Presented by: Y. LEVENT KOÇAĞA

1. A Model for Assessing the Value of Warehouse Risk Pooling: Risk Pooling Over Outside-Supplier Leadtimes Presented by: Y. LEVENT KOÇAĞA

2. THE MODEL • A multi-echelon inventory model • A high service level system • Highlights warehouse risk-pooling • Two alternative configurations

3. Alternative system configurations Outside supplier Outside supplier Leadtime = (LS+LTW ) Warehouse Leadtime = (Ls +Ltr ) Leadtime = (Ltr+LPW ) 1 2 3 1 2 3 Retailers Retailers System1: Direct shipment to Retailers System2: Shipment through Warehouse

4. 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

5. 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

6. Key differences • Time of order allocation • Additional lead time (Ltw + Lpw) in system2 • Pipeline inventory in system 2

7. Comparison of two systems System 1 100 100 30 40 40 30 H H -Ls-Ltr 0 -Ls-Ltr 0 System 2 100 100 30 40 33 37 54 50 -Lpw-Ltr -Lpw-Ltr H H -Ls-Ltw-Lpw-Ltr 0 0 -Ls-Ltw-Lpw-Ltr

8. 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

9. 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

10. 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 +Ltr )

11. Analysis of system 1 • E(Ikl)= S01/N –(k+ Ls +Ltr) μ , k=1,…H • Var(Ikl)= (k+ Ls +Ltr) σ2 , k=1,…H

12. 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=Nt=-(Lpw + Ltr+1)t=k Ikl = {S02 - ∑ ∑ }/N - ∑ Dt , k=1,…,H j=1t=-(Ls +Ltw + Lpw + Ltr ) t= (Lpw + Ltr)

13. 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

14. 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

15. 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

16. 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

17. 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)

18. Inventory cost breakeven leadtimes • Given an inventoryholding 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:

19. Inventory cost breakeven leadtimes • Equating the holding costs gives

20. 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

21. 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

22. 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

23. Holding cost breakeven Lpw values

24. Holding cost breakeven Lpw values

25. Case2: Ltw and Ltr both positive

26. 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

27. Case3: (Ltw,Lpw)-Lines

28. Case3: (Ltw,Lpw)-Lines

29. 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

30. Conclusions • Breakeven values decrease • as N decreases, • as Ltr increases, • as σ/μ decreases, • as H increases.

31. 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

32. Possible extensions • Goods “enter” each system • More complex cost structure • Generalizing transpotation/receiving leadtime • Different H values for different systems

33. Q & A