Question

The Collins Quarter Restaurant in Savannah, Georgia, is a popular place for weekend brunch.

The restaurant serves real maple syrup with French toast and pancakes. The restaurant manager,

Bob, buys the maple syrup from a company in Maine that requires 3 weeks for delivery. The

syrup costs Bobbi $4 a bottle and may be purchased in any quantity. Fixed costs of ordering

amount to about $75 for bookkeeping expenses, and holding costs are based on 20% annual

rate. Based on past experience, the weekly demand for the syrup is normal with mean 12 and

variance 16 (in bottles). Assume that there are 52 weeks in a year and that all excess demand is

backordered.

(a) (5 points) What is the expected demand and variance during lead time?

(b) (10 points) What inventory policy should Bob use if a Type I service level of 95% is desired

and demand is normally distributed?

(c) (10 points) What inventory policy should Bob use if a Type II service level of 95% is desired

and demand is normally distributed?

(d) (5 points) What is the expected number of backordered units under Type I and Type II

service levels?

(e) (10 points) Suppose that Bob’s supplier requires a minimum order size of 500 bottles. Find

the reorder level that Bob should use if he wishes to satisfy type II service level of 99%

assuming normal distribution for the demand.

Answer 1

Average demand, d = 12 per week
Stdev of weekly demand, σ = √Variance = √16 = 4
Average lead time, L = 3 weeks
Unit cost, C = $4
Unit holding cost, h = 20% of C = $0.8 per annum
Fixed ordering cost, K = $75 per order

Annual demand, λ = 12*52 = 624 units

(a)

Expected demand during lead time, d_LT = d.L = 12*3 = 36 units

Variance of demand during lead time, σ_LT² = σ².L = 16*3 = 48 units

Std deviation of lead time demand, σ_LT = √48 = 6.93 units

(b)

In-stock probability, F(z) = 95% which gives z = 1.645

Reorder point, R = d_LT + z.σ_LT = 36 + 1.645*6.93 = 48 units (rounded up)

Optimal order size, Q = (2.λ.K / h)^1/2 = SQRT(2*624*75/0.8) = 342 units

So, the policy variables are Q = 342 and R = 48 i.e. order 342 bottles when the inventory level falls to 48 bottles.

(c)

Fill rate (β) = 95%

DR) Iteration - L-) = (1-B).Q/OLT = R= d.1 + 2.0LT FR) = NORMSDIST (-) n(R) = OLTL) 19 18 342 360 361 361 0.95 0.95 0.95 0.95 2.468 2.597 2.605 2.605 -2.47 -2.60 -2.60 -2.60 0.007 0.005 0.005 0.005 17.10 18.00 18.05 18.05 18 18

The z-values correspoonding to each L(z) has to be found from loss function tables.

Note that the value of Q and R freezes after the third iteration.

So,

the policy variables are Q = 361 and R = 18 i.e. order 361 bottles when the inventory level falls to 18 bottles.

NOTE: One can also use the Q, R values of iteration-0 for simplicity (i.e. Q=342, R=19). Not much difference will be observed in terms of the total cost.

(d)

For the Type-I service level, z = 1.645 leading to L(z) = 0.02 and the expected shortage per cycle = L(z)*σ_LT = 0.02*6.93 = 0.14 bottles per order cycle.

For the Type-II service level, L(z) = 2.605 and the expected shortage per cycle = L(z)*σ_LT = 2.605*6.93 = 18.05 bottles per order cycle.

(e)

R= d LT + 2. OLT L() = (1-B).Q/OLT 0.722 500 0.99 -0.53 32

z = -0.53 for L(z) = 0.722, this has to be found from the loss function tables.

So, the re-order point is 32 bottles