A food processing company uses $$25,000$$ $$kg$$ of corn flour every year. The quantity-discount price of corn flour is provided in the table below:

The order processing charges are Rs. $$500$$/order. The handling plus carry-over charge on an annual basis is $$20\% $$ of the purchase price of the corn flour per $$kg$$. The optimal order quantity (in $$kg$$) is __________.

A project consists of $$14$$ activities, $$A$$ to $$N$$. The duration of these activities (in days) are shown in brackets on the network diagram. The latest finish time (in days) for node $$10$$ is _____________

In a single-channel queuing model, the customer arrival rate is $$12$$ per hour and the serving rate is $$24$$ per hour. The expected time that a customer is in queue is _______ minutes.

A machine element $$XY,$$ fixed at end $$X,$$ is subjected to an axial load $$P,$$ transverse load $$F,$$ and a twisting moment $$T$$ at its free end $$Y.$$ The most critical point from the strength point of view is