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Transcript of Inventory Models for Probabilistic Demand · MIT Center for Transportation & Logistics CTL.SC1x...
MIT Center for Transportation & Logistics
CTL.SC1x -Supply Chain & Logistics Fundamentals
Inventory Models for Probabilistic Demand:
Basic Concepts
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
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)
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µDL = Expected Demand over Lead Time (units/time)
σDL = Standard Deviation of Demand over Lead Time (units/time)
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
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Inventory Replenishment Policies • Policy: How much to order and when
• Five Methods n EOQ Policy – deterministic demand
w Order Q* every T* time periods w Order Q* when IP=µDL
n Single Period Models – variable demand w Order Q* at start of period where P[x≤Q]=CR
n Base Stock Policy – one-for-one replenishment w Order what was demanded when it was demanded
n Continuous Review Policy (s,Q) - event based w Order Q* when IP≤s
n Periodic Review Policy (R,S) – time based w Order up to S units every R time periods.
Recall: Inventory Position (IP) = Inventory on Hand (IOH) + Inventory on Order (IOO) - Backorders Demand over Leadtime=D*L=µDL
(be careful with dimensions)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Quick Aside on Converting Times • What is the µ and σ of demand during the replenishment lead time?
n Lead time for replenishment is 7 days (1 week) n Annual demand is ~N(450,000, 22,000)
• What is the expected demand over lead time? n µDL= (450,000 units/year)(1 week) / (52 weeks/year) n = 8653.8 = 8,654 units
• What is the standard deviation of demand over lead time? n Recall that if we assume that each period (week) is identically and
independently distributed (iid) then; σ21+σ2
2+ . . . σ2n = nσ2
n So, in our problem, σ2year = 52(σ2
week) n Which means σweek=(σyear)/(√52) n σDL= (22,000 units/year) /(√52) = 3,050 units
• Demand over leadtime ~N(8,654, 3,050)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Quick Aside on Converting Times • Suppose we have two periods to consider:
n DS = Demand over short time period (e.g., week) n DL = Demand over long time period (e.g., year) n n = Number of short periods within a long (e.g., 52)
• Converting from Long to Short n E[DS] = E[DL]/n n VAR[DS] = VAR[DL]/n so that σS = σL/√n
• Converting from Short to Long n E[DL] = nE[DS] n VAR[DL] = nVAR[DS] so that σL = √n σS
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Base Stock Policy
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Assumptions: Base Stock Policy • Demand
n Constant vs Variable n Known vs Random n Continuous vs Discrete
• Lead Time n Instantaneous n Constant vs Variable n Deterministic vs Stochastic n Internally Replenished
• Dependence of Items n Independent n Correlated n Indentured
• Review Time n Continuous vs Periodic
• Number of Locations n One vs Multi vs Multi-Echelon
• Capacity / Resources n Unlimited n Limited / Constrained
• Discounts n None n All Units vs Incremental vs One Time
• Excess Demand n None n All orders are backordered n Lost orders n Substitution
• Perishability n None n Uniform with time n Non-linear with time
• Planning Horizon n Single Period n Finite Period n Infinite
• Number of Items n One vs Many
• Form of Product n Single Stage n Multi-Stage
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Base Stock Policy
• One-for-One Order Policy • IP stays constant at Base Stock • When is this used? • How do we set the Base Stock (S*)?
n Cover potential demand over lead time for desired level of service (LOS)
n LOS here is defined as the probability of not stocking out = P[µDL ≤S*]
Inventory
Demand
Order
Receive
Inventory Position (IP) Base Stock or S*
Inventory On Hand (IOH)
Leadtime
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Base Stock Policy
• How do I set the Level of Service (LOS)? n Management decision n Using Critical Ratio
• LOS is the probability that no stock outs will occur during the lead time replenishment period
• Base Stock, S*, is set to this amount:
S*= µDL + kLOSσ DL
LOS*= P µDL ≤ S *!" #$=CR =cs
cs + ce
Base Stock Expected Demand during lead time
Standard deviation of demand during lead time
Safety factor for given LOS
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Base Stock Policy Example • Set the Base Stock Policy for an item:
n Daily demand ~N(100, 15) n Lead time is 2 days n Excess cost is $5 per unit per day n Shortage cost is $25 per unit per day.
• Solution: n Find µDL=100(2) = 200 n Find σDL=15(√2)=21.2 n Find LOS = CR = (25)/(5+25)=0.833 n Find kLOS from Tables or Spreadsheet
w kLOS=NORMSINV(0.833) = 0.967
n Find S*= 200 + (0.967)(21.2) = 220.5 ≅ 221 units
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Continuous Review Policies
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Assumptions: Continuous Review Policies • Demand
n Constant vs Variable n Known vs Random n Continuous vs Discrete
• Lead Time n Instantaneous n Constant vs Variable n Deterministic vs Stochastic n Internally Replenished
• Dependence of Items n Independent n Correlated n Indentured
• Review Time n Continuous vs Periodic
• Number of Locations n One vs Multi vs Multi-Echelon
• Capacity / Resources n Unlimited n Limited / Constrained
• Discounts n None n All Units vs Incremental vs One Time
• Excess Demand n None n All orders are backordered n Lost orders n Substitution
• Perishability n None n Uniform with time n Non-linear with time
• Planning Horizon n Single Period n Finite Period n Infinite
• Number of Items n One vs Many
• Form of Product n Single Stage n Multi-Stage
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Continuous Review Policies • Order-Point, Order-Quantity (s,Q)
n Policy: Order Q if IP ≤ s n Two-bin system
Time
Inve
ntor
y Po
sitio
n
• Order-Point, Order-Up-To-Level (s, S) n Policy: Order (S-IP) if IP ≤ s n Min-Max system
s
s+Q
L L
s
S
L Time
Inve
ntor
y Po
sitio
n
L
Notation s = Reorder Point S = Order-up-to Level L = Replenishment Lead Time Q = Order Quantity R = Review Period IOH= Inventory on Hand IP = Inventory Position = (IOH) + (Inventory On Order) – (Backorders)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Order Point, Order Quantity Policy (s,Q)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Framework for (s, Q) System • Finding Q
n Determines the level of Cycle Stock n Usually from EOQ - but other methods maybe?
• Finding s n Based on expected demand over lead time (forecasted amount) n Added in safety or buffer stock for variability
Reorder Point Forecasted mean demand
over the lead time
Safety Stock
k = SS factor σDL = RMSE of forecast
s = µDL + kσDL
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
What cost and service objectives? 1. Common Safety Factors Approach
n Simple, widely used method n Apply a common metric to aggregated items
2. Customer Service Approach n Establish constraint on customer service n Definitions in practice are fuzzy n Minimize costs with respect to customer service
constraints 3. Cost Minimization Approach
n Requires costing of shortages n Find trade-off between relevant costs
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Service & Cost Metrics
• In this class we will focus on the following: n Performance Metrics
w Cycle Service Level (CSL) w Item Fill Rate (IFR)
n Stockout Cost Metrics w Cost per Stockout Event (CSOE) w Cost per Item Short (CIS)
• Other forms can be used – these are most common.
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TC = cD+ ctDQ!
"#
$
%&+ ce
Q2+ kσ DL
!
"#
$
%&+ csP[StockOutType]
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Cycle Service Level (CSL)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Cycle Service Level (CSL) n Probability of no stockouts per replenishment cycle
n Equal to one minus the probability of stocking out n X is the demand during lead time n = 1 – P[Stockout] = 1- P[X > s] = P[X ≤ s]
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.7%
0.8%
0.9%
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730
Demand
Pro
b (
x)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
0.0%
0.1%
0.2%
0.3%
0.4%
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0.8%
0.9%
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Demand
Pro
b (
x)
Finding P[Stockout]
Reorder Point (s)
Probability of stocking out during an order cycle
Probability of NOT stocking out during
an order cycle
Forecast Demand ~ N(µDL=500, σDL=50)
Forecasted Demand (µDL)
SS = kσDL
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
-
0.10
0.20
0.30
0.40
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0.80
0.90
1.00
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Demand
Pro
b (
x)
Cumulative Normal Distribution
Reorder Point
Probability of Stockout=1-CSL
Cycle Service Level
Forecast Demand ~ N(µDL=500, σDL=50)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Example: Finding (s,Q) Policy with CSL • Problem:
n You are managing the inventory for a production part with annual demand ~N(62,000, 8,000). The cost of the item, c, is $100 and the holding charge is 15% per year. You have determined that the economic order quantity, Q*, is 5,200 units. Lead time is 2 weeks.
n Assuming a CSL of 95%, find the appropriate (s, Q) policy.
• Solution n Find µDL = (62,000)/(26) = 2,384.6 = 2,385 units n Find σDL = (8,000)/(√26) = 1,568.9 = 1,569 units n Find k where CSL = 0.95 or P[x≤k] = 0.95, k=1.644 = 1.64 n Find s = µDL + kσDL = 2,385 + (1.64)(1,569) = 4,958 units
Policy: Order 5,200 when Inventory Position ≤ 4,958 units
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In Spreadsheets: n k=NORMSINV(CSL) n CSL=NORMSDIST(k)
In Standard Normal Tables: n CSL=P[x≤k]
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
k Factor versus Cycle Service Level
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
(3.00) (2.50) (2.00) (1.50) (1.00) (0.50) - 0.50 1.00 1.50 2.00 2.50 3.00
Cyc
le S
ervi
ce L
evel
k-factor
K-Factor versus CSL
23
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Cost Per Stockout Event (CSOE)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
What if I know CSOE? • Consider total costs
n Purchase Price - no change n Order Costs – no change from EOQ n Holding Costs – add in Safety Stock n StockOut Costs product of:
w Cost per stockout event (CSOE), B1 w Number of replenishment cycles w Probability of a stockout per cycle
TC = PurchaseCosts+OrderCosts+HoldingCosts+ StockOutCosts
TC = cD+ ctDQ!
"#
$
%&+ ce
Q2+ kσ DL
!
"#
$
%&+ (B1)
DQ
!
"#
$
%&P x ≥ k() *+
What costs are relevant? How do I solve for k?
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Finding k that minimizes TRC Take First Order Conditions wrt k . . .
Solving for k . . .
TRC k( ) = cekσ DL + (B1)DQ
!
"#
$
%&P x ≥ k() *+
k = 2lnB1D
ceσ DLQ 2π
!
"##
$
%&&
d(P[x ≥ k])dk
= − f x k( ) = −e−k2
2
2π
Recall that for Normal Distribution:
dTRC k( )dk
= ceσ DL + (B1)DQ!
"#
$
%&−e
−k2
2
2π
!
"
###
$
%
&&&= 0
Which gives us:
26
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Cost per Stockout Event (B1)
• Decision Rule for B1 Costs
n If then
n Otherwise, set k as low as management allows
• Questions n Why is the first condition there? n What k would management allow?
B1DceQσ DL 2π
>1 k = 2lnB1D
ceQσ DL 2π
!
"##
$
%&&
27
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Example: Finding (s,Q) Policy with CSOE • Problem:
n You are managing the inventory for a production part with annual demand ~N(62,000, 8,000). The cost of the item, c, is $100 and the holding charge is 15% per year. You have determined that the economic order quantity, Q*, is 5,200 units. Lead time is 2 weeks.
n Assuming CSOE=B1=$50,000 per event since it shuts the production line down, find the appropriate (s, Q) policy.
• Solution n Find µDL = (62,000)/(26) = 2,384.6 = 2,385 units n Find σDL = (8,000)/(√26) = 1,568.9 = 1,569 units n Check that the first condition is met n Solve for k
n Find s = µDL + kσDL = 2,385 + (2.15)(1,569) = 5,758 units
Policy: Order 5,200 when Inventory Position ≤ 5,758 units
28
B1DceQσ DL 2π
=50,000( ) 62,000( )
15( ) 5,200( ) 1,569( ) 2π=10.1>1
k = 2ln 10.1( ) = 2.15
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Item Fill Rate (IFR)
29
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Item Fill Rate (IFR) • Item Fill Rate
n Fraction of customer demand met routinely from IOH n This is equal to one minus the fraction we expect to be short
• Logic for Rule n We order Q each cycle n The fraction we are short = E[US]/Q n Therefore, item fill rate = 1 - E[US]/Q n Assuming ~Normal, E[US]=σDLG(k) n Calculate the desired G(k) n Find appropriate k
IFR =1− E[US]Q
IFR =1−σ DLG[k]Q
G[k]= Qσ DL
1− IFR( )
30
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Example: Finding (s,Q) Policy with IFR • Problem:
n You are managing the inventory for a production part with annual demand ~N(62,000, 8,000). The cost of the item, c, is $100 and the holding charge is 15% per year. You have determined that the economic order quantity, Q*, is 5,200 units. Lead time is 2 weeks.
n Assuming IFR = 99%, find the appropriate (s, Q) policy.
• Solution n Find µDL = (62,000)/(26) = 2,384.6 = 2,385 units n Find σDL = (8,000)/(√26) = 1,568.9 = 1,569 units n Solve for G(k) n Solve for k=1.45 from tables, n Find s = µDL + kσDL = 2,385 + (1.45)(1,569) = 4,660 units
Policy: Order 5,200 when Inventory Position ≤ 4,660 units
31
G[k]= Qσ DL
1− IFR( ) = 5,2001,5691− .99( ) = 0.0331
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Cost per Item Short (CIS)
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CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
What if I know cost per item short? • Consider total costs
n Purchase Price - no change n Order Costs – no change from EOQ n Holding Costs – add in Safety Stock n StockOut Costs product of:
w Cost per item stocked out (cs) w Estimated number of units short w Number of replenishment cycles
TC = PurchaseCosts+OrderCosts+HoldingCosts+ StockOutCosts
TC = cD+ ctDQ!
"#
$
%&+ ce
Q2+ kσ DL
!
"#
$
%&+ csσ DLGu (k)
DQ!
"#
$
%&
• What costs are relevant? • How do I solve for k?
33
P StockOut!" #$= P x ≥ k!" #$=QceDcs
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Cost per Item Short (cs)
• Decision Rule for cs Costs
n If Then
n Otherwise, set k as low as management allows
• Questions n Why is the first condition there? n What k would management allow?
QceDcs
≤1 P StockOut!" #$= P x ≥ k!" #$=QceDcs
34
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Example: Finding (s,Q) Policy with CIS • Problem:
n You are managing the inventory for a production part with annual demand ~N(62,000, 8,000). The cost of the item, c, is $100 and the holding charge is 15% per year. You have determined that the economic order quantity, Q*, is 5,200 units. Lead time is 2 weeks.
n Assuming CIS = cs = 45 $/unit-year, find the appropriate (s, Q) policy.
• Solution n Find µDL = (62,000)/(26) = 2,384.6 = 2,385 units n Find σDL = (8,000)/(√26) = 1,568.9 = 1,569 units n Check decision rule:
n Find k where:
n Find s = µDL + kσDL = 2,385 + (1.91)(1,569) = 5,382 units
Policy: Order 5,200 when Inventory Position ≤ 5,382 units
35
QceDcs
=(5,200)(15)(62,000)(45)
= 0.02795≤1
P x ≥ k!" #$=1− P x ≤ k!" #$= 0.02795 P x ≤ k"# $%= 0.97205 k =1.91
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Key Points from Lesson
36
CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Continuous Review Inventory Policies
Key Points
• Base Stock Policy • Order one for one with initial quantity of S*
• Continuous Review Policy (s,Q) • Order Q when IP ≤ s
• Safety Stock set by service or cost metrics • Cycle Service Level • Item Fill Rate • Cost per Stockout Event • Cost per Item Short
• Safety stock only buffers for demand over lead time
S*= µDL + kLOSσ DL
s = µDL + kσ DL
37
Metric Value k SS
CSL 95% 1.64 $2,573
IFR 99% 1.45 $2,275
CSOE $50,000 2.15 $3,373
CIS $45 1.91 $2,997
MIT Center for Transportation & Logistics
CTL.SC1x -Supply Chain & Logistics Fundamentals
Questions, Comments, Suggestions? Use the Discussion!