Given below is a basic feasible solution to a transportation problem with three supply points $$(A,B,C)$$ and three demand points $$(P, Q, R)$$ that minimizes cost of transportation in the standard tabular format.

$$(a)$$$$\,\,\,\,\,\,\,\,$$ Compute the cost corresponding to the present solution
$$(b)$$$$\,\,\,\,\,\,\,\,$$ It is optimal?
$$(c)$$$$\,\,\,\,\,\,\,\,$$ Does an alternate optimum exist ?

The forecasts for a product for the next three months are given as $$750,850$$ and $$1000$$ units. The number of regular time days and overtime days available are $$22,18,$$ $$22$$ and $$4,$$ $$4,5$$ respectively. With the existing number of employees, the company can produce $$38$$ units per day. To meet the high demand in the third month, the company decides to hire people to increase the daily production to $$45$$ units.

The following costs are given :
Cost of regular time production $$=$$ Rs. $$20$$ per unit
Cost of overtime production $$=$$ Rs. $$25$$ Per unit
Cost of hiring $$=$$ $$200{L^2}$$
where $$'L'$$ is the increase in daily capacity

Inventory $$=$$ Rs. $$10$$ per unit per month (based on average inventory)
Shortage (back-ordering cost) $$=$$ Rs. $$20$$ per unit per month

The beginning inventory is $$100$$ units. The company decides to produce $$800,$$ $$700$$ and $$900$$ units respectively in the three months. Compute the cost of the production plan.

A $$1.5$$ $$kW$$ motor is running at $$1440$$ rev/min. It is to be connected to a stirrer running at $$36$$ rev/min. the gearing arrangement suitable for this application is

The life of a ball bearing at a load of $$10$$ $$kN$$ is $$8000$$ hours. Its life in hours, if the load is increased to $$20$$ $$kN,$$ keeping all other conditions same, is