, and he wrote a book presenting interval mathematics, the concept of interval-valued functions, and several interval-based approaches with a number of real-life applications. Again, he wrote another book [8] by proposing interval order relations with some more real-life applications. Then, Hansen and Walster [9] wrote another excellent book mentioning the technique of global optimization using interval analysis. Ramos et al. [10] introduced the concepts of interval mapping in an arbitrary space. Sahoo et al. [11] developed a modified genetic algorithm for solving a multi-objective-based interval optimization problem. Bhunia and Samanta [12] proposed a complete interval order relation, and using this ordering, they studied the application of the interval metric in a multi-objective interval optimization problem. Malinowski [13] derived some existence theorems on symmetric functional set-valued differential equations. Besides these, there are a lot of works on the developments and applications of Type-1 intervals. Among those, some selected ones were reported here, which were accomplished by Ruidas et al. [14], Mondal and Rahman [15], Yadav et al. [16], and others. In their works, interval uncertainty is represented by fixed lower and upper bounds. However, in reality, several situations arise where representing an imprecise parameter as an interval with fixed bounds is challenging. For instance, the cost of various commodities in a developing country is often depicted as an interval with fixed bounds. In these situations, two fundamental questions often arise regarding the selection of bounds. Firstly, how should one handle the situation if the cost exceeds both bounds? Secondly, what should we do if the cost never reaches either bound? If the first question is ignored, then significant numerical errors may arise during the decision-making process, whereas in the case of the second question, uncertainty may increase, which is contrary to the principles of an optimistic decision-maker.

To overcome the challenges arising in the first approach, Rahman et al. [6] proposed a new interval approach named the Type-2 interval, considering flexibility instead of fixity of the bounds in the interval. Formally, a Type-2 interval is defined as a class of Type-1 intervals where the lower and upper bounds of the intervals of the said class belong to two given Type-1 intervals. With the introduction of the Type-2 interval, very few scholarly articles were written by researchers to enrich the recently developed approach. Rahman et al. [6,17] established Type-2 interval mathematics, Type-2 interval order relations, and the optimality conditions of Type-2 interval optimization problems. Later, Rahman et al. [18] again studied an application of the Type-2 interval approach in the inventory control problem. Das et al. [19] introduced the parametric representations of the Type-2 interval and, using these, derived the optimality theories of Type-2 interval non-linear programming. Then, Rahaman et al. [20] used this Type-2 interval approach in fractional calculus to study some applications in memory-based inventory control. Rahaman et al. [21] again derived the solvability criteria for Type-2 interval differential equations, and they applied these concepts in an economic lot-size model under uncertainty.

In the last-mentioned approach, the fluctuation of both the bounds of an interval is considered. However, in reality, there are also some situations, viz., fluctuation of cost and demand of essential goods from the present to the near future, in which to represent the fluctuating parameters in terms of an interval. In these cases, the lower bounds of the intervals would be known, but the upper bounds may be flexible. Thus, during the modelling of real-life problems in such circumstances, to represent the imprecise parameters more appropriately, one has to require an approach other than both Type-1 and Type-2 intervals. The present work tries to propose such an approach named as Semi-Type-2 interval, which is defined as the class of all Type-1 intervals with common lower bounds and variable upper bounds. Put simply, a Semi-Type-2 interval is an interval whose lower bound is a real number and upper bound is another Type-1 interval. Similarly, the lower bounds of the intervals may be flexible, and the upper bounds may be fixed. The first one is known as a Semi-Type-2 interval for the first kind, and the second one is a Semi-Type-2 interval for the second kind. The main contributions of this paper are presented below:

• Introduction of Semi-Type-2 interval: A new generalization of interval numbers is proposed to capture uncertainty more effectively.
• Development of arithmetic operations and algebraic properties: Arithmetic operations on Semi-Type-2 intervals are developed along with their algebraic properties.
• Introduction of interval ranking: A new order relation to compare Semi-Type-2 interval numbers is introduced, and the related total order properties are established.
• Exploration of potential applications: Potential uses of the Semi-Type-2 interval approach are discussed in fields like supply chain management, engineering, and medical sciences as future research scopes.

The present work proposes a new generalized interval approach named the Semi-Type-2 interval to handle uncertainty involved in real-life decision-making problems. The motivation, literature review, and context of introducing this approach are presented in this section. The rest of the work has been organized as follows: In Section 2, first, the mathematical definition of a Semi-Type-2 interval and all the definitions of different kinds of Semi-Type-2 intervals are introduced. Then, different arithmetic operations of Semi-Type-2 intervals are defined and illustrated with a set of numerical examples. In Section 3.1, the extension of a real-valued function to a Semi-Type-2 interval-valued function and inclusion function are defined. Thereafter, definitions of periodic functions of Semi-Type-2 interval arguments are provided with some illustrative examples. In Section 3.2, some algebraic properties of Semi-Type-2 intervals are discussed with numerical illustration. In Section 4, the Semi-Type-2 interval order relation is introduced by using the score components. Also, it is proved that the proposed order relation is the total order relation. In Section 5, a summary of the findings, with possible applications and future research directions, is presented.

2. Semi-Type-2 Interval and Its Mathematics

In this section, for the first time, we have proposed the mathematical definition of a new Semi-Type-2 interval, equality of two Semi-Type-2 intervals, degenerate Semi-Type-2 intervals, arithmetical operations of Semi-Type-2 intervals, etc.

2.1. Semi-Type-2 Interval with Equality, Degeneracy of Semi-Type-2 Intervals

In this sub-section, we have introduced two kinds of Semi-Type-2 intervals, viz., Semi-Type-2 intervals for the first kind and second kind based on the fixity of the lower or upper bound.

Definition 1. 

(i) 

The Semi-Type-2 interval for the first kind $A_{S2}$ denoted by $\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ is defined in terms of Type-1 intervals given by $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]=\left\{\left[a_L,a_U\right]:a_U\in \left[{\underset{\_}{a}}_U,{\overline{a}}_U\right]\right\}.$

(ii) 

The Semi-Type-2 interval for the second kind $A_{S2}$ denoted by $\left[\left({\underset{\_}{a}}_L,{\overline{a}}_L\right),a_U\right]$ is defined in terms of Type-1 intervals given by $A_{S2}=\left[\left({\underset{\_}{a}}_L,{\overline{a}}_L\right),a_U\right]=\left\{\left[a_L,a_U\right]:a_L\in \left[{\underset{\_}{a}}_L,{\overline{a}}_L\right]\right\}.$

For a better understanding of the concept of a Semi-Type-2 interval, its genesis has been presented graphically in Figure 1.

Figure 1. Genesis of interval uncertainty.

Remark 1. 

Although we have defined Semi-Type-2 intervals for the two kinds in Definition 1, we have discussed more results throughout the paper regarding the first kind only due to symmetry in the definitions. The same for the second kind will be similar. Also, the Semi-Type-2 interval for the first kind will simply be called the Semi-Type-2 interval. We denote the collection of all Semi-Type-2 intervals for the first kind by $I_{S2}(\mathbb{R}).$

Definition 2. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right], B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]$ be two Semi-Type-2 intervals. Then $A_{S2}$ is contained in $B_{S2}$ which is denoted by $A_{S2}{\subseteq }_{S2}{B}_{S2}$ and is defined by $A_{S2}{\subseteq }_{S2}{B}_{S2}\iff \left[a_L,{\overline{a}}_U\right]\subseteq \left[b_L,{\overline{b}}_U\right] \mathrm{and} {\underset{\_}{a}}_U\le {\underset{\_}{b}}_U$.

Definition 3. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right], B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]$ be two Semi-Type-2 intervals. Then $A_{S2}=B_{S2}$ if $a_L=b_L,{\underset{\_}{a}}_U={\underset{\_}{b}}_U,{\overline{a}}_U={\overline{b}}_U.$

Definition 4. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ be a Semi-Type-2 interval. Then $A_{S2}$ is said to be a 1-degenerate if $a_L={\underset{\_}{a}}_U={\overline{a}}_U.$ In this case, a Semi-Type-2 interval looks like $\left[a,\left(a,a\right)\right]$ generated by the real number $a.$

Definition 5. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ be a Semi-Type-2 interval. Then $A_{S2}$ is said to be a 2-degenerate if ${\underset{\_}{a}}_U={\overline{a}}_U.$ This 2-degenerate Semi-Type-2 interval can be identified with a Type-1 interval $\left[a_L,a_U\right].$

Remark 2. 

A Semi-Type-2 interval $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ can be identified by a Type-2 interval $A_{S2}=\left[\left(a_L,a_L\right),\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ with a lower bound as a degenerate Type-1 interval and an upper bound as a proper Type-1 interval.

Remark 3. 

According to Remark 2, a Semi-Type-2 interval can be identified as a Type-2 interval. Now, automatically, a question may arise in the readers’ minds—why does one have to study the Semi-Type-2 approach separately? A partial answer to this question has already been given in the introduction of the Semi-Type-2 interval in Section 1. Another answer can be given from the point of view of the well-defined arithmetical operations of Semi-Type-2 intervals. Actually, the subtraction of two Semi-Type-2 intervals using the definition of Type-2 interval arithmetic is not a Semi-Type-2 interval. Thus, one has to study the mathematics of Semi-Type-2 intervals separately.

2.2. Arithmetic Operations of Semi-Type-2 Interval

In this sub-section, different arithmetic operations, viz., addition, subtraction, scalar multiplication, multiplication, division, etc., are defined with illustrations.

Definition 6. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ and $B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]$ be two Semi-Type-2 intervals.

• (i) Addition: The addition of two Semi-Type-2 intervals $A_{S2}$ and $B_{S2}$ is $A_{S2}+B_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]+\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]=\left[a_L+b_L,\left({\underset{\_}{a}}_U+{\underset{\_}{b}}_U,{\overline{a}}_U+{\overline{b}}_U\right)\right].$

Note 1. 

If we consider $A_{S2}$ and $B_{S2}$ both as Type-2 intervals, then the subtraction formula will be $A_{S2}-B_{S2}=\left[\left(a_L,a_L\right),\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]-\left[\left(b_L,b_L\right),\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]=\left[\left(a_L-{\overline{b}}_U,a_L-{\underset{\_}{b}}_U\right),\left({\underset{\_}{a}}_U-b_L,{\overline{a}}_U-b_L\right)\right]$ which is not a Semi-Type-2 interval, i.e., this operation is not closed. So, we define another subtraction formula as follows:

(ii) 

Subtraction: The subtraction between two Semi-Type-2 intervals $A_{S2}$ and $B_{S2}$ is

$$A_{S2}-B_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]-\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]=\left[a_L-{\overline{b}}_U,\left({\underset{\_}{a}}_U-{\underset{\_}{b}}_U,{\overline{a}}_U-b_L\right)\right]$$

Example 1. 

Let $A_{S2}=\left[-4,\left(3,5\right)\right],$ $B_{S2}=\left[2,\left(7,8\right)\right].$ Then

$$A_{S2}+B_{S2}=\left[-2,\left(10,13\right)\right]\mathrm{and} A_{S2}-B_{S2}=\left[-12,\left(-4,3\right)\right].$$

(iii) Scalar multiplication: Let $\lambda \in \mathbb{R}.$ Then the scalar multiplication of $A_{S2}\in I_{S2}(\mathbb{R})$ with $\lambda$ is defined as follows:

$$\lambda A_{S2}=\lambda \left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]=\left\{\begin{array}{l}\left[\lambda a_L,\left(\lambda {\underset{\_}{a}}_U,\lambda {\overline{a}}_U\right)\right] \mathrm{if} \lambda \ge 0,\\ \left[\lambda {\overline{a}}_U,\left(\lambda {\underset{\_}{a}}_U,\lambda a_L\right)\right] \mathrm{if} \lambda <0.\end{array}\right.$$

(iv) Multiplication: The multiplication of two Semi-Type-2 intervals $A_{S2}$ and $B_{S2}$ is given by

$$A_{S2}{B}_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]=\left[\min C,\left({\underset{\_}{a}}_U{\underset{\_}{b}}_U,\max C\right)\right],$$

where $C=\left\{a_L{b}_L,a_L{\overline{b}}_U,{\overline{a}}_U{b}_L,{\overline{a}}_U{\overline{b}}_U\right\}.$

Example 2. 

Let $A_{S2}=\left[-1,\left(3,6\right)\right],$ $B_{S2}=\left[-2,\left(-1,5\right)\right].$ Then (1) $\lambda =2,$ $2A_{S2}=\left[-2,\left(6,12\right)\right].$ Again for $\lambda =-2,$ $(-2)A_{S2}=\left[-12,\left(-6,2\right)\right].$

• (2) $A_{S2}{B}_{S2}=\left[\min C,\left({\underset{\_}{a}}_U{\underset{\_}{b}}_U,\max C\right)\right],$ where $C=\left\{2,-5,-12,30\right\}$ $=\left[-12,\left(-3,30\right)\right].$

Endpoint formulas for Semi-Type-2 interval multiplication is given in the Table 1.

Table 1. Endpoint formulas for Semi-Type-2 interval multiplication.

Case	$\mathbf{min}\mathit{C}$	$\mathbf{max}\mathit{C}$
$a_L\ge 0, b_L\ge 0$	$a_L{b}_L$	${\overline{a}}_U{\overline{b}}_U$
$a_L\ge 0, {\overline{b}}_U\le 0$	${\overline{a}}_U{b}_L$	$a_L{\overline{b}}_U$
$a_L\ge 0, b_L<0<{\underset{\_}{b}}_U$	${\overline{a}}_U{b}_L$	${\overline{a}}_U{\overline{b}}_U$
$a_L\ge 0, b_L<0, {\underset{\_}{b}}_U<0<{\overline{b}}_U$	${\overline{a}}_U{b}_L$	${\overline{a}}_U{\overline{b}}_U$
${\overline{a}}_U\le 0, b_L\ge 0$	$a_L{\overline{b}}_U$	${\overline{a}}_U{b}_L$
${\overline{a}}_U\le 0, {\overline{b}}_U\le 0$	${\overline{a}}_U{\overline{b}}_U$	$a_L{b}_L$
${\overline{a}}_U\le 0, b_L<0<{\underset{\_}{b}}_U$	$a_L{\overline{b}}_U$	$a_L{b}_L$
${\overline{a}}_U\le 0, b_L<0, {\underset{\_}{b}}_U<0<{\overline{b}}_U$	$a_L{\overline{b}}_U$	$a_L{b}_L$
$a_L<0<{\underset{\_}{a}}_U, b_L\ge 0$	$a_L{\overline{b}}_U$	${\overline{a}}_U{\overline{b}}_U$
$a_L<0<{\underset{\_}{a}}_U, {\overline{b}}_U\le 0$	${\overline{a}}_U{b}_L$	$a_L{b}_L$
$a_L<0<{\underset{\_}{a}}_U, b_L<0<{\underset{\_}{b}}_U$	$\min \left\{a_L{\overline{b}}_U,{\overline{a}}_U{b}_L\right\}$	$\max \left\{a_L{b}_L,{\overline{a}}_U{\overline{b}}_U\right\}$
$a_L<0<{\underset{\_}{a}}_U, b_L<0, {\underset{\_}{b}}_U<0<{\overline{b}}_U$	$\min \left\{a_L{\overline{b}}_U,{\overline{a}}_U{b}_L\right\}$	$\max \left\{a_L{b}_L,{\overline{a}}_U{\overline{b}}_U\right\}$

(v) Division: The division of two Semi-Type-2 intervals $A_{S2}$ and $B_{S2}$ is defined as follows:

$${\displaystyle \frac{A_{S2}}{B_{S2}}}=A_{S2}{\displaystyle \frac{1}{B_{S2}}}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]\left[{\displaystyle \frac{1}{{\overline{b}}_U}},\left({\displaystyle \frac{1}{{\underset{\_}{b}}_U}},{\displaystyle \frac{1}{b_L}}\right)\right] \mathrm{provided} 0\notin B_{S2}.$$

Example 3. 

Let $A_{S2}=\left[-2,\left(-1,3\right)\right],$ $B_{S2}=\left[4,\left(5,6\right)\right].$ Since $0\notin B_{S2},$ ${\displaystyle \frac{A_{S2}}{B_{S2}}}$ is defined. ${\displaystyle \frac{A_{S2}}{B_{S2}}}=\left[-{\displaystyle \frac{1}{2}},\left(-{\displaystyle \frac{1}{5}},{\displaystyle \frac{3}{4}}\right)\right].$

(vi) n-th non-negative power of a Semi-Type-2 interval: The n-th power (n = 0, 1, 2, …) of a Semi-Type-2 interval $A_{S2}$is denoted by $A_{S2}^n$ and is defined as follows:

$$A_{S2}^n=\left\{\begin{array}{l}\left[1,\left(1,1\right)\right] \mathrm{if} n=0 \mathrm{and} 0\notin A\\ \left[a_L^n,\left({{\underset{\_}{a}}_U}^n,{{\overline{a}}^n}_U\right)\right] \mathrm{if} a_L\ge 0 \mathrm{or} n \mathrm{is} \mathrm{odd}\\ \left[{{\overline{a}}^n}_U,\left({{\underset{\_}{a}}_U}^n,a_L^n\right)\right] \mathrm{if} {\overline{a}}_U<0 \mathrm{and} n \mathrm{is} \mathrm{even}\\ \left[0,\left(\max \left\{a_L^n,{{\underset{\_}{a}}_U}^n\right\},\max \left\{a_L^n,{{\overline{a}}^n}_U\right\}\right)\right] \mathrm{elsewhere}\end{array}\right.$$

Example 4. 

(i) 

Let $A_{S2}=\left[-5,\left(-3,-1\right)\right].$

Here ${\overline{a}}_U=-1<0$ and let $n=2$ , an even number. In this case, $A_{S2}^n=\left[{{\overline{a}}^n}_U,\left({{\underset{\_}{a}}_U}^n,a_L^n\right)\right].$

$\therefore$ the square of $A_{S2}$ is given by $A_{S2}^2=\left[1,\left(9,25\right)\right].$

For $n=3$ (which is odd), $A_{S2}^n=\left[a_L^n,\left({{\underset{\_}{a}}_U}^n,{{\overline{a}}^n}_U\right)\right].$

$\therefore$ cube of $A_{S2}$ is given by $A_{S2}^3=\left[-125,\left(-27,-1\right)\right].$

(ii) 

Let $A_{S2}=\left[-10,\left(2,6\right)\right].$

For $n=2$ (which is even), $A_{S2}^n=\left[0,\left(\max \left\{a_L^n,{{\underset{\_}{a}}_U}^n\right\},\max \left\{a_L^n,{{\overline{a}}^n}_U\right\}\right)\right].$

$\therefore$ square of $A_{S2}$ is given by $A_{S2}^2=\left[0,\left(100,100\right)\right].$

Note 2. 

${\displaystyle \frac{1}{A_{S2}}}=\left[{\displaystyle \frac{1}{{\overline{a}}_U}},\left({\displaystyle \frac{1}{{\underset{\_}{a}}_U}},{\displaystyle \frac{1}{a_L}}\right)\right]$ if $0\notin A_{S2}.$

(vii) 

n-th positive root of a Semi-Type-2 interval: The n-th positive root $A_{S2}^{1/n}$ of a Semi-Type-2 interval $A_{S2}$ is defined as follows:

$$A_{S2}^{1/n}=\left[a_L^{1/n},\left({\underset{\_}{a}}_U^{1/n},{{\overline{a}}^{1/n}}_U\hspace{0.17em}\right)\right] \mathrm{if} a_L\ge 0 \mathrm{or} n \mathrm{is} \mathrm{odd}.$$

Example 5. 

Let $A_{S2}=\left[0,\left(1,16\right)\right].$ Then $A_{S2}^{1/2}=\left[0,\left(1,4\right)\right].$

(viii) 

Rational power: Rational power of a Semi-Type-2 interval $A_{S2}$ is defined by $A_{S2}^{p/q}={\left(A_{S2}^p\right)}^{1/q} \mathrm{where} p,q\in \mathbb{N}.$

3. Semi-Type-2 Interval-Valued Functions and Inclusion Functions

The introduction of Semi-Type-2 interval-valued functions enriches the developments of Semi-Type-2 interval analysis beyond interval arithmetic. In this section, the extension of a real-valued function to a Semi-Type-2 interval-valued function and inclusion function has been defined. Then the definitions of values of periodic functions at a Semi-Type-2 interval have been proposed.

Remark 4. 

Semi-Type-2 interval-valued functions can be defined in different ways. We can define a Semi-Type-2 interval-valued function by selecting a real-valued function $f$ and computing the range of $f\left(x\right)$ where $x$ varied through some Semi-Type-2 interval $X.$ Another process is the extension of a given real-valued function $f$ by applying its formula directly to Semi-Type-2 interval arguments, which is shown in Theorem 1.

Definition 7. 

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a real-valued function of real variables $x_1,x_2,\dots ,x_n$ and $F:I_{S2}^n(\mathbb{R})\to I_{S2}(\mathbb{R})$ be a Semi-Type-2 interval-valued function with Semi-Type-2 interval arguments $X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n.$ Then $F$ is said to be a Semi-Type-2 interval extension of $f$ if $F\left(x_1,x_2,\dots ,x_n\right)=f\left(x_1,x_2,\dots ,x_n\right), \forall x_i\in \mathbb{R}, i=1,2,\dots ,n,$ i.e., for 1-degenerate Semi-Type-2 interval arguments, $F$ agrees with $f$.

Again, a Semi-Type-2 interval-valued function $F$ is said to be inclusion monotonic if

$$X_{S2}^i{\subseteq }_{S2}{Y}_{S2}^i, i=1,2,\dots ,n\Rightarrow F\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right){\subset }_{S2}F\left(Y_{S2}^1,Y_{S2}^2,\dots ,Y_{S2}^n\right).$$

Let $D\subseteq {\mathbb{R}}^n$ and $F_{S2}:D\to I_{S2}(\mathbb{R})$ be a Semi-Type-2 interval-valued function defined by $F_{S2}(x)=\left[f_L(x),\left({\underset{\_}{f}}_U(x),{\overline{f}}_U(x)\right)\right],$ where $f_L(x),{\underset{\_}{f}}_U(x),{\overline{f}}_U(x):D\to \mathbb{R}.$

Theorem 1. 

Let $F\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right)$ be an inclusion monotonic Semi-Type-2 interval extension of a crisp function $f\left(x_1,x_2,\dots ,x_n\right).$ Then $f\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right){\subseteq }_{S2}F\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right),$ where $f\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right)=\left\{f\left(x_1,x_2,\dots ,x_n\right):x_i\in X_{S2}^i,i=1,2,\dots ,n\right\}.$ i.e., $F\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right)$ contains the range of values of $f\left(x_1,x_2,\dots ,x_n\right), \forall x_i\in X_i(i=1,2,\dots ,n).$

Proof. 

Since $F$ is a Semi-Type-2 interval extension of $f$ $f\left(x_1,x_2,\dots ,x_n\right)=F\left(x_1,x_2,\dots ,x_n\right),$ $\forall x_i\in \mathbb{R}, i=1,2,\dots ,n.$ Also, for every $\left(x_1,x_2,\dots ,x_n\right)$ in $\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right),$ $F\left(x_1,x_2,\dots ,x_n\right){\subset }_{S2}F\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right),$ as $F$ is inclusion monotonic. Therefore $f\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right){\subseteq }_{S2}F\left(X_{S2}^1,X_{S2}^2,\dots ,X_{S2}^n\right).$ □

Again, monotonicity can be applied directly to calculate the images of Semi-Type-2 intervals under some standard functions, as shown in Proposition 1 and Example 6.

Proposition 1. 

(i) If $f(x)$ is a monotonically increasing in the Semi-Type-2 interval $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right],$ where $x\in \mathbb{R},$ then, $f\left(A_{S2}\right)=\left[f\left(a_L\right),\left(f\left({\underset{\_}{a}}_U\right),f\left({\overline{a}}_U\right)\right)\right].$

(ii) 

If $f(x)$ is a monotonically decreasing in the Semi-Type-2 interval $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right],$ then $f\left(A_{S2}\right)=\left[f\left({\overline{a}}_U\right),\left(f\left({\underset{\_}{a}}_U\right),f\left(a_L\right)\right)\right].$

Proof. 

(i) Here $f$ is monotonically increasing in $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ and $a_L\le {\underset{\_}{a}}_U\le {\overline{a}}_U$ implies that $f\left(a_L\right)\le f\left({\underset{\_}{a}}_U\right)\le f\left({\overline{a}}_U\right).$ Then $f$ maps the Semi-Type-2 interval $A_{S2}$ into the Semi-Type-2 interval $f\left(A_{S2}\right)=\left[f\left(a_L\right),\left(f\left({\underset{\_}{a}}_U\right),f\left({\overline{a}}_U\right)\right)\right].$

(ii)

Let $f$ is monotonically decreasing in $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right].$ Then $f\left(a_L\right)\ge f\left({\underset{\_}{a}}_U\right)\ge f\left({\overline{a}}_U\right).$ Hence $f\left(A_{S2}\right)=\left[f\left({\overline{a}}_U\right),\left(f\left({\underset{\_}{a}}_U\right),f\left(a_L\right)\right)\right].$ □

Example 6. 

Let $f(x)=\exp (x), \forall x\in \mathbb{R},$ the exponential function and also let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]\in I_{S2}(\mathbb{R}).$ Since, $f$ is monotonically increasing in $A_{S2},$ $f\left(A_{S2}\right)=\exp \left(A_{S2}\right)=\left[\exp \left(a_L\right),\left(\exp \left({\underset{\_}{a}}_U\right),\exp \left({\overline{a}}_U\right)\right)\right].$

Similarly, the logarithmic function is also expressed for Semi-Type-2 interval arguments, as it is strictly monotonic function.

3.1. Periodic Functions of Semi-Type-2 Arguments

Definition 8. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ be a Semi-Type-2 interval.

• Then

(i) 

$\sin \left(A_{S2}\right)=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]$,

where

$$b_L=\left\{\begin{array}{l}-1, \mathrm{if} \exists k\in \mathbb{Z}:2k\pi -(\pi /2)\in A_{S2}\\ \min \left\{\sin (a_L),\sin ({\overline{a}}_U)\right\}, \mathrm{otherwise}\end{array}\right.$$

$${\overline{b}}_U=\left\{\begin{array}{l}1, \mathrm{if} \exists k\in \mathbb{Z}:2k\pi +(\pi /2)\in A_{S2}\\ \max \left\{\sin (a_L),\sin ({\overline{a}}_U)\right\}, \mathrm{otherwise}\end{array}\right.$$

$${\underset{\_}{b}}_U=\sin ({\underset{\_}{a}}_U).$$

(ii) 

$\cos \left(A_{S2}\right)=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right],$

where

$$b_L=\left\{\begin{array}{l}-1, \mathrm{if} \exists k\in \mathbb{Z}:(2k+1)\pi \in A_{S2}\\ \min \left\{\cos (a_L),\cos ({\overline{a}}_U)\right\}, \mathrm{otherwise}\end{array}\right.$$

$${\overline{b}}_U=\left\{\begin{array}{l}1, \mathrm{if} \exists k\in \mathbb{Z}:2k\pi \in A_{S2}\\ \max \left\{\cos (a_L),\cos ({\overline{a}}_U)\right\}, \mathrm{otherwise}\end{array}\right.$$

$${\underset{\_}{b}}_U=\cos ({\underset{\_}{a}}_U).$$

(iii) 

$\tan \left(A_{S2}\right)=\left[\tan (a_L),\left(\tan ({\underset{\_}{a}}_U),\tan ({\overline{a}}_U)\right)\right],$ if $(2k+1){\displaystyle \frac{\pi }{2}}\notin A_{S2}, k\in \mathbb{Z}$.

(iv) 

$\cos ec\left(A_{S2}\right)=\left[\min C_1,\left(\cos ec({\underset{\_}{a}}_U),\max C_1\right)\right],$ if $k\pi \notin A_{S2}, k\in \mathbb{Z}$

where $C_1=\left\{\cos ec(a_L),\cos ec({\overline{a}}_U)\right\}.$

(v) 

$\sec \left(A_{S2}\right)=\left[\min C_2,\left(\sec ({\underset{\_}{a}}_U),\max C_2\right)\right],$ if $(2k+1){\displaystyle \frac{\pi }{2}}\notin A_{S2}, k\in \mathbb{Z}$

where $C_2=\left\{\sec (a_L),\sec ({\overline{a}}_U)\right\}.$

(vi) 

$\cot \left(A_{S2}\right)=\left[\cot ({\overline{a}}_U),\left(\cot ({\underset{\_}{a}}_U),\cot (a_L)\right)\right],$ if $k\pi \notin A_{S2}, k\in \mathbb{Z}$.

Example 7. 

Let $A_{S2}=\left[0,\left({\displaystyle \frac{\pi }{6}},{\displaystyle \frac{\pi }{3}}\right)\right].$

• Here $2k\pi -{\displaystyle \frac{\pi }{2}},2k\pi +{\displaystyle \frac{\pi }{2}}\notin A_{S2}.$
  Hence $\sin \left(A_{S2}\right)=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right],$
  where

  $$b_L=\min \left\{\sin \left(0\right),\sin \left({\displaystyle \frac{\pi }{3}}\right)\right\}=0.$$

  $${\overline{b}}_U=\max \left\{\sin \left(0\right),\sin \left({\displaystyle \frac{\pi }{3}}\right)\right\}={\displaystyle \frac{\sqrt{3}}{2}}.$$

  $${\underset{\_}{b}}_U=\sin \left({\displaystyle \frac{\pi }{6}}\right)={\displaystyle \frac{1}{2}}.$$
  Therefore,

  $$\sin \left(A_{S2}\right)=\left[0,\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\sqrt{3}}{2}}\right)\right].$$

Example 8. 

Let $A_{S2}=\left[-\pi ,\left(-{\displaystyle \frac{\pi }{2}},2\pi \right)\right].$

• Then $\cos \left(A_{S2}\right)=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right],$
• where

  $$b_L=-1, \sin \mathrm{ce} \left(2\mathrm{k}+1\right)\pi \in A_{S2}.$$

  $${\overline{b}}_U=1, \sin \mathrm{ce} 2k\pi \in A_{S2},$$

  $${\underset{\_}{b}}_U=\cos \left(-{\displaystyle \frac{\pi }{2}}\right)=0.$$
• Hence

  $$\cos \left(A_{S2}\right)=\left[-1,\left(0,1\right)\right].$$

Example 9. 

Let $A_{S2}=\left[-{\displaystyle \frac{\pi }{3}},\left(-{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{6}}\right)\right].$

• Then

  $$\tan \left(A_{S2}\right)=\left[\tan \left(-{\displaystyle \frac{\pi }{3}}\right),\left(\tan \left(-{\displaystyle \frac{\pi }{4}}\right),\tan \left({\displaystyle \frac{\pi }{6}}\right)\right)\right]=\left[-\sqrt{3},\left(-1,{\displaystyle \frac{1}{\sqrt{3}}}\right)\right].$$

3.2. Algebraic Properties of Semi-Type-2 Intervals

In this section, we have discussed some algebraic properties of Semi-Type-2 intervals with numerical illustration.

(i) Commutative property

For any $A_{S2}, B_{S2}\in I_{S2}$ (a) $A_{S2}+B_{S2}=B_{S2}+A_{S2}.$ (b) $A_{S2}{B}_{S2}=B_{S2}{A}_{S2}.$

Proof. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right], B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right].$

(a)

Now

$$A_{S2}+B_{S2}=\left[a_L+b_L,\left({\underset{\_}{a}}_U+{\underset{\_}{b}}_U,{\overline{a}}_U+{\overline{b}}_U\right)\right]=\left[b_L+a_L,\left({\underset{\_}{b}}_U+{\underset{\_}{a}}_U,{\overline{b}}_U+{\overline{a}}_U\right)\right]=B_{S2}+A_{S2}$$

(b)

Now

$$A_{S2}{B}_{S2}=\left[\min E,\left({\underset{\_}{a}}_U{\underset{\_}{b}}_U,\max E\right)\right],$$

where $E=\left\{a_L{b}_L,a_L{\overline{b}}_U,{\overline{a}}_U{b}_L,{\overline{a}}_U{\overline{b}}_U\right\}$

• Again, $B_{S2}{A}_{S2}=\left[\min F,\left({\underset{\_}{b}}_U{\underset{\_}{a}}_U,\max F\right)\right],$
• where $F=\left\{b_L{a}_L,b_L{\overline{a}}_U,{\overline{b}}_U{a}_L,{\overline{b}}_U{\overline{a}}_U\right\}.$
• Since $E=F,$ hence $A_{S2}{B}_{S2}=B_{S2}{A}_{S2}.$ □

(ii) Associative property

• Let $A_{S2}, B_{S2}, C_{S2}\in I_{S2}.$ Then
• (a) $\left(A_{S2}+B_{S2}\right)+C_{S2}=A_{S2}+\left(B_{S2}+C_{S2}\right).$
• (b) $\left(A_{S2}{B}_{S2}\right)C_{S2}=A_{S2}\left(B_{S2}{C}_{S2}\right).$

Proof. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right], B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right], C_{S2}=\left[c_L,\left({\underset{\_}{c}}_U,{\overline{c}}_U\right)\right]\in I_{S2}.$

(a)

Now $\left(A_{S2}+B_{S2}\right)+C_{S2}$

$=\left[\left(a_L+b_L\right)+c_L,\left(\left({\underset{\_}{a}}_U+{\underset{\_}{b}}_U\right)+{\underset{\_}{c}}_U,\left({\overline{a}}_U+{\overline{b}}_U\right)+{\overline{c}}_U\right)\right]$

$=\left[a_L+\left(b_L+c_L\right),\left({\underset{\_}{a}}_U+\left({\underset{\_}{b}}_U+{\underset{\_}{c}}_U\right),{\overline{a}}_U+\left({\overline{b}}_U+{\overline{c}}_U\right)\right)\right]$

$=A_{S2}+\left(B_{S2}+C_{S2}\right)$

(b)

Now $A_{S2}{B}_{S2}=\left[\min G,\left({\underset{\_}{a}}_U{\underset{\_}{b}}_U,\max G\right)\right],$

where $G=\left\{a_L{b}_L,a_L{\overline{b}}_U,{\overline{a}}_U{b}_L,{\overline{a}}_U{\overline{b}}_U\right\}$

and $B_{S2}{C}_{S2}=\left[\min H,\left({\underset{\_}{b}}_U{\underset{\_}{c}}_U,\max H\right)\right],$

where $H=\left\{b_L{c}_L,b_L{\overline{c}}_U,{\overline{b}}_U{c}_L,{\overline{b}}_U{\overline{c}}_U\right\}.$

Then $\left(A_{S2}{B}_{S2}\right)C_{S2}=\left[\min G^{\prime },\left(\left({\underset{\_}{a}}_U{\underset{\_}{b}}_U\right){\underset{\_}{c}}_U,\max G^{\prime }\right)\right],$

where $G^{\prime }=\left\{c_L\min G,{\overline{c}}_U\min G,c_L\max G,{\overline{c}}_U\max G\right\}$

and $A_{S2}\left(B_{S2}{C}_{S2}\right)=\left[\min H^{\prime },\left({\underset{\_}{a}}_U\left({\underset{\_}{b}}_U{\underset{\_}{c}}_U\right),\max H^{\prime }\right)\right],$

where $H^{\prime }=\left\{a_L\min H,a_L\max H,{\overline{a}}_U\min H,{\overline{a}}_U\max H\right\}.$

Clearly, $\min G^{\prime }=\min H^{\prime },$ $\left({\underset{\_}{a}}_U{\underset{\_}{b}}_U\right){\underset{\_}{c}}_U={\underset{\_}{a}}_U\left({\underset{\_}{b}}_U{\underset{\_}{c}}_U\right),$ $\max G^{\prime }=\max H^{\prime }.$

Therefore, $\left(A_{S2}{B}_{S2}\right)C_{S2}=A_{S2}\left(B_{S2}{C}_{S2}\right).$ □

(iii) Existence of additive, multiplicative identities

• The degenerate Semi-Type-2 intervals $O_{S2}=\left[0,\left(0,0\right)\right]$ and $I_{S2}=\left[1,\left(1,1\right)\right]$ are additive and multiplicative identity elements respectively because
  $\forall A_{S2}\in I_{S2},$ $A_{S2}+O_{S2}=O_{S2}+A_{S2}=A_{S2}$ and $A_{S2}. I_{S2}=I_{S2}. A_{S2}=A_{S2}.$

(iv) Non-existence of inverse

• Let $A_{S2}\in I_{S2}.$ Now, $A_{S2}-A_{S2}=\left[a_L-{\overline{a}}_U,\left(0,{\overline{a}}_U-a_L\right)\right]\ne \left[0,\left(0,0\right)\right].$ So the additive inverse of $A_{S2}$ does not exist in general. $A_{S2}-A_{S2}=O_{S2}$ holds only if $A_{S2}$ is degenerate.
  Similarly, $A_{S2}/A_{S2}=\left[{\displaystyle \frac{a_L}{{\overline{a}}_U}},\left(1,{\displaystyle \frac{{\overline{a}}_U}{a_L}}\right)\right]\ne \left[1,\left(1,1\right)\right]$ in general. $A_{S2}/A_{S2}=\left[1,\left(1,1\right)\right]$ holds only if $A_{S2}$ is degenerate.

(iv) Sub-distributive property

• Let $A_{S2}, B_{S2}, C_{S2}\in I_{S2}.$ Then $A_{S2}\left(B_{S2}+C_{S2}\right){\subseteq }_{S2}{A}_{S2}{B}_{S2}+A_{S2}{C}_{S2}.$

Proof. 

Proof is trivial. It follows from the definitions of Semi-Typ-2 interval arithmetic. □

Note 3. 

The equality $A_{S2}\left(B_{S2}+C_{S2}\right)=A_{S2}{B}_{S2}+A_{S2}{C}_{S2}$ does not hold in general.

We have shown this with the help of an example as follows:

Example 10. 

For example, let $A_{S2}=\left[-1,\left(2,3\right)\right], B_{S2}=\left[-5,\left(-2,1\right)\right], C_{S2}=\left[2,\left(3,4\right)\right].$

• Then

  $$A_{S2}\left(B_{S2}+C_{S2}\right)=\left[-1,\left(2,3\right)\right]\left[-3,\left(1,5\right)\right]=\left[-9,\left(2,15\right)\right]$$

  $$\begin{array}{ll}{A}_{S2}{B}_{S2}+A_{S2}{C}_{S2}& =\left[-1,\left(2,3\right)\right]\left[-5,\left(-2,1\right)\right]+\left[-1,\left(2,3\right)\right]\left[2,\left(3,4\right)\right]\\ & =\left[-15,\left(-4,5\right)\right]+\left[-4,\left(6,12\right)\right]\\ & =\left[-19,\left(2,17\right)\right].\end{array}$$
  Therefore, $A_{S2}\left(B_{S2}+C_{S2}\right)\ne A_{S2}{B}_{S2}+A_{S2}{C}_{S2}$ in general.

Note 4. 

(1) If $A_{S2}$ is degenerate Semi-Type-2, then equality holds.

• (2) Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right], B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right], C_{S2}=\left[c_L,\left({\underset{\_}{c}}_U,{\overline{c}}_U\right)\right]\in I_{S2}.$ If $a_L,b_L,c_L\ge 0$ then $A_{S2}\left(B_{S2}+C_{S2}\right)=A_{S2}{B}_{S2}+A_{S2}{C}_{S2}.$

(v) Cancellation property

• Let $A_{S2}, B_{S2}, C_{S2}\in I_{S2}.$ Then $A_{S2}+C_{S2}=B_{S2}+C_{S2}\Rightarrow A_{S2}=B_{S2}.$

Proof. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right], B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right], C_{S2}=\left[c_L,\left({\underset{\_}{c}}_U,{\overline{c}}_U\right)\right].$

• Now, $A_{S2}+C_{S2}=B_{S2}+C_{S2}$
  $\Rightarrow \left[a_L+c_L,\left({\underset{\_}{a}}_U+{\underset{\_}{c}}_U,{\overline{a}}_U+{\overline{c}}_U\right)\right]=\left[b_L+c_L,\left({\underset{\_}{b}}_U+{\underset{\_}{c}}_U,{\overline{b}}_U+{\overline{c}}_U\right)\right]$
  $\Rightarrow a_L+c_L=b_L+c_L, {\underset{\_}{a}}_U+{\underset{\_}{c}}_U={\underset{\_}{b}}_U+{\underset{\_}{c}}_U, {\overline{a}}_U+{\overline{c}}_U={\overline{b}}_U+{\overline{c}}_U$
  $\Rightarrow a_L=b_L, {\underset{\_}{a}}_U={\underset{\_}{b}}_U, {\overline{a}}_U={\overline{b}}_U.$
  Hence $A_{S2}=B_{S2}. \square$

Note 5. 

Cancellation Law for multiplication does not hold in Semi-Type-2 interval arithmetic, i.e., $A_{S2}{C}_{S2}=B_{S2}{C}_{S2}$ may not imply $A_{S2}=B_{S2}.$

We have shown this with the help of an example as follows:

Example 11. 

For example, let $A_{S2}=\left[3,\left(3,3\right)\right],$ $B_{S2}=\left[-3,\left(3,3\right)\right]$ and $C_{S2}=\left[-4,\left(4,4\right)\right].$ Then $A_{S2}{C}_{S2}=B_{S2}{C}_{S2}=\left[-12,\left(12,12\right)\right].$ But $A_{S2}\ne B_{S2}.$

4. Order Relations of Semi-Type-2 Interval

In this section, we have proposed a Semi-Type-2 interval order relation to compare any two Semi-Type-2 intervals. Then, we have shown that the proposed order relation is a total order relation. Before proposing the said order relation, some score components have been defined to determine a Semi-Type-2 interval uniquely. We have introduced score components to determine any Semi-Type-2 interval uniquely and compare any two such intervals easily, similar to the centre radius in the case of an interval. However, two Semi-Type-2 intervals can also be compared by their bounds; score components are introduced to define a complete interval order relation in an easy way. Also, the order relation based on score components will be very user-friendly in real-life scenarios.

Definition 9. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ be a Semi-Type-2 interval. Then a set of score components of $A_{S2}$ is defined by the set $\left\{a^{\prime },a^{″},{\overline{a}}_U\right\}$ which uniquely determines $A_{S2}$ , where $a^{\prime }={\displaystyle \frac{a_L+{\underset{\_}{a}}_U+{\overline{a}}_U}{3}},$ $a^{″}={\displaystyle \frac{{\underset{\_}{a}}_U+{\overline{a}}_U}{2}}$.

Proposition 2. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ and $B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]$ be two Semi-Type-2 intervals with corresponding sets of score components $\left\{a^{\prime },a^{″},{\overline{a}}_U\right\}$ and $\left\{b^{\prime },b^{″},{\overline{b}}_U\right\}$ . Then, $A_{S2}=B_{S2}$ iff $a^{\prime }=b^{\prime },a^{″}=b^{″} \mathrm{and} {\overline{a}}_U={\overline{b}}_U.$

Proof. 

Follows from Definitions 3 and 9. □

Definition 10. 

Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ and $B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]$ be two Semi-Type-2 intervals with corresponding sets of score components $\left\{a^{\prime },a^{″},{\overline{a}}_U\right\}$ and $\left\{b^{\prime },b^{″},{\overline{b}}_U\right\}$ . Then the ordering relation ${\le }_{S2}$ between two Semi-Type-2 intervals $A_{S2}$ and $B_{S2}$ is defined as follows:

$$A_{S2}{\le }_{S2}{B}_{S2}\iff \left\{\begin{array}{l}{a}^{\prime }\le b^{\prime }, \mathrm{if} a^{\prime }\ne b^{\prime }\\ a^{″}\le b^{″}, \mathrm{if} a^{\prime }=b^{\prime }, a^{″}\ne b^{″}\\ {\overline{a}}_U\le {\overline{b}}_U, \mathrm{if} a^{\prime }=b^{\prime }, a^{″}=b^{″}\end{array}\right.$$

and $A_{S2}{<}_{S2}{B}_{S2}\iff A_{S2}{\le }_{S2}{B}_{S2} \& A_{S2}\ne B_{S2}.$

Example 12. 

Compare the following pairs of Semi-Type-2 intervals using Definition 10:

(1) 

$A_{S2}=\left[-1,\left(4,6\right)\right]$ and $B_{S2}=\left[-2,\left(-1,5\right)\right]$

(2) 

$A_{S2}=\left[2,\left(4,6\right)\right]$ and $B_{S2}=\left[3,\left(5,7\right)\right]$

(3) 

$A_{S2}=\left[1,\left(3,5\right)\right]$ and $B_{S2}=\left[2,\left(3,4\right)\right]$

(4) 

$A_{S2}=\left[-1,\left(4,6\right)\right]$ and $B_{S2}=\left[-1,\left(-1,11\right)\right]$

Solution. 

Depending upon the corresponding set of score components of $A_{S2}$ and $B_{S2}$ given Semi-Type-2 intervals are compared for each case and the results have been presented in the Table 2.

Table 2. Comparisons between $A_{S2}$ and $B_{S2}$ w.r.t Example-13.

Example-13	${\mathit{A}}_{\mathit{S}\mathbf{2}}$$\mathbf{and} {\mathit{B}}_{\mathit{S}\mathbf{2}}$	$\left\{{\mathit{a}}^{\prime }\mathbf{,}{\mathit{a}}^{″}\mathbf{,}{\overline{\mathit{a}}}_{\mathit{U}}\right\}$	$\left\{{\mathit{b}}^{\prime }\mathbf{,}{\mathit{b}}^{″}\mathbf{,}{\overline{\mathit{b}}}_{\mathit{U}}\right\}$	Score Component Comparison	Comment
(1)	$A_{S2}=\left[-1,\left(4,6\right)\right]$ $B_{S2}=\left[-2,\left(-1,5\right)\right]$	$\left\{3,5,6\right\}$	$\left\{{\displaystyle \frac{2}{3}},2,5\right\}$	$b^{\prime }={\displaystyle \frac{2}{3}}<3=a^{\prime }$	$B_{S2}{<}_{S2}{A}_{S2}$
(2)	$A_{S2}=\left[2,\left(4,6\right)\right]$ $B_{S2}=\left[3,\left(5,7\right)\right]$	$\left\{4,5,6\right\}$	$\left\{5,6,7\right\}$	$a^{\prime }=4<5=b^{\prime }$	$A_{S2}{<}_{S2}{B}_{S2}$
(3)	$A_{S2}=\left[1,\left(3,5\right)\right]$ $B_{S2}=\left[2,\left(3,4\right)\right]$	$\left\{3,4,5\right\}$	$\left\{3,3.5,4\right\}$	$\begin{array}{l}{a}^{\prime }=3=b^{\prime }\\ \mathrm{and}\\ b^{″}=3.5<4=a^{″}\end{array}$	$B_{S2}{<}_{S2}{A}_{S2}$
(4)	$A_{S2}=\left[-1,\left(4,6\right)\right]$ $B_{S2}=\left[-1,\left(-1,11\right)\right]$	$\left\{3,5,6\right\}$	$\left\{3,5,11\right\}$	$\begin{array}{l}{a}^{\prime }=3=b^{\prime }, a^{″}=5=b^{″}\\ \mathrm{and}\\ {\overline{a}}_U=6<11={\overline{b}}_U\end{array}$	$A_{S2}{<}_{S2}{B}_{S2}$

Theorem 2. 

Let $A_{S2},$ $B_{S2},$ $C_{S2}$be any three Semi-Type-2 intervals.

Then

(i) 

$A_{S2}{\le }_{S2}{A}_{S2}$ (Reflexivity)

(ii) 

$A_{S2}{\le }_{S2}{B}_{S2}$ and $B_{S2}{\le }_{S2}{A}_{S2}$ $\Rightarrow A_{S2}=B_{S2}$ (Anti-symmetric)

(iii) 

$A_{S2}{\le }_{S2}{B}_{S2}$ and $B_{S2}{\le }_{S2}{C}_{S2}$ $\Rightarrow A_{S2}{\le }_{S2}{C}_{S2}$ (Transitive)

(iv) 

$A_{S2}{\le }_{S2}{B}_{S2}$ or $B_{S2}{\le }_{S2}{A}_{S2}$ (Comparability).

Proof. 

(i) Let $\left\{a^{\prime },a^{″},{\overline{a}}_U\right\}$ be the set of score components of $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right].$

• From Definition 10 it is clear that $A_{S2}{\le }_{S2}{A}_{S2},$ for any Semi-Type-2 interval.
• (ii) Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right]$ and $B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right]$ be any two Semi-Type-2 intervals with set of score functions $\left\{a^{\prime },a^{″},{\overline{a}}_U\right\}$ and $\left\{b^{\prime },b^{″},{\overline{b}}_U\right\}$.
• Also, let $A_{S2}{\le }_{S2}{B}_{S2}$ and $B_{S2}{\le }_{S2}{A}_{S2}$ hold.
• Then $A_{S2}{\le }_{S2}{B}_{S2}\Rightarrow a^{\prime }\le b^{\prime }$ and $B_{S2}{\le }_{S2}{A}_{S2}\Rightarrow b^{\prime }\le a^{\prime }$ and so $a^{\prime }=b^{\prime }.$

Now,

$$a^{\prime }=b^{\prime }\mathrm{and} A_{S2}{\le }_{S2}{B}_{S2}\Rightarrow a^{″}\le b^{″},$$

(1)

On the other side,

$$b^{\prime }=a^{\prime }\mathrm{and} B_{S2}{\le }_{S2}{A}_{S2}\Rightarrow b^{″}\le a^{″}$$

(2)

(1) and (2) $\Rightarrow a^{″}=b^{″}.$

Again,

$$a^{\prime }=b^{\prime }, a^{″}=b^{″} \mathrm{and} A_{S2}{\le }_{S2}{B}_{S2}\Rightarrow {\overline{a}}_U\le {\overline{b}}_U$$

(3)

On the other hand,

$$b^{\prime }=a^{\prime }, b^{″}=a^{″} \mathrm{and} B_{S2}{\le }_{S2}{A}_{S2}\Rightarrow {\overline{b}}_U\le {\overline{a}}_U$$

(4)

(3) and (4) $\Rightarrow {\overline{a}}_U={\overline{b}}_U.$

• Therefore, $a^{\prime }=b^{\prime },$ $a^{″}=b^{″}$ and ${\overline{a}}_U={\overline{b}}_U$ $\Rightarrow A_{S2}=B_{S2}.$
• (iii) Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right],$ $B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right],$ $C_{S2}=\left[c_L,\left({\underset{\_}{c}}_U,{\overline{c}}_U\right)\right].$ Also, let $A_{S2}{\le }_{S2}{B}_{S2}$ and $B_{S2}{\le }_{S2}{C}_{S2}$ hold.
  Then, $A_{S2}{\le }_{S2}{B}_{S2}\Rightarrow a^{\prime }\le b^{\prime }$ and $B_{S2}{\le }_{S2}{C}_{S2}\Rightarrow b^{\prime }\le c^{\prime }.$
  Therefore $a^{\prime }\le b^{\prime }\le c^{\prime }$ $\Rightarrow \mathrm{either} a^{\prime }<c^{\prime } \mathrm{or} a^{\prime }=b^{\prime }=c^{\prime }.$
  Now, if $a^{\prime }<c^{\prime },$ then $A_{S2}{\le }_{S2}{C}_{S2}.$
  On the other hand, if $a^{\prime }=b^{\prime }=c^{\prime },$ then $A_{S2}{\le }_{S2}{B}_{S2}\Rightarrow a^{″}\le b^{″}$ and $B_{S2}{\le }_{S2}{C}_{S2}\Rightarrow b^{″}\le c^{″}.$
  Therefore $a^{″}\le b^{″}\le c^{″}$ $\Rightarrow \mathrm{either} a^{″}<c^{″} \mathrm{or} a^{″}=b^{″}=c^{″}.$
  Now, if $a^{″}<c^{″},$ then $A_{S2}{\le }_{S2}{C}_{S2}.$
  On the other hand, if $a^{″}=b^{″}=c^{″},$ then $A_{S2}{\le }_{S2}{B}_{S2}\Rightarrow {\overline{a}}_U\le {\overline{b}}_U$ and $B_{S2}{\le }_{S2}{C}_{S2}\Rightarrow {\overline{b}}_U\le {\overline{c}}_U$ $\Rightarrow {\overline{a}}_U\le {\overline{c}}_U.$
  Now, $a^{\prime }=c^{\prime },$ $a^{″}=c^{″}$ and ${\overline{a}}_U\le {\overline{c}}_U$ $\Rightarrow A_{S2}{\le }_{S2}{C}_{S2}.$
• (iv) Let $A_{S2}=\left[a_L,\left({\underset{\_}{a}}_U,{\overline{a}}_U\right)\right],$ $B_{S2}=\left[b_L,\left({\underset{\_}{b}}_U,{\overline{b}}_U\right)\right].$
  Now, from the law of trichotomy of real numbers one has $a^{\prime }<b^{\prime }$ or $b^{\prime }<a^{\prime }$ or $a^{\prime }=b^{\prime }.$
  If, $a^{\prime }<b^{\prime } \mathrm{then} A_{S2}{\le }_{S2}{B}_{S2}$ or if $b^{\prime }<a^{\prime } \mathrm{then} B_{S2}{\le }_{S2}{A}_{S2}.$
  Again if $a^{\prime }=b^{\prime },$ then either, $a^{″}<b^{″}$ or $b^{″}<a^{″}$ or $a^{″}=b^{″}.$
  If, $a^{\prime }=b^{\prime }$ and $a^{″}<b^{″}$ $\mathrm{then} A_{S2}{\le }_{S2}{B}_{S2}$ or $a^{\prime }=b^{\prime }$ and $b^{″}<a^{″}$ $\Rightarrow B_{S2}{\le }_{S2}{A}_{S2}.$
  Finally, if $a^{\prime }=b^{\prime }$ and $a^{″}=b^{″},$ then either ${\overline{a}}_U\le {\overline{b}}_U$ or ${\overline{b}}_U\le {\overline{a}}_U.$
  Now, $a^{\prime }=b^{\prime },$ $a^{″}=b^{″}$ and ${\overline{a}}_U\le {\overline{b}}_U$ $\Rightarrow A_{S2}{\le }_{S2}{B}_{S2}.$
  OR, $a^{\prime }=b^{\prime },$ $a^{″}=b^{″}$ and ${\overline{b}}_U\le {\overline{a}}_U$ $\Rightarrow B_{S2}{\le }_{S2}{A}_{S2}.$ □

Combining all of the above, we can say that for any two Semi-Type-2 intervals $A_{S2}$ and $B_{S2}$, either $A_{S2}{\le }_{S2}{B}_{S2}$ or $B_{S2}{\le }_{S2}{A}_{S2}$ hold.

5. Conclusions

In this work, the concept of a Semi-Type-2 interval is proposed as a new uncertainty handling approach. Then, the definitions of interval mathematics and ranking of Semi-Type-2 intervals are established. Also, some other related properties are introduced. This proposed interval framework extends classical interval mathematics by introducing flexibility in one of the interval bounds. This modification addresses a limitation of both Type-1 and Type-2 intervals, where either full rigidity or complete flexibility may not adequately reflect real-world uncertainty. By bridging this gap, Semi-Type-2 intervals offer a more balanced mathematical structure that strengthens accuracy in uncertainty modelling and provides a richer foundation for future theoretical development in interval analysis and Operational Research/Management Sciences.

Beyond the mathematical contributions, Semi-Type-2 intervals carry important implications for practical decision-making under uncertainty. For instance, in financial forecasting, the method allows managers/system analysts to represent price fluctuations where lower thresholds are relatively stable but upper limits are uncertain. This leads to more robust predictions and reduced forecasting errors. In risk management and avoidance, Semi-Type-2 intervals can model asymmetric uncertainties such as fluctuating demand in supply chains/inventory, unpredictable patient responses in healthcare, or variable energy consumption in power systems. These applications highlight how the proposed approach can help decision-makers to minimize risk, optimize resource allocation, and improve resilience in uncertain environments.

With the introduction of the Semi-Type-2 interval approach, this work opens up several avenues for further research. Firstly, integrating it into optimization models, machine learning algorithms, and simulation-based decision systems could further enhance its utility in the area of financial risk, smart grids, epidemics, and climate modelling. Secondly, one can also extend the optimality theory for interval non-linear programming into the Semi-Type-2 interval case. Also, one can generalize the gH difference with the theory of interval differential equations in a Semi-Type-2 interval environment.