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 __________________.

A product made in two factories $$P$$ and $$Q,$$ is transported to two destinations, $$R$$ and $$S.$$ The per unit costs of transportation (in Rupees) from factories to destinations are as per the following matrix.

Factory $$P$$ produces $$7$$ units and factory $$Q$$ produces $$9$$ units of the product. Each destination requires $$8$$ units. If the north-west corner method provides the total transportation cost as $$X$$ (in Rupees) and the optimized (the minimum) total transportation cost $$Y$$ (in Rupees), then $$(X$$-$$Y),$$ in Rupees, is

The principal stresses at a point in a critical section of a machine component are $${\sigma _1} = 60\,\,MPa,\,\,{\sigma _2} = 5\,\,MPa$$ and $${\sigma _3} = - 40\,\,MPa.$$ For the material of the component, the tensile yield strength is $${\sigma _y} = 200\,\,MPa.$$ According to the maximum shear stress theory, the factor of safety is