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Find the economic lot size.

The annual demand of a product is 10,000 units. Each unit costs Rs. 100 if orders are placed in quantities below 200 units, but for orders of 200 and above, the price is Rs. 95. The annual inventory holding costs are 10 per cent of the value of the item and ordering cost is Rs. 5 per order. Find the economic lot size. -

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$\text{Demand} \hspace{3.1cm} D = 10,000 \ \text{units} \\ \text{Unit cost} \hspace{3cm} Cp = Rs. 100 \ \text{(orders below 200 units)} \\ \hspace{4.6cm} Cp = Rs. 95 \ \text{(orders of 200 and above)} \\ \text{Inventory holding costs} \hspace{0.5cm} Ch = 10 \% \ of \ Cp \\ \text{Cost of each order} \hspace{1.6cm} Co = Rs. 5$


  • Case 1:

    Finding EOQ (i.e. Q*) when Cp = Rs. 100

    $Q*$ $= \sqrt{(2.D.CoCh)} \\ = \sqrt{(2.D.CoCp.I )} \\ = \sqrt{(2×10000×5100×0.1 )} \\ = 100 \ units$

    Total minimum cost $= \sqrt{(2.D.Ch.Co) + Cp.D} \\ = \sqrt{(2×10000×100×0.1×5) + 100×10000} \\ = Rs. 10,01,000$

  • Case 2:

    Finding EOQ (i.e. Q*) when Cp = Rs. 95

    $Q*$ $= \sqrt{(2.D.CoCh)} \\ = \sqrt{(2.D.CoCp.I )} \\ = \sqrt{(2×10000×595×0.1 )} \\ = 102.6 \ units$

    But we cannot use this, since Cp = 95 is allowed only for orders for 200 and above.

  • Case 3:

    Cp = Rs. 95 and Q = 200 units

    Total minimum cost $= \sqrt{(2.D.Ch.Co) + Cp.D} \\ = \sqrt{(2×10000×95×0.1×5) + 95×10000} \\ = Rs. 9,50,974.7$

    So the economic lot size should be 200 units, and (10,000÷200) 50 orders need to be placed each year.


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