Question

Weekly demand of a good is normally distributed with mean 305 and variance of 1,280. c= $8 each. The processing cost for each order is $120 and the lead time is 5 weeks.

The store uses a 25% annual inventory holding cost rate. Under the current inventory policy, the store purchases 1,250 bottles whenever the inventory falls below 1,650. Assume that there are 50 weeks in a year.

(a) What is the fill rate under the current inventory policy?

(b) What order quantity and reorder point would minimize the total annual setup, holding and shortage costs subject to having a fill rate at least equal to 98%?

(c) Would the solution in part (b) change if the store has a minimum Type I service level constraint of 80%? Explain why.

Answer 1

Average demand, d = 305 per week
Stdev of weekly demand, σ = √1280 = 35.78
Unit cost, c = $8
Ordering cost, K = $120
Unit carrying cost, h = 25% of c = $2 per annum
Lead time, L = 5 weeks

(a)

Present order size, Q = 1250 units
Reorder point, R = 1650 units

So, Safety stock = z.σ.√L = R - d.L = 1650 - 305*5 = 125 units

or, z = 125 / σ.√L = 125 / (35.78*√5) = 1.56

The corresponding normal loss function, L(z) = 0.026

So, Fill rate, β = 1 - L(z)*σ.√L / Q = 1 - 0.026*35.78*√5/1250 = 0.9983 or 99.83%

(b)

Annual demand, λ = 305*50 = 15250 units

The starting point is Q = EOQ = (2.K.λ/h)^1/2 = (2*120*15250/2)^0.5 = 1353 units

σ_LTD = σ.√L = 35.78*√5 = 80
d_LTD = d.L = 305*5 = 1525

So,

The optimal value of Q and R is 1,416 units and 1,532 units respectively where the total cost will be minimized.

(c)

If we achieve just a 98% fill rate, the type-I service level F(R) will just be 53.7% (look at the table).

But if we increase the fill rate to 99.38% as shown below, the F(R) can be 80%

The optimal Q and R will be 1398 and 1594 respectively.