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


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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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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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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Base Stock Policy

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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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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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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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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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Continuous Review Policies

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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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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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Order Point, Order Quantity Policy (s,Q)

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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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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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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]


Cycle Service Level (CSL)

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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%

250

280

310

340

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490

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670

700

730

Demand

Pro

b (

x)

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0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

250

280

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730

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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-

0.10

0.20

0.30

0.40

0.50

0.60

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0.90

1.00

250

280

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520

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580

610

640

670

700

730

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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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]


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

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Cost Per Stockout Event (CSOE)

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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?

25


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:

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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

!

"##

$

%&&

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Example: Finding (s,Q) Policy with CSOE •  Problem:


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


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B1DceQσ DL 2π

=50,000( ) 62,000( )

15( ) 5,200( ) 1,569( ) 2π=10.1>1

k = 2ln 10.1( ) = 2.15


Item Fill Rate (IFR)

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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( )

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Example: Finding (s,Q) Policy with IFR •  Problem:


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


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G[k]= Qσ DL

1− IFR( ) = 5,2001,5691− .99( ) = 0.0331


Cost per Item Short (CIS)

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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?

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P StockOut!" #$= P x ≥ k!" #$=QceDcs


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

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Example: Finding (s,Q) Policy with CIS •  Problem:


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


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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


Key Points from Lesson

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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

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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

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inventory models for probabilistic demand · mit center for transportation & logistics ctl.sc1x...

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