Notes
• Quiz This Friday
• Covers 13 March through today
MGTSC 352
Lecture 21:
Inventory Management
A&E Noise example
Methods for finding good inventory policies:
1) simulation
2) EOQ + LTD models
Using EOQ for the Distribution Game:
Multi-Echelon Systems
Why Keep Inventory?
1. Seasonality (anticipated variation)
2. Provide flexibility (unanticipated
variation) a.k.a.:
3. Economies of scale
4. Price speculation (not an ops reason)
5. Something to work on
6. NDR,JP
Inventory By Where it IS
•
•
•
•
Raw Materials
Finished Goods
Work in Process
Or, with apologies to PS, “One man’s
ceiling is another man’s floor.”
Inventory
Approximation 1: constant demand
Therefore: We let inventory drop to
zero just before an order arrives
Time
Acquisition Costs (pg. 142)
No matter what the inventory policy,
acquisition costs = Demand X Cost
They don’t change,
So they don’t go in the model
(Unless you get quantity discounts, then it matters.)
Order Costs
•
Number of orders per year 
(3695 VCRs / year)/(80 VCRs / order)
= 46.2 orders / year
•
Total order cost per year 
(46.2 orders / year)($30 / order)
= $1385.63 / year
•
Total Order Costs = S * D/Q
Holding Costs (pg. 143)
• Minimum inventory  0 for now
Later = Safety Stock
• Maximum inventory = Q (+SS)
• Average inventory 
Q/2 = (80)/2 = 40 VCRs
• Total holding cost per year 
(40 VCR-years)($37.5 / VCR / year) = $1500 / year
• Total Holding Costs = H*Q/2
pg. 144
EOQ = Economic Order Quantity Model
• Given demand is constant
• Find the Q that minimizes total cost
Acquisition costs don’t
depend on Q
Total cost = acquisition cost + order cost +
carrying cost + shortage cost No shortages, by
assumption
• Total relevant cost = order cost + carrying cost
EOQ Derivation
S = order cost ($/order)
H = carrying cost ($/item/year)
D = demand (units/year)
Relevant cost
RC
RC(Q)
=
=
=
order cost
SN
SD/Q
pg. 147
Q = order quantity
N = number of orders per year
Iavg = average inventory
+
+
+
carrying cost
H  Iavg
HQ/2
Note: you can change
year to day, week, or any
other time unit, as long as
you are consistent
Common mistake:
inconsistent time units
To Excel
EOQ Formula
Relevant cost
RC
=
RC(Q)
=
pg. 147
=
ordering cost +
carrying cost
SN
+
H  Iavg
SD/Q
+
HQ/2
The magic part (optional)
pg. 147
Using EOQ for A&E Noise YNOS XD
D = 10.12 VCRs/day,
S = $30/order,
H = $0.10/VCR/day
 Q* = SQRT(210.1230/0.10) = 77.9
 round to Q* = 78
N* = 10.12/78 = 0.13 orders/day = 47.4 orders/year
Order every 365/47.4 = 8 days
Relevant cost:
RC(Q*) = S  (D/Q*) + H  (Q*/2)
= 30  (10.12/78) + 0.10  (78/2)
= 3.90 + 3.90
= $7.80 / day = $2,847 / year
Common mistake:
using inconsistent time units
D = 10.12 VCRs/day, S = $30/order, H =
$37.5/VCR/year
 Q* = SQRT(210.1230/37.5) = 4
• Off by (77.9 – 4)/77.9 = 95%
• Will not be worth a lot of part marks
Pg. 149
More on EOQ: Economies of Scale
The Capital Health Region* operates four hospitals.
Presently each hospital orders its own supplies and
manages its inventory. A common item used is a sterile
intravenous (IV) kit, with a weekly demand of 600 per
week at each hospital. Each IV kit costs $5 and incurs a
holding cost of 30% per year. Each order incurs a fixed
cost of $150 regardless of order size. The supplier takes
one week to deliver an order. Currently, each hospital
orders 6,000 kits at a time.
Question 1: Could costs be decreased by ordering more
often?
Question 2: Would it make sense to centralize inventory
management for the four hospitals?
* Fictional data
Analysis for one Hospital
• D = 600 / week = (600 / week)  (52 weeks/year)
= 31,200 / year
• S = $150 / order
• H = 0.3  5 = $1.50 / kit / year
• Q = SQRT(2  D  S / H) = 2,498 ≈ 2,500
• Costs:
– Q = 6,000:
S  D / Q + H  Q / 2 = $780 + $4,500 = $5,280
– Q = 2,500:
S  D / Q + H  Q / 2 = $1,872 + $1,875 = $3,747
– 29% savings
Analysis for one Hospital
• D = 600 / week = (600 / week)  (52 weeks/year)
= 31,200 / year
• S = $150 / order
• H = 0.3  5 = $1.50 / kit / year
• Q = SQRT(2  D  S / H) = 2,498 ≈ 2,500
• Close your course pack
• Active Learning: How do we change the
analysis if inventory management were
centralized for the four hospitals?
Analysis for four hospitals
managed together
•
•
•
•
•
D = 4  31,200 / year = 124,800 / year
S = $150 / order
H = $1.50 / kit / year
Q = SQRT(2  124,800  150 / 1.5) = 4,996 ≈ 5,000
Costs:
– Each hospital operated independently:
4  $3,747 = $14,988 / year
– All four together:
S  D / Q + H  Q / 2 = $3,744 + $3,750 = $7,494 / year
– 50% savings
• Quadrupling demand doubles the optimal order
quantity and doubles the total relevant cost
Four hospitals managed together
• Costs:
– Each hospital operated independently:
4  $3,747 = $14,988 / year
– All four together:
S  D / Q + H  Q / 2 = $3,744 + $3,750 = $7,494 / year
– 50% savings
• Quadrupling demand doubles the optimal order
quantity and doubles the total relevant cost
Determining ROP with EOQ model
Lead time = 5 days
Demand during lead time = (5 days)  (10.12 VCRs / day)  51 VCRs
 Set ROP = 51 VCRs
Inventory
Problem: this
calculation
assumes constant
demand. May lead
to shortages too
frequently
ROP
demand during
lead time
lead time
Time
Pg. 149
What happens to Holding Cost
when we Increase ROP?
• EOQ: constant demand, zero safety stock
– ROP = avg. demand during lead time
– Iavg = (min + max)/2 = (0+Q)/2 = Q/2
– Holding cost = H  Q / 2
• If we add safety stock = SS, then:
– ROP = avg. demand during lead time + SS
– Iavg = Q/2 + min = SS + Q/2
– Holding cost = H  (SS + Q / 2)
Pg. 152
Inventory
How Shortages Happen
Active learning:
How could we
have avoided the
shortage?
ROP
Demand
during
leadtime
Leadtime
Demand that was not met
Time
Inventory
The demand during the lead
time is uncertain. Here are 4
possibilities.
ROP
We’ll see how to pick
ROP so as to provide
a specified fill rate
… to Excel
Time
LTD Recap
• “LTD” worksheet in A&E Noise workbook
– Purpose: vary ROP (and Q, if desired) and
see what happens to the fill rate
• “LTD-exotic version”: can vary the lead
time
– Useful for comparing suppliers that provide
different lead times
pg. 151
Simulation versus EOQ
Dimension
Simulation
EOQ + LTD
Ease of evaluating a
policy
Need to build model –
time consuming
Simple formula for RC
– back of an envelope
Finding the optimum
Trial and error / data
table
Plug into formula for
Q*
Random demand
fluctuations
Taken into account
Ignored in EOQ
Seasonal demand
fluctuations
Can be taken into
account
Ignored
Shortages
Taken into account
Ignored in EOQ
Likely errors
(common mistakes)
Errors in formulas
Inconsistent units
Pg. 158
Back to the Distribution Game: Can
we use EOQ here?
A “multi-echelon” system
Supplier
Warehouse
Retailer
Retailer
Retailer
Using EOQ for a two-echelon
system
• Upper echelon:
– Use warehouse holding cost rate
• Ignore higher cost of holding inventory at retailers
– Lead time
= 15 (supplier  warehouse) + 5 (warehouse  retailer)
= 20 days
• Lower echelon:
– Use incremental retailer holding cost rate
– Lead time = 5 days
• Coordination: warehouse order size should be a
multiple of the sum of the retailer order sizes
Data
Assume open 250 days / year
•
•
•
•
•
•
•
•
•
Supplier to warehouse transit time: 15 days
Warehouse to retailer transit time: 5 days
Demand per retailer: 500 per year
Selling price: $100/unit
Purchase price: $70/unit
Supplier to warehouse order cost: $200
Warehouse to retailer order cost: $2.75
Warehouse holding cost: $10/unit/year
… To Excel
Retailer holding cost: $12/unit/year
Upper echelon:
Use warehouse holding cost rate
(Ignore higher cost of holding inventory at retailers)
Lead time = 15 (supplier  warehouse) + 5 (warehouse 
retailer)
= 20 days
Upper echelon
Supplier
Warehouse
Retailer
Retailer
Retailer
Lower echelon:
Use incremental retailer holding cost rate
= retailer holding cost rate – warehouse holding cost rate
Lead time = 5 days
Lower echelon
Retailer
Supplier
Warehouse
Retailer
Retailer
Coordination
• Suppose each retailer uses QLower = 20. If
all retailers order at once, the total is 60.
• Active learning: you are the warehouse
manager. Knowing the retailer order
sizes, how would you pick the warehouse
order size?
Using EOQ for a 2-echelon system:
the details
• Upper echelon:
–
–
–
–
–
DUpper = 3  DRetailer
SUpper = SWarehouse
HUpper = HWarehouse
LTUpper = LTSupplier  Warehouse + LTWarehouse  Retailer
ROPUpper = DUpper  LTUpper
• Lower echelon
–
–
–
–
–
DLower = DRetailer
SLower = SRetailer
HLower = HRetailer - HWarehouse
LTLower = LTWarehouse  Retailer
ROPLower = DLower  LTLower
• Coordination: QUpper = n  SUM(QLower)
• Choose n (an integer) and QLower to minimize total cost for
the whole system
Data
Assume open 250 days / year
•
•
•
•
•
•
•
•
•
Supplier to warehouse transit time: 15 days
Warehouse to retailer transit time: 5 days
Demand per retailer: 500 per year
Selling price: $100/unit
Purchase price: $70/unit
Supplier to warehouse order cost: $200
Warehouse to retailer order cost: $2.75
Warehouse holding cost: $10/unit/year
… To Excel
Retailer holding cost: $12/unit/year