The standard deviation of linear dimensions $$P$$ and $$Q$$ are $$3\mu m$$ and $$4\mu m$$, respectively. When assembled, the standard deviation (in $$\mu m$$) of the following linear dimension $$(P+Q)$$ is ___________.

A project starts with activity $$A$$ and ends with activity $$F.$$ The precedence relation and durations of the activities are as per the following table:

The minimum project completion time (in days) is ___________

For a single server with Poisson arrival and exponential service time, the arrival rate is $$12$$ per hour. Which one of the following service rates will provide a steady state finite queue length?

Maximize $$\,\,\,\,\,\,\,\,\,Z = 5{x_1} + 3{x_2}$$
Subject to $$\,\,\,\,\,\,\,\,\,\,{x_1} + 2{x_2} \le 10,$$
$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_1} - {x_2} \le 8, \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_1},{x_2} \ge 0 \cr} $$

In the starting simplex tableau, $${x_1}$$ and $${x_2}$$ are non-basic variables and the value of $$Z$$ is zero. The value of $$Z$$ in the next simplex tableau is __________________.