*5.1. Numerical Example*

The long-term stable development of enterprise hampered due to these issues: Development of production, environmental pollution, poor quality production, waste of resources, and lack of protection of the interests of the employees, as a result shareholders lose interest to invest their wealth, and they urge to special-purpose investment to the company and bear the investment risk. Thus, an enterprise's growth and survival depends on its ability to effectively deal with the relationship among various shareholders. The strategic managemen<sup>t</sup> experts gradually realized that it is a small-minded behavior for enterprises if they want to achieve the goal of shareholder value in the production of process, regardless of the interest of other stakeholders requirements. From the standpoint of stakeholders, as a supervision and managemen<sup>t</sup> system, the enterprise's financial performance is not only an enterprise's important self-monitoring, self-restraint, self-evaluation, but also have a vital instrumentation to effectively communicate with stakeholders, coordinating each stakeholder's interest, and finally achieving the strategic managemen<sup>t</sup> goal of enterprise. In this part, we shall present a project for the selection of best enterprise alternative(s) on the basis of the present trend of enterprise financial performances in order to investigate our proposed method. Here, we have evaluated the enterprise overall performance of five possible enterprises *Qt* (*t* = 1, 2, 3, 4, <sup>5</sup>). A company invests its money to an enterprize with the enterprise performances, and seeks to maximize the expected profit. In that view, it is required to calculate the enterprise performance of five possible enterprises as to select the desirable one. The whole decision-making process is presented by a flow-chart in Figure 1. The investment company take a decision depending on the following four attributes:


**Figure 1.** A flow chart of PFNs based on multiple attribute decision making (MADM) problem.

To keep away from dominating each other, decision makers are required to exempted the five possible enterprises *Qa* (*q* = 1, 2, 3, 4, 5) under the considered attributes whose weight vector (0.2, 0.1, 0.3, 0.4) determined by decision makers. According to opinions of decision makers, decision matrix *R* = (*β*ˆ *ab*)<sup>5</sup>×<sup>4</sup> is constructed under picture fuzzy information as in Table 1.

• **Step 1.** Decision matrix *R* is constructed by decision maker or expert under PF information as follows:


**Table 1.** Decision matrix *R* under picture fuzzy (PF)-information.

• **Step 2.** Let *ξ* = 3. By using the PFHWA operator of the overall performance values *β* ˆ *a* of enterprises, *Qa* (*a* = 1, 2, 3, 4, 5) are obtained as follows:

$$\hat{\mathcal{B}}\_{1} = \begin{array}{c} [(1+2\times0.96)^{0.2}\times(1+2\times0.90)^{0.1}\times(1+2\times0.40)^{0.3}\times(1+2\times0.09)^{0.4}] - [(1-0.96)^{0.2}\times(1-0.90)^{0.1}\times(1-0.40)^{0.3}\times(1-0.09)^{0.4}] \\\hline [(1+2\times0.96)^{0.2}\times(1+2\times0.90)^{0.1}\times(1+2\times0.40)^{0.3}\times(1+2\times0.09)^{0.4}] + 2\times[(1-0.96)^{0.2}\times(1-0.90)^{0.3}\times(1-0.40)^{0.3}\times(1-0.09)^{0.4}] \end{array}$$

$$\frac{3 \times [(0.34)^{0.2} \times (0.06)^{0.3} \times (0.33)^{0.3} \times (0.79)^{0.4}]}{[(1 + 2 \times (1 - 0.34))^{0.2} \times (1 + 2 \times (1 - 0.06))^{0.3} \times (1 + 2 \times (1 - 0.39))^{0.3} \times (1 + 2 \times (1 - 0.79))^{0.4}] - 2 \times [(0.34)^{0.2} \times (0.06)^{0.3} \times (0.33)^{0.3} \times (0.79)^{0.4}]} = 0.023$$

$$\frac{3 \times [(0.10)^{0.2} \times (0.04)^{0.1} \times (0.19)^{0.3} \times (0.03)^{0.4}]}{[(1 + 2 \times (1 - 0.10))^{0.2} \times (1 + 2 \times (1 - 0.19))^{0.3} \times (1 + 2 \times (1 - 0.03))^{0.4}] - 2 \times [(0.10)^{0.2} \times (0.04)^{0.1} \times (0.19)^{0.3} \times (0.03)^{0.4}]} = 0.027$$

= (0.394, 0.434, 0.070)

by a similar way, *β* ˆ 2, *β* ˆ 3, *β* ˆ 4, and *β* ˆ 5 are obtained as follows: *β* ˆ 2 = (0.492, 0.294, 0.107), *β* ˆ 3 = (0.520, 0.298, 0.063), *β*ˆ4 = (0.301, 0.517, 0.052), *β*ˆ5 = (0.351, 0.462, 0.067).

• **Step 3.** By using Equation (3) the score values *S* <sup>ˆ</sup>(*β*<sup>ˆ</sup> *a*) (*a* = 1, 2, 3, 4, 5) of the overall PFNs *β*ˆ *a* (*a* = 1, 2, 3, 4, 5) are obtained as follows:

*S* <sup>ˆ</sup>(*β*ˆ1) = 0.394 − 0.070 = 0.324. By a similar way, *S*<sup>ˆ</sup>(*β*ˆ2) = 0.386, *S*<sup>ˆ</sup>(*β*ˆ3) = 0.457, *S*<sup>ˆ</sup>(*β*ˆ4) = 0.249, *S* <sup>ˆ</sup>(*β*ˆ5) = 0.284.


Figures 2 and 3 show the graph of score values of *β* ˆ *a* obtained by two different operators.

**Figure 2.** Graph of score values of *β* ˆ *a* obtained by picture fuzzy Hamacher weighted averaging (PFHWA) operator.

**Figure 3.** Graph of score values of *β* ˆ *a* obtained by picture fuzzy Hamacher weighted geometric (PFHWG) operator.

If PFHWG operator is implemented instead, then the problem can be solved similarly as above.


*β* ˆ 1 = <sup>3</sup>×[(0.56)0.2×(0.90)0.1×(0.40)0.3×(0.09)0.4] [(<sup>1</sup>+2×(<sup>1</sup>−0.56))0.2×(<sup>1</sup>+2×(<sup>1</sup>−0.90))0.1×(<sup>1</sup>+2×(<sup>1</sup>−0.40))0.3×(<sup>1</sup>+2×(<sup>1</sup>−0.09))0.4]−2×[(0.56)0.2×(0.90)0.1×(0.40)0.3×(0.09)0.4] [(<sup>1</sup>+2×0.34)0.2×(<sup>1</sup>+2×0.06)0.1×(<sup>1</sup>+2×0.33)0.3×(<sup>1</sup>+2×0.79)0.4]−[(<sup>1</sup>−0.34)0.2×(<sup>1</sup>−0.06)0.1×(<sup>1</sup>−0.33)0.3×(<sup>1</sup>−0.79)0.4] [(<sup>1</sup>+2×0.34)0.2×(<sup>1</sup>+2×0.06)0.1×(<sup>1</sup>+2×0.33)0.3×(<sup>1</sup>+2×0.79)0.4]+<sup>2</sup>×[(<sup>1</sup>−0.34)0.2×(<sup>1</sup>−0.06)0.1×(<sup>1</sup>−0.33)0.3×(<sup>1</sup>−0.79)0.4] , [(<sup>1</sup>+2×0.10)0.2×(<sup>1</sup>+2×0.04)0.1×(<sup>1</sup>+2×0.19)0.3×(<sup>1</sup>+2×0.03)0.4]−[(<sup>1</sup>−0.10)0.2×(<sup>1</sup>−0.04)0.1×(<sup>1</sup>−0.19)0.3×(<sup>1</sup>−0.03)0.4] [(<sup>1</sup>+2×0.10)0.2×(<sup>1</sup>+2×0.04)0.1×(<sup>1</sup>+2×0.19)0.3×(<sup>1</sup>+2×0.03)0.4]+<sup>2</sup>×[(<sup>1</sup>−0.10)0.2×(<sup>1</sup>−0.04)0.1×(<sup>1</sup>−0.19)0.3×(<sup>1</sup>−0.03)0.4]

,

```
= (0.281, 0.531, 0.092)
```
by a similar way, *β* ˆ 2, *β* ˆ 3, *β* ˆ 4, and *β* ˆ 5 are obtained as *β* ˆ 2 = (0.327, 0.435, 0.110), *β*ˆ 3 = (0.312, 0.430, 0.067), *β* ˆ 4 = (0.129, 0.704, 0.054), *β*ˆ5 = (0.200, 0.669, 0.078).

• **Step 3.** Calculate the values of the score functions *S* <sup>ˆ</sup>(*β*<sup>ˆ</sup> *a*) (*a* = 1, 2, 3, 4, 5) of the overall picture fuzzy numbers *β* ˆ *a* (*a* = 1, 2, 3, 4, 5) as follows: *S* <sup>ˆ</sup>(*β*ˆ1) = 0.281 − 0.092 = 0.189, by a similar way, the other score values are obtained as follows *S* <sup>ˆ</sup>(*β*ˆ2) = 0.218, *S*<sup>ˆ</sup>(*β*ˆ3) = 0.246, *S*<sup>ˆ</sup>(*β*ˆ4) = 0.075, *S*<sup>ˆ</sup>(*β*ˆ5) = 0.122.


From the analysis, it is clear that although overall rating values of the alternatives are different for these two operators, graphically presented in Figures 2 and 3, the ranking orders of the alternatives are similar, and the most desirable enterprise is *Q*3.
