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GATE Mechanical Engineering (ME) 2020 Shift 2 Solved Paper

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Question : 56 of 65

Marks: +1, -0

For a single item inventory system, the demand is continuous, which is 10000 per year. The replenishment is instantaneous and backorders (S units) per cycle are allowed as shown in the figure.

As soon as the quantity (Q units) ordered from the supplier is received, the backordered quantity is issued to the customers. The ordering cost is Rs. 300 per order. The carrying cost is Rs. 4 per unit per year. The cost of backordering is Rs. 25 per unit per year. Based on the total cost minimization criteria, the maximum inventory reached in the system is _______ (round off to nearest integer).

Your Answer:

Solution:

In back order model,

(Q^{*}-S^{*}) C_{n}=S^{*} \times C_{b}

S^{*}=\left(\frac{Q^{*} C_{h}}{C_{h}+C_{b}}\right)

Maximum inventory level

=(Q^{*}-S^{*})

=Q^{*}-\frac{Q^{*} C_{h}}{C_{h}+C_{b}}

=\left(\frac{Q^{*} C_{b}}{C_{h}+C_{b}}\right)

M^{*}=\sqrt{\frac{2 D C_{0}}{C_{h}} \times \left(\frac{C_{b} \times C_{h}}{C_{b}}\right)} \times \left(\frac{C_{b}}{C_{h}+C_{b}}\right)

M^{*}=\sqrt{\frac{2 D C_{0}}{C_{h}} \times \left(\frac{C_{b}}{C_{b}+C_{b}}\right)}

=\sqrt{\frac{2 \times 10000 \times 300}{4} \times \left(\frac{25}{29}\right)}

=1137.147

unit

\cong 1137

units

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