The throughput rate of a production system is $$20$$ units per hour. The average flow time is $$30$$ minutes and the cycle time is $$3$$ minutes. The average inventory (in units) in the system is

Consider a network with nodes $$1, 2, 3, 4, 5$$ and $$6.$$ The nodes are connected with directed arcs as shown in the table below. The respective costs (in $$INR$$) incurred while traversing the directed arcs are also mentioned.

The second shortest path from node $$1$$ to node $$6$$ (i.e. the path that has the second least total cost and does not use any part of the shortest path) has a total cost (in $$INR$$) of

In linear gas tungsten arc welding of two plates of the same material, the peak temperature $$T$$ (in $$K$$) is given as ܶ$$T = {{{c_1}q} \over \alpha },$$ where $$q$$ is the heat input per unit length (in $$J/m$$) of weld, $$\alpha $$ is the thermal diffusivity (in $${m^2}/s$$) of the plate materials and $${C_1}$$ is a constant independent of process parameters and material types. Two welding cases are given below.

where, $$V$$ is welding voltage, $$I$$ is welding current, $$v$$ is welding speed, and $$k,$$ $$\rho $$ and $$C$$ refer to the thermal conductivity, the density and the specific heat of the plate materials, respectively. Consider that electrical energy is completely converted to thermal energy. All other conditions remain same.

The ratio of the peak temperature in Case $${\rm I}$$ to that in Case $${\rm I}{\rm I}$$ is

Two flat steel sheets, each of $$2.5$$ $$mm$$ thickness, are being resistance spot welded using a current of $$6000$$ $$A$$ and weld time of $$0.2$$ $$s.$$ The contact resistance at the interface between the two sheets is $$200\,\,\mu \Omega $$ and the specific energy to melt steel is $$10 \times {10^9}\,\,J/{m^3}.$$ A spherical melt pool of diameter $$4$$ $$mm$$ is formed at the interface due to the current flow. Consider that electrical energy is completely converted to thermal energy. The ratio of the heat used for melting to the total resistive heat generated is _____________