Question

Question 5. (Risk Pooling)

Canadian Tires owns and operates two warehouses in Canada to serve two independent markets: Alberta and British Columbia. At present, Warehouse A serves Alberta and Warehouse B serves British Columbia. Demand in Alberta is normal with mean 100 and standard deviation 20. The demand in British Columbia is normal with mean 200 and standard deviation 30. The demand for each day is also independent. Suppose the lead time is 1 day.

(a) How much safety stock should each warehouse keep to ensure a 90% cycle service level? What is the re-order point (ROP) for each warehouse? What is the total safety stock for the two warehouses?

(b) Canadian Tires is considering replacing the two warehouses by building a central warehouse to serve the two markets. If the lead time remains unchanged, what is the new safety stock and ROP at the central warehouse to ensure the same service level of 90%?

(c) Why the safety stock at the central warehouse is smaller than the safety stock at the two warehouses combined?

Answer 1

The unit of demand is important. We assume that the demands given are daily demand.

(a)

Alberta

Mean demand, μ_A = 100
Stdev of demand, σ_A = 20
Average lead time, L = 1 day

Average lead time demand, μ_LTD = μ_A.L = 100*1 = 100
Stdev of lead time demand, σ_LTD = σ_A.√L = 20√1 = 20

Z = normsinv(0.90) = 1.28

Safety stock = Z.σ_LTD = 1.28*20 = 25.6 or 26 units

ROP = μ_LTD + Safety stock = 100+26 = 126 units

British Columbia

Mean demand, μ_B = 200
Stdev of demand, σ_B = 30
Average lead time, L = 1 day

Average lead time demand, μ_LTD = μ_B.L = 200*1 = 200
Stdev of lead time demand, σ_LTD = σ_B.√L = 30√1 = 30

Z = normsinv(0.90) = 1.28

Safety stock = Z.σ_LTD = 1.28*30 = 38.4 or 39 units

ROP = μ_LTD + Safety stock = 200+39 = 239 units

Total safety stock of two warehouses = 26 + 39 = 65 units

(b)

Mean demand, μ = μ_A + μ_B = 100+200 = 300
Stdev of demand, σ = [σ_A²+ σ_B²]^1/2 = sqrt(20^2 + 30^2) = 36.06
Average lead time, L = 1 day

Average lead time demand, μ_LTD = μ.L = 300*1 = 300
Stdev of lead time demand, σ_LTD = σ.√L = 36.06√1 = 36.06

Safety stock = Z.σ_LTD = 1.28*36.06 = 46.2 or 47 units

(c)

The safety stock is smaller becasue the standard deviation of demand is getting 'pooled' with the formula σ = [σ_A²+ σ_B²]^1/2 where we simply are not summing up the standard deviations.