*3.3. Gray-Based Combined Compromise Solution (CoCoSo)*

For the first time in the literature related to MCDM methods, [6] proposed the CoCoSo method as an alternative ranking tool for multi-criteria problems. Since its introduction, the CoCoSo method was used for multiple problems in its initial form or in an extended form. In this paper, we use the gray-based CoCoSo method. The steps of the gray-based CoCoSo are described below.

**Step 1**. The decision-maker identifies decision factors and alternatives.

**Step 2**. In this step, the decision-maker constructs the initial gray-based decision matrix.

$$X = \begin{bmatrix} \bigcup \left[a\_{11}, b\_{11}\right] & \bigcup \left[a\_{12}, b\_{12}\right] & \cdots & \bigcup \left[a\_{1m}, b\_{1m}\right] \\ \bigcup \left[a\_{21}, b\_{21}\right] & \bigcup \left[a\_{22}, b\_{22}\right] & \cdots & \bigcup \left[a\_{2m}, b\_{2m}\right] \\ \vdots & \vdots & \cdots & \vdots \\ \bigcup \left[a\_{n1}, b\_{n1}\right] & \bigcup \left[a\_{n2}, b\_{n2}\right] & \cdots & \bigcup \left[a\_{nm}, b\_{nm}\right] \end{bmatrix} \tag{18}$$

where *aij* represents the lower bound, while *bij* represents the upper bound, for *i* = 1, 2, . . . , *m*, *j* = 1, 2, . . . , *n*.

**Step 3**. The constructed initial decision matrix is normalized using Equations (18) and (19) considering benefit and cost criteria.

$$r = \bigcup \left[ c\_{ij}, d\_{ij} \right] = \frac{\bigcup \left[ a\_{ij}, b\_{ij} \right] - \min\_{i} \bigcup \left[ a\_{ij}, b\_{ij} \right]}{\max\_{i} \bigcup \left[ a\_{ij}, b\_{ij} \right] - \min\_{i} \bigcup \left[ a\_{ij}, b\_{ij} \right]}; \text{ for the benefit criterion}. \tag{19}$$

$$\tau = \bigcup \left[ c\_{ij}, d\_{ij} \right] = \frac{\max \bigcup\_{i} \left[ a\_{ij}, b\_{ij} \right] - \bigcup \left[ a\_{ij}, b\_{ij} \right]}{\max \bigcup\_{i} \left[ a\_{ij}, b\_{ij} \right] - \min \bigcup\_{i} \left[ a\_{ij}, b\_{ij} \right]}; \text{ for the cost criterion}.\tag{20}$$

**Step 4**. The weighted normalized matrix and the sum of power weights of comparability sequences for each alternative are calculated using Equations (20) and (21).

$$S\_{\bar{l}} = \sum\_{j=1}^{n} (w\_j \bigcup [c\_{ij}, d\_{\bar{l}j}]).\tag{21}$$

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

$$P\_i = \sum\_{j=1}^{n} (\bigcup [c\_{ij}, d\_{ij}]^{w\_j}) . \tag{22}$$

**Step 5**. The relative weights of alternatives using three aggregation strategies are calculated in various ways. We use three appraisal score strategies to calculate the relative weights of other options using Equations (22)–(24).

$$H\_{\rm in} = \left[ h\_{1\rm ij}, h\_{2\rm ij} \right] = \frac{P\_i + S\_i}{\sum\_{i=1}^{m} (P\_i + S\_i)}.\tag{23}$$

$$L\_{\rm in} = \left[ l\_{1\bar{i}j\prime} l\_{2\bar{i}j} \right] = \frac{S\_{\bar{i}}}{\min\_{\bar{i}} S\_{\bar{i}}} + \frac{P\_{\bar{i}}}{\min\_{\bar{i}} P\_{\bar{i}}}.\tag{24}$$

$$M\_{\rm ini} = \left[ m\_{1\bar{1}\bar{\mu}}, m\_{2\bar{1}\bar{j}} \right] = \frac{\lambda S\_{\bar{i}} + (1 - \lambda)(P\_{\bar{i}})}{\lambda \max\_{\bar{i}} S\_{\bar{i}} + (1 - \lambda) \left( \max\_{\bar{i}} P\_{\bar{i}} \right)}, \text{ for } 0 \le \lambda \le 1. \tag{25}$$

The final preference order of the alternatives according to the CoCoSo-G method is calculated using Equation (25).

$$K\_i = (H\_{\rm ia} \ast L\_{\rm ia} \ast M\_{\rm ia})^{\frac{1}{3}} + \frac{1}{3}(H\_{\rm ia} + L\_{\rm ia} + M\_{\rm ia}),\tag{26}$$

where λ (normally λ = 0.5) is chosen by the decision-makers. Different values of λ can have a significant effect on the flexibility and stability of the proposed CoCoSo.

**Step 6**. To rank the alternatives, we obtain the length of the gray values shown by the above equation (based on Definition 5).
