CTL.SC1x -Supply Chain & Logistics Fundamentals
Inventory Models for
Probabilistic Demand:
Cost & Service Trade-offs
MIT Center for
Transportation & Logistics
Items to Cover
•  Comparing Inventory Performance Metrics
•  Inputted versus Implied Metrics
•  Periodic Review Policies
•  Example Problem: ShopCo
•  Trading off Lead Time and Review Period
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Notation
D = Average Demand (units/time)
c = Variable (Purchase) Cost ($/unit)
h = Carrying or Holding Charge ($/
inventory $/time)
ct = Fixed Ordering Cost ($/order)
ce = c*h = Excess Holding Cost ($/unit/
time)
cs = Shortage Cost ($/unit/time)
Q = Replenishment Order Quantity
(units/order)
L = Replenishment Lead Time (time)
T = Order Cycle Time (time/order)
N = 1/T = Orders per Time (order/time)
IP = Inventory Position (units)
IOH = Inventory on Hand (units)
IOO = Inventory On Order (units)
CTL.SC1x - Supply Chain and Logistics Fundamentals
µDL , σDL = Expected and Standard
Deviation of Demand over Lead Time
(units)
µDL+R , σDL+R= Expected and Standard
Deviation of Demand over Lead Time
plus Review Period (units)
k = Safety Factor
s = Reorder point (units)
S = Order up to Point (units)
R = Review Period (time)
IFR = Item Fill Rate (%)
CSL = Cycle Service Level (%)
CSOE = Cost of Stock Out Event ($/event)
CSI = Cost per Item Short
E[US] = Expected Units Short (units)
G(k) = Unit Normal Loss Function
Lesson: Cost & Service Trade-offs
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Comparing Inventory Performance Metrics
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Inventory Performance Metrics
•  Establishes Safety Stock
• 
• 
s = µ DL + kσ DL
Finds k value
Expected Cost of Safety Stock = cekσDL
•  Service Based Metrics – set k to meet expected LOS
• 
Cycle Service Level (CSL)
CSL = P "# x ≤ k $%
•  Probability of not stocking out during cycle
• 
Item Fill Rate (IFR)
•  Expected percentage of demand met during each cycle
σ DLG[k]
IFR = 1−
Q
•  Cost Based Metrics – find k that minimizes total costs
• 
Cost per Stockout Event (CSOE)
•  Penalty of B1 if any stock out occurs
• 
Cost per Item Short (CIS)
•  Penalty of cs for each item short per cycle
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
& D)
!
#
!
#
E "CSOE $ = (B1 )P " x ≥ k $( +
'Q*
% D(
!
#
E "CIS $ = csσ DLG(k) ' *
&Q)
5
Performance Metrics v. Safety Stock Costs
100%
$325,000
95%
$300,000
Level of Service % (CSL or IFR)
85%
80%
75%
Demand ~N(62,000, 8,000)
ce = 15 $/unit-year
Q*= 5,200 units/order
L = 2 weeks
B1= 50,000 $/event
cs= 45 $/unit-year
µDL = 2,385 units
σDL = 1,569 units
70%
$275,000
$250,000
$225,000
$200,000
$175,000
$150,000
$125,000
65%
$100,000
$75,000
60%
Total Annual Shortage Cost (CSOE or CIS)
• 
• 
• 
• 
• 
• 
• 
• 
90%
$50,000
55%
$25,000
50%
$0
$$5,000 $10,000 $15,000 $20,000 $25,000 $30,000 $35,000 $40,000 $45,000 $50,000 $55,000 $60,000 $65,000 $70,000
Expected Annual Cost of Safety Stock
CSL
CTL.SC1x - Supply Chain and Logistics Fundamentals
IFR
TotCost_CSOE
Lesson: Cost & Service Trade-offs
TotCost_CIS
E !"Cost of SS#$ = ce kσ DL
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Lead Time v. Safety Stock Costs
Level of Service % (CSL)
100%
95%
90%
85%
80%
75%
CSL_LT=1 wk
70%
CSL_LT=2 wk
65%
CSL_LT=4 wk
60%
CSL_LT=8 wk
55%
50%
$-
$10,000
$20,000
$30,000
$40,000
$50,000
$60,000
$70,000
$80,000
$90,000
$100,000
Expected Annual Cost of Safety Stock
Level of Service % (IFR)
100%
98%
96%
IFR_LT=1 wk
94%
IFR_LT=2 wk
92%
IFR_LT=4 wk
90%
IFR_LT=8 wk
88%
$-
$10,000
$20,000
$30,000
$40,000
$50,000
$60,000
$70,000
$80,000
$90,000
$100,000
Expected Annual Cost of Safety Stock
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Inputted vs. Implied Metrics
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Safety Stock Logic
CSL
CSL = P "# x ≤ k $%
E !"Cost of SS#$ = ce kσ DL
Re-order
Point
s=µDL+kσDL
Safety
Stock
SS=kσDL
TRC(s,Q)
P[SO]=1-CSL
()
E !"US #$ = σ DLG k
P[SO]
Qc
P !" SO#$ = e
Dcs
CIS
(cs)
k
!
$
B1 D
&
k = 2ln #
# c Qσ
&
" e DL 2π %
CSOE
(B1)
G(k)
E[US]
G(k)=(1-IFR)Q/σDL
IFR
CIS
(cs)
Figure adapted from Dr. Jim Master’s ESD.260 Course Notes (2002)
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Periodic Review Policies
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Assumptions: Periodic Review Policies
•  Demand
n 
n 
n 
•  Discounts
Constant vs Variable
Known vs Random
Continuous vs Discrete
n 
n 
•  Excess Demand
•  Lead Time
n 
n 
n 
n 
n 
Instantaneous
Constant vs Variable
Deterministic vs Stochastic
Internally Replenished
•  Dependence of Items
n 
n 
n 
Independent
Correlated
Indentured
n 
n 
n 
Unlimited
Limited / Constrained
n 
None
Uniform with time
Non-linear with time
•  Planning Horizon
n 
n 
n 
Single Period
Finite Period
Infinite
•  Number of Items
n 
One vs Many
•  Form of Product
n 
n 
CTL.SC1x - Supply Chain and Logistics Fundamentals
None
All orders are backordered
Lost orders
Substitution
•  Perishability
n 
One vs Multi vs Multi-Echelon
•  Capacity / Resources
n 
n 
Continuous vs Periodic
•  Number of Locations
n 
n 
•  Review Time
n 
None
All Units vs Incremental vs One Time
Single Stage
Multi-Stage
Lesson: Cost & Service Trade-offs
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Periodic Review Policies
•  Order-Up-To-Level (R, S)
n 
n 
•  Hybrid (R, s, S) System
Policy: Order S-IP every
R time periods
Replenishment cycle system
n 
S
S
Inventory Position
Inventory Position
n 
L
L
R
Policy: Every R time periods,
Order S-IP if IP ≤ s,
if IP>s then do not order
General case for many policies
s
Time
R
L
Time
L
R
R
R
Notation
s = Reorder Point
Q = Order Quantity
S = Order-up-to Level
R = Review Period
IP = Inventory Position
= (IOH) + (Inventory On Order) – (Backorders)
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
L = Replenishment Lead Time
IOH= Inventory on Hand
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Periodic Review Policies (R, S)
Inventory Position
S
L
R
Time
L
R
Differences from Continuous Review Policy (s, Q)
•  How much to order?
•  How long should safety stock cover?
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
13
Periodic vs Continuous Review
•  Convenient transformation of (s, Q) to (R, S)
n 
n 
(s, Q)= Continuous, order Q when IP≤s
(R, S)= Periodic, order up to S every R time periods
•  Allows for the use of all previous (s, Q) decision rules
n 
n 
n 
s for continuous system becomes S for periodic system
Q for continuous system becomes D*R for periodic system
L for a continuous system becomes R+L for periodic system
•  Approach
n 
n 
n 
Make transformations
Solve for (s, Q) using transformations
Determine final policy, so that
S = xDL+R + kσDL+R
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
(s, Q)
(R, S)
s
ó
S
Q
ó
D*R
L
ó
R+L
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Example: ShopCo
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
Example: ShopCo
•  Background:
n 
n 
ShopCo is a North America based large store
format retailer of home improvement products
with >2,000 stores. Each ShopCo store generally operates independently:
ordering and receiving product directly from its suppliers.
One supplier (Hurricane Drills) sells a portfolio of electric drills that, on average,
cost ShopCo $75 each. Each store uses periodic review policies to order
directly from Hurricane and uses an annual holding charge of 15%. Assume 52
week year.
•  Problem:
n 
Find the (R, S) ordering policy for Hurricane drills for store #1301 given:
w  Forecasted annual demand of Hurricane drills is ~N(3,400, 400)
w  Lead Time is 1 week
w  Review Period is 4 weeks
w  Desired CSL = 95%
w  Hurricane has a minimum order quantity (MOQ) of 240 drills
w  Orders need to be in multiples of 12 drills to fit on pallets
n 
What is the expected annual cost of cycle and safety stock?
Case adapted from Anand, S. and Song, X. (2011) “Supply Chain Responsiveness for a Large Retailer,” MIT Supply Chain Management Program Thesis.
Image Source:http://commons.wikimedia.org/wiki/File:Hardware_Store.jpg
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Example: ShopCo
•  Finding Order Policy:
n 
n 
n 
n 
n 
n 
Find Q = D*R = (3,400 units/year)(4/52 years) = 261.5 ≈ 264 units (why?)
Find R+L = 4 weeks + 1 week = 5 weeks or 0.0962 years
so that, n= 52/5 = 10.4 “coverage” periods per year
Find µDL+R = (3,400)/(10.4) = 326.9 ≈ 327 units
Find σDL+R = (400)/(√10.4) = 124.03 ≈124 units
Find k where CSL = 0.95 or P[x≤k] = 0.95, k=1.644 = 1.64
Find S = µDL+R + kσDL+R = 327 + (1.64)(124) = 530.4 ≈ 530 units
Policy: Order up to 530 units every 4 weeks.
•  Finding Cost of Cycle & Safety Stock:
n 
n 
Cost of Cycle Stock = ce(DR/2) = (75)(0.15)(264/2) = $1,485 per year
Cost of Safety Stock = cekσDL+R = (75)(0.15)(1.64)(124) = $2,288 per year
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Trading Off Lead Time and Review
Period
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Example: ShopCo continued
•  New Mixing Center Strategy:
n 
n 
ShopCo has decided to deploy a fulfillment strategy
where each store orders from its Regional Distribution Center (RDC), instead
of directly to the supplier.
Each ShopCo RDC then consolidates orders from its dedicated stores and
places a combined order to the vendor. The vendor will then ship to each of
the RDCs where ShopCo “mixes” the products from multiple suppliers to
distribute a single combined load to each store.
Case adapted from Anand, S. and Song, X. (2011) “Supply Chain Responsiveness for a Large Retailer,” MIT Supply Chain Management Program Thesis.
Image Sources: http://commons.wikimedia.org/wiki/File:Hardware_Store.jpg and Anand & Song (2011)
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Example: ShopCo continued
•  Revised Problem with Mixing Centers:
n 
Find the (R, S) ordering policy for Hurricanes for store #1301 given:
w  Forecasted annual demand of Hurricane drills is ~N(3,400, 400)
w  Desired CSL = 95%
w  Lead Time is now 10 days (call this 1.5 weeks for simplicity)
w  Review Period is reduced to 2 weeks
w  ShopCo’s RDCs do not have a minimum order quantity (MOQ) to stores (why?)
w  Orders still need to be in multiples of 12 drills to fit on pallets (why?)
n 
What is the expected annual cost of cycle and safety stock?
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Example: ShopCo with Mixing Strategy
•  Finding Order Policy:
n 
n 
n 
n 
n 
n 
Find Q = D*R = (3,400 units/year)(2/52 years) = 130.8 ≈ 132 units (why?)
Find R+L = 2 weeks + 1.5 week = 3.5 weeks or 0.0673 years
so that, n= 52/3.5 = 14.86 “coverage” periods per year
Find µDL+R = (3,400)/(14.86) = 228.8 ≈ 229 units
Find σDL+R = (400)/(√14.86) = 103.76 ≈ 104 units
Find k where CSL = 0.95 or P[x≤k] = 0.95, k=1.644 = 1.64
Find S = µDL+R + kσDL+R = 229+ (1.64)(104) = 399.56 ≈ 400 units
Policy: Order up to 400 units every 2 weeks.
•  Finding Cost of Cycle & Safety Stock:
n 
n 
Cost of Cycle Stock = ce(DR/2) = (75)(0.15)(132/2) ≈ $743 per year
Cost of Safety Stock = cekσDL+R = (75)(0.15)(1.64)(104) ≈ $1,919 per year
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Example: ShopCo continued
Strategy
Lead
Time
(weeks)
Review
Period
(weeks)
Cycle
Stock
($/year)
Safety
Stock
($/year)
Avg. Inventory
Costs
($/year)
Direct-to-Store
1
4
1,485
2,288
3,773
Mixing Centers
1.5
2
743
1,919
2,662
Which is better? Which did the store managers prefer?
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Relationship Between L & R
•  Average Inventory Costs = ce[DR/2 + kσDL+R + LD]
•  Individual Impacts
n 
n 
Increasing Lead Time L
è Increases Safety Stock non-linearly
è Increases Pipeline Stock linearly
Increasing Review Period, R:
è Increases Safety Stock non-linearly
è Increases Cycle Stock linearly
•  Combined Impacts
n 
n 
Can be used to trade Replenishment speed (L) for frequency (R)
Determine which is the right mix
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
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Key Points from Lesson
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
24
Key Points
•  Inventory Performance Metrics
• 
• 
Service Based: IFR vs. CSL
Cost Based: CSOE vs. CIS
•  Inputted vs. Implied Metrics
• 
• 
Designing to one metric sets the others
Can backwards calculate implied values
•  Periodic Review (R, S)
• 
• 
• 
Very commonly used
Use the (s, Q) rules with simple transformations
Q è D*R, sèS, LèL+R
Changing L & R have different impacts on inventory
CTL.SC1x - Supply Chain and Logistics Fundamentals
Lesson: Cost & Service Trade-offs
25
CTL.SC1x -Supply Chain & Logistics Fundamentals
Questions, Comments, Suggestions?
Use the Discussion!
MIT Center for
Transportation & Logistics
[email protected]