Question

The best-selling product of ABC Manufacturing is Product X. ABC uses a (Q, R)- policy to manage the inventory level of Product X. If the on-hand inventory is not enough to meet demand, the excess demand is back-ordered. ABC’s service goals are twofold:(1) limit stock-outs to at most 5% of the cycles, and (2) limit the demand met through backorders to at most 1% of the total demand.ABC would like to pick the smallest reorder level R that achieves both of these goals. The lead time for producing X is 4 hours. It costs $10 to produce one unit of Product X. In addition, there is a fixed cost of producing a batch, equal to $400.The holding cost incurred by ABC is $0.02 per unit per hour. The hourly demand for product X is normally distributed with a mean of 25 and a standard deviation of 5. Product X sells for $15 per unit.

(a) What are the mean and the standard deviation of demand for Product X during the lead time of 4 hours?

(b)What are the order quantity (Q) and the reorder level (R) the purchasing department should use to achieve its goal?

(c)What is the resulting expected profit per hour?

Answer 1

Lead time, L = 4 hours

Unit cost, c = $ 10

Fixed setup cost, k = $ 400

Holding cost, h = $ 0.02 per unit per hour

Hourly demand, d = 25

Standard deviation of hourly demand, = 5

Selling price, p = $ 14

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(a)

Mean demand during lead time, d_L = d*L = 25*4 = 100

Standard deviation of demand during lead time, = *sqrt(L) = 5*sqrt(4) = 10

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(b)

Order quantity (Q) = sqrt(2dk/h)

= sqrt(2*25*400/0.02)

= 1000

(1) Limit stock-outs to at most 5% of the cycles

z = NORMSINV(.95) = 1.645

Reorder level (R) = d_L + z*

= 100+10

= 110

(2) Limit the demand met through back-orders to at most 1% of the total demand.  

This means fill-rate, = 99%

L(z) = (1-)*Q/

= (1-.99)*1000/10

= 1

Corresponding value of z = -0.8995   (from standard normal loss table)

Safety stock = z

= -0.8995*10

= -9

Reorder level (R) = d_L + safety stock

= 100 - 9

= 91

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