In a counter flow heat exchanger, hot fluid enters at $${65^ \circ }C$$ and cold fluid leaves at $${30^ \circ }C.$$ mass flow rate of the hot fluid is $$1$$ $$Kg/s$$ and that of cold fluid is $$2$$ $$kg/s$$. Specific heat of the hot fluid is $$10$$ $$kgK$$ and that of cold fluid is $$5$$ $$kj/kgK.$$ The $$LMTD$$ for the heat ecchanger is

The maximum level of inventory of item is $$100$$ and it is achieved with infinite replenishment rate. The inventory becomes zero over one and half month due to consumption at a uniform rate. This cycle continuous throughout the year. Ordering cost is Rs.$$100$$ per order and inventory carrying cost is Rs. $$10$$ per item per month. Annual cost (in Rs. ) of the plan, neglecting material cost, is

In a machine shop, pins of $$15mm$$ diameter are produced at a rate of $$1000$$ per month and the same is consumed at a rate of $$500$$ per month. The production and consumption continue simultaneously till the maximum inventory is reached. Then inventory is allowed to reduced to zero due to consumption. The lot size of production is $$1000.$$ If backlog is not allowed, the maximum inventory levels is

Capacities of production of an item over $$3$$ consecutive months in regular time are $$100,$$ $$100$$ and $$80$$ and in overtime are $$20,$$ $$20$$ and $$40.$$ The demands over those $$3$$ months are $$90,$$ $$130$$ and $$110.$$ The cost of production in regular time and overtime are respectively Rs. $$20$$ per item and Rs. $$24$$ per item. Inventory carrying cost is Rs. $$2$$ per item per month. The levels of starting and final inventory are nil. Backorder is not permitted. For minimum cost of plan, the level of planned production in overtime in the third month is