The simplified value of $$\frac{1}{2}  of  \frac{8}{5} \div \left\{2\frac{1}{5} - \left(\frac{5}{16} + \frac{3}{5} \times 1\frac{7}{8} \div \frac{2}{3}\right)\right\}$$ is:

$$\frac{1}{2}  of  \frac{8}{5} \div \left\{2\frac{1}{5} - \left(\frac{5}{16} + \frac{3}{5} \times 1\frac{7}{8} \div \frac{2}{3}\right)\right\}$$

$$\frac{1}{2}\times\frac{8}{5}\div\left\{\frac{11}{5}-\left(\frac{5}{16}+\frac{3}{5}\times\frac{15}{8}\times\frac{3}{2}\right)\right\}$$

$$\frac{4}{5}\div\left\{\frac{11}{5}-\left(\frac{5}{16}+\frac{27}{16}\right)\right\}$$

$$\frac{4}{5}\div\left\{\frac{11}{5}-2\right\}$$

$$\frac{4}{5}\times5 = 4$$