1. Page 1 Stephen C. Graves Copyright 2010 All Rights Reserved Strategic Safety Stocks in Supply Chains: Update on Recent Work Stephen C. Graves MIT, E53-347 sgraves@mit.edu http://web.mit.edu/sgraves/www/ Joint work with Sean Willems, Boston University, Katerina Lesnaia, Oracle, Tor Schoenmeyr, FirstSolar

2. Page 2 Stephen C. Graves Copyright 2010 All Rights Reserved Overview Motivation and assumptions for SIP model Prior work – review of base model & example (joint with Willems) Recent work – extend to account for capacity (joint with Schoenmeyr) Recent work – extend to include evolving forecasts (joint with Schoenmeyr) Summary Papers available on request!

3. Page 3 Stephen C. Graves Copyright 2010 All Rights Reserved Strategic Safety Stock Model: Intent Tactical model to determine the amount and positioning of safety stocks in supply chains Tactical model to support supply chain improvement teams Simple model, easily accessible, runs on PC, understandable inputs/outputs; academic version available from www.sipmodel.comwww.sipmodel.com Commercialized by Optiant; applications support both tactical and operational decisions

4. Page 4 Stephen C. Graves Copyright 2010 All Rights Reserved Assumptions Supply chain modeled by an acyclic graph Deterministic processing time for each stage No capacity constraints Deterministic yield Periodic review, base stock control for each stage; common review period and no lot sizing

5. Page 5 Stephen C. Graves Copyright 2010 All Rights Reserved Assumptions Fixed service time between stages where service time is the decision variable Each stage quotes same service time to all adjacent downstream stages Stationary, bounded demand process for each end item Each stage provides 100% service: “Guaranteed service model”

6. Page 6 Stephen C. Graves Copyright 2010 All Rights Reserved Stage k Inventory Processing Orders d(t) Service time At time node k must deliver d(t) to the downstream node from its inventory At time, d(t) units are delivered as raw material from node k + 1, and at time the d(t) units are ready as inventory at node k Service timeProcessing time Stage k must have a base stock level equal to max demand over the net replenishment time Review of guaranteed-service base-stock problem

7. Page 7 Stephen C. Graves Copyright 2010 All Rights Reserved Base stock mechanics B is base stock level.I (t) is inventory at end of time t. Demand arrives: d0d0 …d t-SI-T-1 d t-SI-T d t-S dtdt …… shipped received I d0d0 …d t-SI-T received d0d0 …d t-S shipped

8. Page 8 Stephen C. Graves Copyright 2010 All Rights Reserved Key results for guaranteed service, bounded demand

9. Page 9 Stephen C. Graves Copyright 2010 All Rights Reserved Stage k InventoryProcessing Stage k+1 InventoryProcessingInventoryProcessing Stage k-1 Safety stock If we view the service times as decision variables we get a global optimization problem: Review of guaranteed-service base-stock problem

10. Page 10 Stephen C. Graves Copyright 2010 All Rights Reserved Simpson (1958): Solve serial system through enumeration. “All-or-nothing” property of optimal solution (i.e., either ) Graves and Willems (2000): Solve spanning tree system through polynomial-time dynamic programming (Lesnaia, 2004). Fast enough for large, real-life applications. Review of guaranteed-service base-stock problem

11. Page 11 Stephen C. Graves Copyright 2010 All Rights Reserved Algorithmic Results For serial systems, Simpson (1958) showed the all or nothing property for solution Graves and Willems (2000) developed a pseudo- polynomial DP for spanning trees; also Graves (1988), Inderfurth (1991) and Inderfurth and Minner (1998) Lesnaia (2004) provides polynomial DP for spanning tree and specialized algorithm for any two layer network General network is NP hard (Lesnaia, 2004); optimum occurs at an extreme point for concave bound function Several exact and heuristic algorithms for general networks: Humair and Willems (2006, 2008); Lesnaia (2004); Minner (2000); Magnanti et al. (2006)

12. Page 12 Stephen C. Graves Copyright 2010 All Rights Reserved KIMES 100

13. Page 13 Stephen C. Graves Copyright 2010 All Rights Reserved Supply Chain: Before

14. Page 14 Stephen C. Graves Copyright 2010 All Rights Reserved Supply Chain: Lead Times

15. Page 15 Stephen C. Graves Copyright 2010 All Rights Reserved Supply Chain: Costs

16. Page 16 Stephen C. Graves Copyright 2010 All Rights Reserved Supply Chain: Optimized

17. Page 17 Stephen C. Graves Copyright 2010 All Rights Reserved Supply Chain: Implemented

18. Page 18 Stephen C. Graves Copyright 2010 All Rights Reserved Supply-Chain

19. Page 19 Stephen C. Graves Copyright 2010 All Rights Reserved KIMES 100 Project results –Sizing finished goods inventory –Assess where to target lead-time reduction efforts –Framework to work with suppliers on purchasing long lead-time parts

20. Page 20 Stephen C. Graves Copyright 2010 All Rights Reserved Key Benefits & Learning Shows value from “holistic” perspective Formalizes inventory-related supply chain costs, and provides an optimal benchmark Provides framework and standard terminology for cross- functional debate Shows the effectiveness of inventory, strategically positioned in a few places to de-couple the supply chain De-couple supply chain prior to a high-cost added stage; and prior to product explosion Most leverage from lead time reduction

21. Page 21 Stephen C. Graves Copyright 2010 All Rights Reserved Battery Supply Chain

22. Page 22 Stephen C. Graves Copyright 2010 All Rights Reserved HP Supplies Inventory Modeling Project RegionsFactoriesSuppliers Customers Answer the bulk pen inventory question… … in the context of what is best for the system

23. IPG Builds Roughly 1000 Network Models per Year standard training class and tier based support

24. Page 24 Stephen C. Graves Copyright 2010 All Rights Reserved Key Limitations, circa 2000 Stationary demand assumptions No capacity constraints DP algorithm for spanning tree only Deterministic lead times Common review period Common service time to all downstream customers

25. Page 25 Stephen C. Graves Copyright 2010 All Rights Reserved Strategic safety stocks in supply chains with evolving forecasts Same assumptions as for base case, but now there is an evolving forecast for the end item demand Guaranteed service model – each stage commits to a guaranteed service but now for a bound on forecast errors Each stage uses a forecast-based ordering policy, rather than a base stock policy

26. Page 26 Stephen C. Graves Copyright 2010 All Rights Reserved Forecast evolution model Graves et al. (1986), Heath and Jackson (1994) is our forecast, in period t, for demand in period t + i Each period the forecasts are revised Assumptions (“rational forecasts”): (A1) are i.i.d. R.V. (A2) Unlike previous authors, we make no assumptions about We show an equivalence with this model and general, state-space models of demand (e.g., ARIMA) The forecast is initialized as at the horizon forecast revision Current forecast is demand

27. Page 27 Stephen C. Graves Copyright 2010 All Rights Reserved Forecast-based order policy: For zero service times, this corresponds to orders in a simple (no lot sizing, etc) MRP system. Both the forecast evolution model, and similar order mechanisms have been considered before; new contribution is to consider non-zero service times in a global optimization problem cumulative lead time Order placed by stage k For zero service times, the forecast-based orders have some local optimality properties (Aviv, 2003)

28. Page 28 Stephen C. Graves Copyright 2010 All Rights Reserved We can use the equations for the evolving forecast and the order policy to derive the inventory Safety stock constant Forecast revisions If we can find a bound on the sum: and set the safety stock level to then the stage can guarantee service; i.e.,

29. Page 29 Stephen C. Graves Copyright 2010 All Rights Reserved For the cumulative forecast error: Define can be calculated from historical data on demand and forecasts we have a valid (probabilistic) bound. Loosely speaking, the stages will provide guaranteed service as long as the cumulative forecast errors are smaller than We find that which we propose as a bound. By setting the safety stock level How might we set the bound? D is demand RV; σ() is now a function too!

30. Page 30 Stephen C. Graves Copyright 2010 All Rights Reserved j Optimization problem: how do we find the least cost safety stock configuration that maintains guaranteed service for any forecast/demand realization within the bounds

31. Page 31 Stephen C. Graves Copyright 2010 All Rights Reserved Forecast problemBase stock problem The problem is very similar to the base stock problem solved by Simpson (1958), and extended by Graves and Willems (2000) and others. Under some mild assumptions about the forecasts, we show that the all- or-nothing property holds We can use existing, effective algorithms to find optimal service times, after modifying the bound function

32. Page 32 Stephen C. Graves Copyright 2010 All Rights Reserved Managerial insight nugget Assuming forecast revisions at time t are independent and with proposed bound on forecast errors:

33. Page 33 Stephen C. Graves Copyright 2010 All Rights Reserved Electronic test system manufactured by Teradyne, Inc. 3,866 part/locations Used real data on supply chain topology, lead times, costs of parts

34. Page 34 Stephen C. Graves Copyright 2010 All Rights Reserved Schedule contained booked and “preliminary” orders, and got increasingly locked down as the date of delivery approach The schedule was effectively a forecast, and we used data on past schedule changes to calculate F(L) As a forecast of actual demand, it was fairly accurate in the short term but useless >10 weeks out

35. Page 35 Stephen C. Graves Copyright 2010 All Rights Reserved 25.5% improvement Total cost ? ?? ? Difficult to compare with current situation because no consistent optimization procedure/ service level used ? In the forecasted case, most savings were far downstream, where forecasts were accurate Optimization time ~1 minute on a laptop computer Schoenmeyr thesis discusses generalizations for multi-product networks

36. Page 36 Stephen C. Graves Copyright 2010 All Rights Reserved Base stock policy (Graves-Willems) 5 week forecast 10 week forecast (current) 20 week forecast

37. Page 37 Stephen C. Graves Copyright 2010 All Rights Reserved We have shown how to map the optimization method used for base stock systems, so that it can be used for forecast-driven (push) systems This approach enables optimization of large system with Evolving schedule in make-to-order context Evolving demand forecast in make-to-stock context Benefit relative to base stock case depends on forecast quality; in one case study it was ~25% Summary of results

38. Page 38 Stephen C. Graves Copyright 2010 All Rights Reserved Strategic safety stocks in supply chains with capacity constraints Same assumptions as before, but now there can be a capacity constraint at each stage Guaranteed service model – each stage commits to a guaranteed service for bounded demand Deterministic production lead time T Each stage follows a base stock policy, subject to capacity constraint

39. Page 39 Stephen C. Graves Copyright 2010 All Rights Reserved Open problem: what if there are capacity constraints? Now may not be enough safety stock, because any units that get “stuck” will be delayed. Q: How much extra inventory do we need? Q: How do we optimize a supply chain with one or more capacity constraints? Q: Do the structural results from before (“all or nothing”) hold up? Stage k Inventory Processing Orders d(t) Service time Processing time “stuck” units

40. Page 40 Stephen C. Graves Copyright 2010 All Rights Reserved Q: How much extra inventory do we need? A: The base stock level for the node with capacity constraint can be calculated with a functional transformation of the unconstrained base stock level. Original base stock level/order bound: Base stock with capacity constraint: In general

41. Page 41 Stephen C. Graves Copyright 2010 All Rights Reserved Q: How much extra inventory do we need? A: The base stock level for the node with capacity constraint can be calculated with a functional transformation of the unconstrained base stock level. Original base stock level/order bound: Common example Base stock with capacity constraint: The base stock grows hyperbolically as we decrease capacity. On the other hand, if the capacity constraint is large enough it becomes irrelevant.

42. Page 42 Stephen C. Graves Copyright 2010 All Rights Reserved For sufficiently large net replenishment times the capacity constraint does not matter A stage with a capacity constraint needs safety stock even at zero net replenishment time With a capacity constraint, we permit negative net replenishment time

43. Page 43 Stephen C. Graves Copyright 2010 All Rights Reserved Q: How do we optimize a supply chain with one or more capacity constraints? A: We have shown how to calculate new base stock levels for a single stage. Other stages are not affected (orders placed/delivered as before). Hence after transforming affected bounds, we can use existing optimization procedures. A: The functional transformation preserves concavity, and hence the “all or nothing” property holds. Q: Do the structural results from before (“all or nothing”) hold up?

44. Page 44 Stephen C. Graves Copyright 2010 All Rights Reserved Stage k Inventory Processing Orders Service time Processing time “stuck” units But can we do better? “Why should we ever order units if we cannot process them when they arrive?” Place the censored order Censored orders where we keep a backlog of delayed orders

45. Page 45 Stephen C. Graves Copyright 2010 All Rights Reserved Q: But how much inventory do we need/ how do we optimize supply chain? A: We find that base stock transformation remains the same but we need another functional transformation to obtain a new bound for orders (demand) placed by a censoring node Q: Do the structural results from before (“all or nothing”) hold up? A: Yes. (Φ also preserves concavity) Order bound Base stock

46. Page 46 Stephen C. Graves Copyright 2010 All Rights Reserved Average Inventory 87654321 Total cost No capacity constraint 00017.9000 2,377 Capacity constraint, no censorship 0015.500 012.62,433 Capacity constraint, censorship 555556.5012.62,233 Serial system with 8 nodes and capacity constraint at node 3. Assumed processing time 5 at each node; holding costs increase with 40% per stage. Censorship reduces cost impact of constraint Censorship cost is sometimes even lower than uncapacitated problem! “Paradox”: Under censorship, add constraint → better solution Explanation 1: Censorship smoothes demand and reduces safety stocks upstream Explanation 2: The (uncensored) local base stock policy is not optimal in a multi- stage system with guaranteed service It may be of interest to censor even in the absence of actual capacity constraints

47. Page 47 Stephen C. Graves Copyright 2010 All Rights Reserved Summary of results for capacity constraints We can generalize the base stock model to incorporate capacity constraints. For serial systems, we find exact analytical transformations, under which existing algorithms can be used with small modifications Known structural results (“all-or-nothing”) hold. These results also hold if we censor orders with the capacity. The necessary safety stocks are reduced. Censored orders sometimes lead to costs that are even lower than for the same problem without capacity constraints (in many examples 30-40% reductions by censoring the right amount at the right location) Development is for serial systems, and extends immediately to assembly structures; more general networks require a calculus to combine bounds

48. Page 48 Stephen C. Graves Copyright 2010 All Rights Reserved Overall Summary Motivation, assumptions and review of guaranteed service supply chain model Extension for capacity –Requires transformation of base stock and of demand bound –Structural results and algorithms extend directly –Capacity constraint can lead to lower cost solution –Multi-item supply chains requires more work Extensions for evolving forecast –Requires forecast-based ordering and bound on forecast errors –Structural results and algorithms extend directly –Incorporating forecast can lead to lower costs –Multi-item supply chains requires more work

49. Page 49 Stephen C. Graves Copyright 2010 All Rights Reserved

50. Page 50 Stephen C. Graves Copyright 2010 All Rights Reserved We then set the safety stock level When we have no forecast to get the base-stock model !