Supply Chain Management
Lecture 19
Outline
• Today
– Finish Chapter 10
– Start with Chapter 11
• Sections 1, 2, 3, 7, 8
– Skipping 11.2 “Evaluating Safety Inventory Given Desired Fill rate”
• Friday
– Homework 4 online tomorrow
• Due Thursday April 8 before class
• Next week
– Spring break
Economic Order Quantity
Annual cost
Total cost
CD + (D/Q)S + (Q/2)H
Holding cost
(Q/2)H
Order cost
(D/Q)S
Q* 
2 DS
hC
Order quantity
Material cost
CD
Example: Economic Order Quantity
• Example 10-2
– The store manager at Best Buy would like to reduce
the optimal lot size from 980 to 200. For this lot size
reduction to be optimal, the store manager wants to
evaluate how much the order cost per lot should be
reduced (currently $4,000)
Q* = sqrt((2DS)/(hC))
200 = sqrt((2 x 12,000 x S)/(0.2 x 500))
S = (hC(Q*)2)/2D
= (0.2 x 500 x 2002)/(2 x 12,000) = $166.7
Example: Economic Order Quantity
• How can the store manager reduce the fixed
ordering cost?
– Aggregate multiple products in a single order
• Can possibly combine shipments of different products from the
same supplier
• Can also have a single delivery coming from multiple suppliers
Aggregating replenishment across products in a single
order allows for a reduction in lot size for individual
products because fixed ordering and transportation
cost are now spread across multiple products
Lot Sizing with Multiple Products or
Customers
• Multiple products
– Independent orders
• No aggregation: Each product ordered separately
1
2
3
– Joint order of all products
• Complete aggregation: All products delivered on each truck
1 2 3
1 2 3
1 2 3
– Joint order of a subset of products
• Tailored aggregation: Selected subsets of products on each truck
1
1 2
Which option will likely have the lowest cost?
1 2 3
Aggregating Replenishment Across
Products
• Ordering cost has two components
– Common (to all products)
– Individual (to each product)
• Example
– It is cheaper for Wal-Mart to receive a truck containing
a single product than a truck containing many different
products
• Inventory and restocking effort is much less for a single product
Lot Sizing with Multiple Products or
Customers
• Example 10-3
– Best buy sells three models of computers, the Litepro,
the Medpro, and the Heavypro. Annual demands for
the three products are DL = 12,000 units for the
Litepro, DM = 1,200 units for the Medpro, and DH =
120 units for the Heavypro. Each model costs Best
Buy $500. A fixed transportation cost of $4,000 is
incurred each time an order is delivered. For each
model ordered and delivered on the same truck, an
additional fixed cost of $1,000 is incurred for receiving
and storage. Best Buy incurs a holding cost of 20
percent. Evaluate the lot sizes that the Best Buy
manager should order if lots for each product are
ordered and delivered independently.
Independent Orders
• Ordering cost is considered independent for each
product
– Apply EOQ to each product
Independent Orders
• Example 10-3
–
–
–
–
–
–
–
DL = 12,000
DM = 1,200
DH = 120
S = $4,000
sL = sM = sH = $1,000
h = 0.2
cL = cM = cH = $500
(demand per year)
(common order cost)
(product specific order cost)
(holding cost)
(material cost)
2 D  S
EOQ
Q* 
H
hC
Independent Orders
1
2
3
Litepro
Medpro
Heavypro
Demand per year D
12,000
1,200
120
Fixed cost/order S
$5,000
$5,000
$5,000
Optimal order size Q*
1,095
346
110
Cycle inventory Q/2
548
173
55
Order frequency n*
11.0/year
3.5/year
1.1/year
Average flow time
2.4 weeks
7.5 weeks
23.7 weeks
Annual holding cost
$54,772
$17,321
$5,477
Annual order cost
$54,772
$17,321
$5,477
Total cost
$109,544
$34,642
$10,945
Total cost = $155,140
Joint Orders of all Products
• Joint order of all products
– Complete aggregation: All products delivered on each
truck
• An order frequency is calculated by aggregating the ordering
costs and assuming that all products will be ordered at the
same time
Supply Chain Cost Influenced by Lot
Size
Annual cost
Holding cost
(Q1/2)H1 + (Q2/2)H2 + (Q3/2)H3
Order cost
(D1/Q1)S + (D2/Q2)S + (D3/Q3)S
Material cost
C1D1 + C2D2 + C3D3
Order quantity
Supply Chain Cost Influenced by Lot
Size
Annual cost
Holding cost
(D1/2n)H1 + (D2/2n)H2 + (D3/2n)H3
Order cost
nS*
Material cost
C1D1 + C2D2 + C3D3
Order quantity
Joint Orders of all Products
• Joint order of all products
– Complete aggregation: All products delivered on each
truck
• An order frequency is calculated by aggregating the ordering
costs and assuming that all products will be ordered at the
same time
n* 

k
i 1
Di hi Ci
2S *
Joint Orders of all Products
n* 

k
i 1
2S *
• Inputs
–
–
–
–
–
–
–
•
•
•
•
•
DL = 12,000
DM = 1,200
DH = 120
S = $4,000
sL = sM = sH = $1,000
h = 0.2
cL = cM = cH = $500
(demand per year)
(common order cost)
(product specific order cost)
(holding cost)
(material cost)
S* = S + sL + sM + sH = $7000
n* = SQRT((DLhCL+ DMhCM + DHhCH)/2S*) = 9.75
QL = DL/n* = 12000/9.75 = 1230
QM = DM/n* = 1200/9.75 = 123
QH = DH/n* = 120/9.75 = 12.3
Di hi Ci
Joint Orders of all Products
1 2 3
Litepro
1 2 3
Medpro
1 2 3
Heavypro
Demand per year
12,000
1,200
120
Order frequency
9.75/year
9.75/year
9.75/year
1,230
123
12.3
$61,512
$6,151
$615
Optimal order size
Annual holding cost
Annual order cost
$68,278
Total cost = $136,556
Joint Order of a Subset of Products
• Joint order do not include all products
• Ordering frequency may be different for each
product
– It is based on the product that has the highest
frequency
Total cost = $130,767
Lessons From Aggregation
• Complete aggregation is effective if product
specific fixed cost is a small fraction of joint fixed
cost
• Tailored aggregation is effective if product
specific fixed cost is a large fraction of joint fixed
cost
EOQ in Practice
• 1961 survey
– A majority cited “pure judgment” as the method for determining
inventory ordering
• 1973 survey
– 56% of the respondents were using EOQ
• 1978 survey
– 85% of the respondents were using EOQ
• 1983 survey
– 57% of the respondents were using EOQ
• 2007 textbook
– “The EOQ is extremely valuable, but it is rarely used in
practice because of the difficulty in implementing it and
capturing the requirement elements”
EOQ in Practice
• Criticism
– Difficult to accurately estimate holding and ordering
costs
– Demand is assumed to be constant
– Lead time is assumed to be zero or constant
– Order is assumed to arrive in one batch at one point in
time
– Costs are assumed stationary
– Quantity discounts are not possible (basic EOQ)
Measuring Demand Uncertainty
Inventory
Demand (D)
Order quantity/lot size (Q)
Reorder point (ROP)
Cycle
Inventory
Time
Lead time (L)
Cycle
Measuring Demand Uncertainty
Inventory
Demand (D)
Order quantity/lot size (Q)
Reorder point (ROP)
Cycle
Inventory
Average
Inventory
Safety
Inventory
Time
Lead time (L)
Cycle
What is Safety Inventory?
• Safety inventory
– Safety inventory is inventory carried for the purpose of
satisfying demand that exceeds the amount forecasted
for a given period
– Safety inventory is the average inventory remaining
when the replenishment lot arrives
Role of Safety Inventory
• There is a fundamental tradeoff
– Raising the level of safety inventory provides higher
levels of product availability and customer service
– Raising the level of safety inventory also raises the
level of average inventory and therefore increases
holding costs
Two Key Questions when Planning
Safety Inventory
1. What is the appropriate level of safety inventory
to carry?
2. What actions can be taken to improve product
availability while reducing safety inventory?
Determining Appropriate Level of Safety
Inventory
•
The appropriate level of safety inventory is
determined by the following three factors
1. The uncertainty of both demand and supply
– Higher levels of uncertainty require higher levels of safety
inventory
2. The desired level of product availability
– Higher levels of desired product availability require higher
levels of safety inventory
3. The replenishment policy
– Different replenishment policies lead to different levels of
safety inventory
Measuring Demand Uncertainty
Inventory
Demand (D)
Order quantity/lot size (Q)
Reorder point (ROP)
Cycle
Inventory
Average
Inventory
Safety
Inventory
Time
Lead time (L)
Cycle
Demand during lead time DL = LD
Standard deviation of demand over lead time L = (L)D
Measuring Product Availability
1. Cycle service level (CSL)
•
Fraction of replenishment cycles that end with all
customer demand met
2. Product fill rate (fr)
•
•
Fraction of demand that is satisfied from product in
inventory
Probability that product demand is supplied from
available inventory
3. Order fill rate
•
Fraction of orders that are filled from available
inventory
CSL and fr are different!
inventory
CSL is 0%, fill rate is almost 100%
0
inventory
0
time
CSL is 0%, fill rate is almost 0%
time
Replenishment Policies
• Continuous review
– Inventory is continuously
monitored and an order of size
Q is placed when the inventory
level reaches the reorder
point (ROP)
• Periodic review
– Inventory is checked at regular
(periodic) intervals and an order
is placed to raise the inventory
to a specified threshold, the
order-up-to level (OUL)
Safety Inventory
What actions can be taken to improve product
availability while reducing safety inventory?
Why is it that successful retailers and manufacturers
(i.e. Wal-Mart, Seven-Eleven Japan, Dell) carry only
little inventory but still have high levels of product
availability?
“RFID reduced Out-of-Stocks by
30 percent for products selling
between 0.1 and 15 units a day
at Wal-Mart”
Continuous Review Policy: Safety
Inventory and Cycle Service Level
L: Lead time for
replenishment
D: Average demand per unit
time
D:Standard deviation of
demand per period
DL: Mean demand during
lead time
L: Standard deviation of
demand during lead time
CSL: Cycle service level
ss: Safety inventory
ROP: Reorder point
D


DL
L
L
 L D
ss  F S (CSL)  L
1
ROP  D L  ss
CSL  F ( ROP , D L , L )
Average Inventory = Q/2 + ss