MSCI 432/633: Production and Service Operations Management, Winter 2010
MSCI 432.001/633: Assignment # 3; MSCI 432.002: Assignment # 2
Q1. Discounting schemes.
a. Find the optimal ordering quantities for each of the items and the corresponding annual total cost.
[3 marks X3]
b. What is the primary difference between the ordering decisions made in a? Illustrate your answer
graphically. [2 marks] Q2. Limit on shelf life.
Simply find T EOQ=EOQ/ߣ and if T EOQ>SL then Q =SL*λ
Q3. Uncertain demand.
a. Assume that the firm has centralized all inventories in a single warehouse and that the probability
of stocking out in a cycle can still be no more than 5%. Ideally, how much average inventory can
the company now expect to hold, and at what cost? In this case, how long will a unit spend in the
warehouse before being sold? [5 marks] Solution:
(Since pipeline inventory is not influenced, we do not include it in the calculations below.)
(a) To determine the optimal order quantity, we will use the EOQ formula. Observe that
H = $10 * 25%/yr = $2.5/yr, d = 10,000 /wk, (i.e., annual demand, D = 500,000/yr), and S =
Optimal order quantity at each warehouse
2 DS 2×500,000×1000
Q = H 2.5 = 20,000.
Demand is normally distributed with:
Mean weekly demand = 10,000 units
Std dev. = 2000 units
Since replenishment lead time is 1 week:
LT = 1 week
And, the standard deviation of demand during lead time at each warehouse:
σLT = √LT*σ d 2,000.
For a 95% desired service level, z = 1.65.
Safety stock (SS) at each warehouse for 95% level of service
= z* σLT=1.65*2,000 = 3,300.
Average lead time demand (σ ) = 10,000.
Then, the reorder point (ROP) = σLT*z+ SS = 13,300. (Not necessary to calculate ROP in this
Average inventory at each warehouse (I) = Q/2 + SS = (20,000/2) + 3,300 = 13,300 units.
Average inventory holding cost per warehouse = H*I = $2.5*13,300 = $33,250.
Number of orders per year at each warehouse = D/Q = 500,000/20,000 = 25 order/year.
Annual Order cost per warehouse = S*R/Q = $1,000 * 25 = $25,000.
Average time unit spends in warehouse (by Little’s Law) = I/R = 13,300/10,000 = 1.33 weeks.
Since each warehouse is identical, the total average inventory across four warehouses
= 4* Average inventory in each warehouse
= 4*(13,300) = 53,200.
Annual order cost for all four warehouses
= 4* Annual order cost per warehouse
= 4* $25,000 = $100,000.
Annual holding cost for all four warehouses
= 4 * Average annual holding cost per warehouse
= 4* $33,250 = $133,000.
[b] The total average demand that the centralized warehouses faces = 4* Average demand per
warehouse = 4 * 10,000 = 40,000 units per week. We assume that the centralized warehouse has the same
cost structure as the individual warehouses. The optimal order quantity for the centralized warehouse is
determined using the EOQ formula with d = 40,000 per week (i.e., Annual Demand D = 50*40,000 =
2,000,000) and the remaining parameters as in [a]. This gives Q = 40,000 units.
Standard deviation of weekly demand at the central warehouse
= SQRT(# of warehouses) * std dev of weekly demand at one warehouse
= SQRT(4)* 2000 = 4,000 / week.
With a replenishment lead time of 1 week, the standard deviation of lead time demand at central
warehouse ( σLT = SQRT(LT)*( σ )d= 4,000.
Safety stock (SS) at central warehouse for 95% level of service = z*σ LT = 1.65*4,000 = 6,600.
Average lead time demand ( σ ) = 40,000 units.
Reorder point (ROP) = σ LT + SS = 46,600 units. (Not necessary to calculate ROP in this
Average inventory in central warehouse
= Q/2 + SS = (40,000/2) + 6,600 = 26,600.
Number of orders per year the central places
= D/Q = 50*40,000/40,000 = 50 orders/year.
Annual order cost for central warehouse = 50*$1,000 = $50,000.
Annual holding cost for central warehouse = H*I = $2.5 * 26,600 = $66,500.
Average time unit spends in warehouse = I/R = 26,600/40,000 = 0.67 weeks.
Q4 is a bonus question!
Q4. Order quantity under inflation. Q5 and Q6 are required only from MSCI432.001/633 students
Q5 [Nahmias 4.20] EPQ.
(a) Calculate the number of JJ39877 filters that Filter Systems should produce in each production run of
this particular part, so as to minimize annual holding and set-up costs? [2 marks]
(b) Assuming that it produces the optimal number of filters in each run, compute the