*3.2. Gray Numbers*

In this section, we provide the fundamental principles and functions of gray numbers and their integration with the recently developed CoCoSo method [6].

The exact value of a gray number is uncertain; however, we know that it is within a range or a closed interval. Gray numbers can be continuous gray numbers within an interval. On the other hand, values from a finite number or a set of numbers are labeled as discrete gray numbers. A combined approach for both continuous and discrete gray numbers provided a new definition for gray numbers [33,34].

**Definition 1.** *Suppose that <sup>S</sup> is a gray value. If* <sup>∀</sup>e*<sup>s</sup>* <sup>∈</sup> *<sup>S</sup> and* <sup>e</sup>*<sup>s</sup>* <sup>=</sup> [*a*, *<sup>b</sup>*]*, then* <sup>e</sup>*<sup>s</sup> is known as an interval gray number, where a and b are the upper and lower bounds of* <sup>e</sup>*s and a*, *<sup>b</sup>* <sup>∈</sup> *R.*

**Definition 2.** *Assume that* <sup>e</sup>*s*<sup>1</sup> <sup>=</sup> [*a*, *<sup>b</sup>*] *and* <sup>e</sup>*s*<sup>2</sup> <sup>=</sup> [*c*, *<sup>d</sup>*] *are two interval gray numbers, and* λ > <sup>0</sup>*,* <sup>λ</sup> <sup>∈</sup> *R. The arithmetic operations are defined as follows [34,35]:*

$$1\,\widetilde{s}\_1 + \widetilde{s}\_2 = [a+c, b+d];\tag{6}$$

$$\text{l.2.} - s\_1 = [-b\_\prime - a];\tag{7}$$

$$[3.\ \overline{s}\_1 - \overline{s}\_2 = [a-d, b-c];\tag{8}$$

$$[\mathbf{4}, \lambda \overline{\mathbf{s}}\_1 = [\lambda \mathbf{a}, \lambda \mathbf{b}]. \tag{9}$$

**Definition 3.** *For a gray number S, if S* = S*n i*=1 [*ai* , *bi* ]*, then S is called an extended gray number (EGN). We consider S as a union of a set of closed or open intervals, while n is an integer and* 0 < *n* < ∞*, a<sup>i</sup>* , *b<sup>i</sup>* ∈ *R, and bi*−<sup>1</sup> < *a<sup>i</sup>* ≤ *b<sup>i</sup>* < *ai*+<sup>1</sup> *[33].*

**Theorem 1.** *If S is an EGN, then the following properties hold:*


**Definition 4.** *For two EGNs S* <sup>1</sup> = S*n i*=1 [*ai* , *bi* ] *and S*<sup>2</sup> = S*m j*=1 h *cj* , *d<sup>j</sup>* i *, let a<sup>i</sup>* ≤ *bi*(*i* = 1, 2, . . . , *n*)*,*

$$c\_i \le d\_i(j = 1, 2, \dots, m), \lambda \ge 0,\\
\text{and } \lambda \in \mathbb{R}. \text{ Then, the arithmetic operations can be defined as follows [34]:}\\
c\_i \le d\_i(j = 1, 2, \dots, m), \lambda \in \mathbb{R}. \text{ For each } i \in \{1, 2, \dots, m\}, \text{ we have } \lambda \le \lambda\_i \le \lambda\_{i+1}, \text{ where } \lambda\_i \text{ is the } i\text{th row of the } i\text{th row of the } i\text{th row of the } i\text{th row of the } i\text{th row of the } i\text{th row of the } i\text{th row of the } i\text{th row of the } i\text{th}$$

$$\mathbf{1}. \mathbf{S}\_1 + \mathbf{S}\_2 = \bigcup\_{i=1}^n \bigcup\_{j=1}^m \left[ a\_i + c\_j, b\_i + d\_j \right] \tag{10}$$

$$\text{L2.} - \text{S} = \bigcup\_{i=1}^{n} [-b\_{i\prime} - a\_{i\prime}]\_{\prime} \tag{11}$$

$$\text{G.S.} - \text{S}\_2 = \bigcup\_{i=1}^{n} \bigcup\_{j=1}^{m} [a\_i - d\_j, b\_i - c\_j]\_{\text{'}} \tag{12}$$

*Symmetry* **2020**, *12*, 886

$$4. \frac{S\_1}{S\_1} = \bigcup\_{i=1}^n \bigcup\_{j=1}^m [\min\{\frac{a\_i}{c\_j}, \frac{a\_i}{d\_j}, \frac{b\_i}{c\_j}, \frac{b\_i}{d\_j}\}, \max\{\frac{a\_i}{c\_j}, \frac{a\_i}{d\_j}, \frac{b\_i}{c\_j}, \frac{b\_i}{d\_j}\}] ,\tag{13}$$

*where c<sup>j</sup>* , 0*, d<sup>j</sup>* , 0*, and (j*=*1,2,* . . . *,m).*

$$\mathbf{5.8}\_1 \star \mathbf{S}\_2 = \bigcup\_{i=1}^n \bigcup\_{j=1}^m \left[ \min \{ a\_i c\_j, a\_i d\_j, b\_i c\_j, b\_i d\_j \}, \max \{ a\_i c\_j, a\_i d\_j, b\_i c\_j, b\_i d\_j \} \right], \tag{14}$$

$$\text{6.} \lambda \text{S}\_1 = \bigcup\_{i=1}^{n} [\lambda a\_i, \lambda b\_i]\_\prime \tag{15}$$

$$\nabla S\_1^{\ \lambda} = \bigcup\_{i=1}^n \left[ \min \left( a\_i^{\ \lambda}, b\_i^{\ \lambda} \right), \max \left( a\_i^{\ \lambda}, b\_i^{\ \lambda} \right) \right]. \tag{16}$$

**Definition 5.** *The length of a gray value S* = [*a*, *b*] *is measured as*

$$L(\mathcal{S}) = [b - a] / b. \tag{17}$$
