Question

A product with an annual demand of 1000 units has S = $25.50 and H = $8. The demand exhibits some variability such that the lead-time demand follows a normal probability distribution with μ = 25 and σ = 5.

1. What is the recommended order quantity? (1 mark)
2. What are the reorder point and safety stock if the firm desires at most a 2% probability of stock-out on any given cycle? (1.5 marks)
3. If the manager sets the reorder point at 30, what is the probability of a stock-out on any given cycle? How many times would you expect a stock-out during the year if this reorder point were used? (1.5 marks)

Answer 1

(a)

Annual demand, D = 1000 units
Fixed ordering cost, S = 25.50
Unit carrying cost, H = 8 per annum

So, the recommended order qty, EOQ = (2.D.S / H)^0.5 = (2*1000*25.5/8)^0.5 = 80 units

(b)

In-stock probability, F(z) = 1 - 0.02 = 0.98 which gives: z = 2.05

Safety stock = z*σ = 2.05*5 = 10.25 or 11 units (rounded up)

Reorder point, R = μ + safety stock = 25 + 11 = 36 units

(c)

R = μ + safety stock = 30

or, Safety stock = 30 - 25 = 5

or, z*σ = 5

or, z = 5/5 = 1

So, F(z) = 0.50 and 1 - F(z) = 0.5

So, the probability of stock-out in a given cycle will be 50%

With Q=80 units, the number of cycles per annum = 1000/80 = 12.5 cycles.

50% of these 12.5 cycles will observe stock-outs, So, the number of stock-outs per annum = 12.5*50% = 6.25