Two models, $$P$$ and $$Q,$$ of a product earn profits of Rs. $$100$$ and Rs. $$80$$ per piece, respectively. Production times for $$P$$ and $$Q$$ are $$5$$ hours and $$3$$ hours, respectively, while the total production time available is $$150$$ hours. For a total batch size of $$40,$$ to maximize profit, the number of units of $$P$$ to be produced is ____________.

Following data refers to the jobs $$(P, Q, R, S)$$ which have arrived at a machine for scheduling. The shortest possible average flow time is ___________ days.

Consider the schematic of a riveted lap joint subjected to tensile load $$F,$$ as shown below. Let $$d$$ be the diameter of the rivets, and $${S_f}$$ be the maximum permissible tensile stress in the plates. What should be the minimum value for the thickness of the plates to guard against tensile failure of the plates? Assume the plates to be identical.

A machine element has an ultimate strength $$\left( {{\sigma _u}} \right)$$ of $$600\,\,N/m{m^2},$$ and endurance limit $$250$$ $$\,N/m{m^2}.$$ The fatigue curve for the element on log-log plot is shown below. If the element is to be designed for a finite of $$10000$$ cycles, the maximum amplitude of a completely reversed operating stress is _________ $$\,N/m{m^2}.$$