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A Model for Assessing the Value of Warehouse Risk Pooling: Risk Pooling Over Outside-Supplier Leadtimes. Presented by: Y. LEVENT KOÇAĞA. THE MODEL. A multi-echelon inventory model A high service level system Highlights warehouse risk-pooling Two alternative configurations. - PowerPoint PPT Presentation
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A Model for Assessing the Value of Warehouse Risk Pooling: Risk Pooling Over Outside-Supplier Leadtimes
Presented by:
Y. LEVENT KOÇAĞA
THE MODEL
A multi-echelon inventory model A high service level system Highlights warehouse risk-pooling Two alternative configurations
Alternative system configurations
1 2 3
Leadtime = (Ls + Ltr )
Retailers
System1: Direct shipment to Retailers
Warehouse
Outsidesupplier
Outsidesupplier
1 2 3
Retailers
System2: Shipment through Warehouse
Leadtime = (Ltr+LPW )
Leadtime = (LS+LTW )
Assumptions
Retailers supply a normal identical demand Periodic-review demand replenishment Fixed lead times IT system to track inventory(at order time) No interchange of goods between retailers
Assumptions( specific to system2)
Warehouse does not hold inventory Arriving orders are allocated at warehouse Allocation only at the order receipt Equalization of retailers` inventory Cost of allocation avoided
Key differences
Time of order allocation Additional lead time (Ltw + Lpw) in system 2
Pipeline inventory in system 2
-Ls-Ltr
100
30
30
100
40
40
H0 -Ls-LtrH0
-Ls-Ltw-Lpw-Ltr
30
37
40
33
H0 H0-Ls-Ltw-Lpw-Ltr
-Lpw-Ltr -Lpw-Ltr
50 54
100100
Comparison of two systems
System 2
System 1
Scope of the Study
Derive expressions for means and variances Formulate the performance measures Analysis to find breakeven lead times Sensitivity analysis Conclusions and managerial insights Further extensions
Two alternative systems
N identical retailers Identical demand is N~(μ, σ) Drawings are independent(iid) Review period is H (system cycle) Order up to S0i every H periods, i=1,2
Analysis of system 1
System order up to level is S01
Order placed (Ls + Ltr ) periods before period 1
Then retailer end-of-period k net inventory is: t=k
Ikl = S01/N -∑Dt , k=1,…,H
t=-(Ls
+ L
tr )
Analysis of system 1
E(Ikl) = S01/N –(k+ Ls + Ltr) μ , k=1,…H
Var(Ikl) = (k+ Ls + Ltr) σ2 , k=1,…H
Analysis of system 2
System order up to level is S02
Order placed at (Ls + Ltw + Lpw + Ltr ) periods before
period 1 Then retailer end-of-period k net inventory is:
j=N t=-( Lpw + Ltr+1) t=k
Ikl = {S02 - ∑ ∑ }/N - ∑ Dt , k=1,…,H
j=1 t=-(Ls
+ L
tw + L
pw + L
tr ) t= (L
pw + L
tr)
Analysis of system 2
E(Ik2) =S02/N –(k+ Ls + Ltw + Lpw + Ltr )μ, k=1,…H
Var(Ik2) = [k+ Ls/N+(Ltw/N+Lpw) + Ltr] σ2, k=1,…H
Service level measures
Retailer expected end-of-period backorders is :
EUBki = √var(Iki) . G[ E(Iki) / √Var(Iki)] , k=1,...H
P = EUBki /(Hμ) Observe that P = 1 – fill rate
Risk pooling: incentive quantified
Warehouse serves to pool risk over outside-supplier leadtime
The incentive is reduced overall variance of inventory process
RPI = Var(Ik1) - Var(Ik2)
RPI = [(N-1) Ls – Ltw – NLpw)] σ2 / N
SS Breakeven Leadtimes
How large can (Ltw ,Lpw) be given that Retailers have the same safety stock Both systems provide the same service level
This yields:
Ltw + N.Lpw = (N-1).Ls
Inventory cost breakeven leadtimes
System 2 incurs pipeline stock due to its internal lead time (Lpw + Ltr)
Change the question to address this issue:
How large can (Ltw ,Lpw) be given that Both systems provide the same service level The same safety holding cost ( plus pipeline holding
cost for System 2)
Inventory cost breakeven leadtimes
Given an inventory holding cost h, the safety stock holding cost for system1 per cycle is:
Whereas the safety stock plus pipeline inventory holding cost for system 1 is:
Inventory cost breakeven leadtimes
Equating the holding costs gives
Inventory cost breakeven leadtimes
Average cycle inventories ignored Safety stocks approximated by end-of-period
expected on-hand inventory System 2 retailer stock is set to system 1
retailer stock less the retailer pipeline inventory Determination of breakeven points trades the
reduction in variance against this reduction
Computational studies
Holding cost breakeven (Ltw ,Lpw) lead times for representative sets
Ls used as a scale parameter to assess the breakevens
H is set to 1
Case1: Ltw and Ltr both set to zero
If transportation / receiving times are set to zero
RPI = [(N-1) Ls – NLpw)] σ2 / N
system 2’s pipeline inv. hldng. cost is LpwNμHh
Holding cost breakeven Lpw values
Holding cost breakeven Lpw values
Case2: Ltw and Ltr both positive
Case3: (Ltw,Lpw)-Lines
Lpw incurs pipeline inventory holding cost of
LpwNμHh per system cycle
System does not pool risk over Lpw
Therefore holding cost breakeven Lpw’s will be smaller than h. Cost breakeven Ltw’s
As Ltr increases both should decrease
Case3: (Ltw,Lpw)-Lines
Case3: (Ltw,Lpw)-Lines
Conclusions
Overall value of using System to pool risk critically depends on System 2’s pipeline stock
Holding cost breakeven (Lpw,Ltw) values:
Very small values of Lpw but larger for Ltw
Conclusions
Breakeven values decrease as N decreases, as Ltr increases, as σ/μ decreases, as H increases.
Managerial Interpretations
If System 2 is to outperform System 1 Lpw must be quite small compared to Ls
Ltw may be considerably larger than Ls
Limited to high service level systems due to the allocation assumption
Possible extensions
Goods “enter” each system More complex cost structure Generalizing transpotation/receiving leadtime Different H values for different systems
Q & A