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