1BUS 1700 Managing
Operations
Class 6: Managing Inventory
Multi-Period Models
Hallie S. Cho
1
What Is Bad about Inventory?
• Opportunity cost of capital
• Physical holding cost
• Warehouse space
• Insurance
• Risk of obsolescence/Value loss
• Ordering cost
• Record keeping
• Transportation
• Receiving and material handling
2
2So Why Hold Inventory?
3
Types of Inventory by Stage in Process
• Raw Materials (RM)
• Inputs in the state in which they arrive at the firm
• Work in Process (WIP)
• Product at an intermediate stage of processing
• Finished Good (FG)
• Completed products ready for sale/shipping
4
3Types of Inventory by Function
Address the mismatch between supply and demand:
• Pipeline (in Transit) Inventory
• Movement of goods
• Decoupling Inventory
• Buffer between interdependent operations (breakdowns, uneven
production rates)
• Safety Stock
• Cushion against fluctuations in supply or demand or uncertain lead
times
• Anticipation Inventory
• Cushion for expected changes in supply or demand (e.g., seasonal
fluctuations)
5
Can We Replenish Inventory?
• Yes
• Long product life cycles
• E.g. coffee
• Multi-Period Models
• Today
• Economic Order Quantity
(EOQ)
• Safety Stock
• No
• Short product life cycles
• E.g. newspaper
• Single-Period Models
• Next class
• Newsvendor
6
4Multi-Period Models: When We Can Replenish
• Economic Order Quantity (EOQ) Model – How much
should we order each time we replenish?
• Safety Stock Model – How much should we buffer when
demand is uncertain?
7
EOQ Model: Inventory Replenishment
Time
Inventory
Q
Demand
Rate (D)
If the optimal order size Q = 10,
why not order 100?
why not order 5?
• Perfect World: No Demand Uncertainty
• Two Inventory Related Costs: Holding Cost and Ordering Cost
8
5Holding cost includes cost of capital, warehouse space, insurance, deterioration, …
ℎ: Annual holding cost per unit (cost of storing one unit in inventory for one year)
Total holding cost: ℎ× !"
Increasing order quantity, Q, will increase holding cost
Holding Cost
Time
Q
Average Inventory,
!"
9
Ordering cost includes fixed transportation costs, order processing/receiving costs
Annual demand is D. Order size is Q. Number of orders per year is: D/Q
: Cost per order
Total ordering cost: × #!
Increasing order quantity, Q, will decrease ordering cost
Ordering Cost
Time
Q
…
Demand
Rate (D)
1st order 2nd order 3rd , 4th , 5th … orders (D/Q)th order
10
6Calculating Q*
Order Quantity
Cost
Holding Cost: ℎ×2
11
Calculating Q*
Order Quantity
Cost
Holding Cost: ℎ×2Ordering Cost: ×
12
7Calculating Q*
Order Quantity
Cost
∗
Holding Cost: ℎ×2Ordering Cost: ×
Total Cost: ℎ×2 + ×
∗ = 2ℎ
13
Total Cost Curve is Relatively Flat Around Q*
Order Quantity
Cost
∗
Holding Cost: ℎ×2Ordering Cost: ×
Total Cost: ℎ×2 + ×
If you make mistakes in estimating any of the parameters h, K, or D, the EOQ model
will still produce costs close to the optimum → EOQ is a robust solution
→ Frequently used in practice
14
8EOQ Model with Lead Times > 0
Time
Inventory
Q*
Reorder
Point, ROP
Lead
time, LT
Lead
time, LT
Lead
time, LT
Order
here
Inventory Policy:
Order Q* units when inventory level drops to the reorder point, ROP
= !" (Demand during lead time) = ×
15
Music City Coffee (MCC) is a new independent coffee
shop targeting Nashville tourists. MCC predicts a
perfectly steady demand of 20 bags of coffee every week
throughout the year. MCC purchases its coffee from a
supplier in Honduras who charges $12.5 per bag and an
$8 fixed cost for delivery independent of the order size.
The delivery lead time is two weeks, and MCC estimates
its annual cost of capital to be 20%. Assume MCC is
open 50 weeks a year.
a) What is the optimal order quantity (EOQ), Q*?
b) What is the reorder point, ROP?
Music City Coffee Co.
16
9Music City Coffee Co. – TOPHAT
a) What is the optimal order quantity (EOQ), Q*?
D =
K =
h =
17
Music City Coffee Co. – TOPHAT
a) What is the optimal order quantity (EOQ), Q*?
b) What is the reorder point, ROP?
D = 20 bags/week × 50 weeks/year = 1,000 bags/year
K = $8.00
h = 20% per year × $12.50/unit = $2.50/unit/year ∗= '()* = '×+×,,...'.0 = 80 bags
LT = 2 weeks
ROP = 20 bags/week × 2 weeks = 40 bags
18
10
Time
Inventory
Q*
ROP = DLT
When demand is
perfectly predictable
(perfect world), you
can provide 100%
service level.
LT
LT
When demand is uncertain, setting the reorder point equal to D×LT can result in
poor service (i.e., stock-outs)
What if Demand is Uncertain?
19
Time
Inventory
Q*
How can we avoid stock-outs?
stock-outs
What if Demand is Uncertain?
ROP = DLT
20
11
Safety Stock
Time
Inventory
New
Reorder
Point
How much safety stock do I need?
Reorder Point = Expected demand over LT + Safety Stock
Expected
demand over
the lead time
Safety Stock
Hold Safety Stock to Reduce Stock-outs
21
A: Availability (In-stock rate)
LT: Lead time
Demand forecast per period
µ : Expected demand per period
s : Standard deviation of demand per period
Determining Safety Stock – Model Inputs
22
12
Å ÅÅ ...
Period 1 Period 2 Period LT
If demand in each period is normally distributed with mean and standard
deviation , then demand over the lead time will be normally distributed with
mean and standard deviation "
Demand Distribution over the Lead Time
!" = !" = $
23
Safety
Stock
Probability of
a stock-out
Expected demand
over LT
Availability
Step 1: Compute the z-value such that the area to the left of the z-value equals A
(z = NORM.S.INV(A) in Excel)
Step 2: Find the reorder point
Expected
demand over LT Safety stock
Determining the Reorder Point
$% = $% = "
= ($%) + SS = $% + $%= + "
24
13
• Given
• Annual demand D, annual holding cost per unit h, and cost per order K
• Availability A (e.g. 95%)
• Expected demand per period µ and standard deviation of demand per period s
• Lead time LT
• Find the economic order quantity
• Find the expected demand and its standard deviation over the lead time
• Compute the z-value such that the area to the left of the z-value equals A
(z=NORM.S.INV(A) in Excel)
• Find the reorder point
Summary
∗ = 2ℎ
$% = and $% = "
= + "
25
Demand Uncertainty at MCC
After operating for a few months, MCC executives realize
that hipsters’ demand for coffee is not as predictable as
they initially thought (i.e., demand is not perfectly stable)
They forecast that the average demand for coffee is 20
bags per week and the standard deviation of weekly
demand is 4 bags (i.e., = 20 and = 4)
Since customer satisfaction is the top priority for MCC, the
executives would like to have 98% availability (z = 2.05 for
A = 98%)

a) What is the optimal order quantity?
b) What is the reorder point?
26
14
Tasty Dutch Coffee Co. – Solution
a) What is the optimal order quantity?
b) What is the reorder point, ROP?
Q* = 80 bags. Demand uncertainty does not affect Q*.
= + "= 20×2 + 2.05 2×4" = 52 bags
27
Final Exam Formula Sheet: EOQ, Safety
Stock
Economic Order Quantity
D = annual demand
K = setup or ordering cost
h = annual holding cost
Reorder Point
μ = expected demand per period
s = standard deviation of demand per period
LT = lead time
z = z-score for a given availability A
hDKQ 2=
2sµ LTzLTROP +=
28
15
Quiz 3 Material
• Understand the trade-off in the EOQ model
• Find EOQ parameters
• ,, and ℎ
• Calculate ∗
• Understand the ROP and the ROP components
• Find ROP parameters
• , , , and
• Calculate expected demand over the lead time, safety stock,
and ROP
29

学霸联盟