Control of Production-Inventory Systems with Multiple Echelons

1. Control of Production-Inventory Systems with Multiple Echelons

2. Characteristics • Demand is recurrent and stationary (in distribution) over time • Demand occurs continuously over time with stochastic inter-arrival times between consecutive orders • The production and inventory systems are tightly linked • The production system has a finite capacity with stochastic production times • Inventory replenishment leadtimes are load-dependent • Inventory is reviewed continuously

3. Example 1: A Single Stage Production-Inventory System Customer demand Raw material Work-in-process Production system Finished goods inventory

4. Example 2: A Series System Customer demand Stage N-1 Stage N Stage 1

5. Example 3: An Assembly System Customer demand External supply

6. The State of the System • The state of the system is described by the amount of finished-goods inventory (FGI) and work-in-process (WIP) at every stage. • The state of the system changes with either the arrival of an order or the completion of production at one of the stages.

7. Costs, Decisions, and Objectives • Example costs: • inventory holding cost at every stage • backorder cost at stage N • Decisions (actions): Given the current state of the system, which of the production stages should be producing. • Example objectives: • Expected total cost (sum of inventory holding and backorder costs) • Inventory holding cost subject to a service level constraint

8. The Optimal Production Policy • Decisions at any stage affect all other stages. • The optimal decision at any stage must take into account the current state of the entire system. • Solutions that decompose the problem into problems involving single stages can lead to bad decisions. • Coordination among the stages is important.

9. Challenges • The optimal policy is difficult to characterize in general and the optimal cost difficult to compute. • In some cases, the problem can be formulated as a stochastic optimal control and solved using dynamic programming. • For multi-dimensional problems (several stages, several products, and complex routing structures), the problem becomes computationally intractable.

10. Heuristic (but Common) Policies • Make-to-order (MTO) systems • Make-to-stock (MTS) system with only FGI inventory • MTS systems with inventories at every stage • MTS/MTO systems with inventory at only stage • MTS systems with limits on WIP (pull systems such as Kanban, extended Kanban, and CONWIP)

11. MTO Systems Customer demand Stage N-1 Stage N Stage 1

12. MTO Systems Appropriate when • WIP and FGI holding costs are high • backorder costs are low (customers tolerate delays) • production capacity is uniformly high • product variety is high with little commonalities among products

13. MTO Systems with Limits on WIP • Limits on total WIP • Limits on WIP at individual stages (or groups of stages) Total WIP K WIPN-1kN-1 WIPNkN WIP1k1

14. MTO/MTS Systems Customer demand Stage 5 Stage 1 Stage 2 Stage 3 Stage 4 Make-to-stock segment Make-to-order segment

15. MTO/MTS Systems (Continued…) Appropriate when • capacity is tight upstream in the production process • there is an identifiable bottleneck • holding costs are high downstream in the production process • customers tolerate some amount of delay • there are multiple products with common components or processes (e.g., MTO/MTS systems enable delayed differentiation)

16. Base-Stock Systems Customer demand s1 sN-1 sN Demand signal

17. Base-Stock Systems Each stage manages an output buffer according to a base-stock policy with base-stock level si at stage i (each stage keeps a constant inventory position IPi= si = Ii + IOi – Bi). Production at each stage occurs only in response to external demand (or equivalently demand from a downstream stage). If demand at any stage cannot be satisfied from on-hand inventory, it is backordered. Base-stock levels at each stage can be optimized to reflect the corresponding holding costs and production capacity.

18. Advantages of Base-Stock Systems Production is driven by actual consumption of finished goods. Backlogging at every stage reduces the likelihood that the bottleneck is starved for parts allows the bottleneck to occasionally work ahead of downstream stages (the bottleneck is never blocked) maximizes utilization of production resources by eliminating blocking and starvation

19. Disadvantages of Base-Stock Systems Backlogging at every stage could lead to excessive work-in-process (WIP). Every stage responds to consumption of finished goods instead of consumption of its output by the immediate downstream stages. Production stages are decoupled, making it more difficult to uncover sources of inefficiency in the system.

20. Reorder Point/Order Quantity Systems Each stage manages an output buffer according to a (Q, r) policy with parameters ri and Qi at stage i. By placing orders in batches setup costs and setup times are reduced. Similar advantages and disadvantages to base-stock policy.

21. Kanban Systems • A “kanban” is a sign-board or card in Japanese and is the name of the flow control system developed by Toyota.

22. Kanban Systems (Continued…) Similar to a base-stock system, except that backlogged demand does not trigger a replenishment order. The maximum amount of inventory on order (WIP) at every stage is limited to the maximum output buffer size at that stage. Total WIP in the system is capped.

23. Implementation • One card systems • Two card systems

24. One-Card Kanban Outbound stockpoint Outbound stockpoint Completed parts with cards enter outbound stockpoint. Production cards When stock is removed, place production card in hold box. Production card authorizes start of work.

25. Two-Card Kanban Inbound stockpoint Outbound stockpoint Move stock to inbound stock point. Move card authorizes pickup of parts. When stock is removed, place production card in hold box. Remove move card and place in hold box. Production cards Move cards Production card authorizes start of work.

26. Signaling • Cards • Lights & sounds • Electronic messages • Automation

27. The Main Design Issue • How many Kanbans should we have at each stage of the process and for each product?

28. Tradeoffs • Too many Kanbans lead to too much WIP and long cycle times. • Too few Kanbans lead to lower throughput and vulnerability to demand and process variability.

29. Advantages of Kanban • Attempts to coordinate production at various stages • Limits WIP accumulation at all production stages • Improves performance predictability and consistency • Fosters communication between neighboring processes • Encourages line balancing and process variability reduction

30. Limitations of Kanban • Possibility of starving bottlenecks • Vulnerable to fluctuations in demand volume and product mix • Vulnerable to process variability and machine breakdowns • Vulnerability to raw material shortages and variability in supplier lead times • Ideal for high volume and low variety manufacturing (becomes unpractical when product variety is high)

31. Constant Work-In-Process (CONWIP) System Customer demand Total WIP K • Basic CONWIP • Multi-loop CONWIP • Kanban

32. CONWIP Mechanics • A new job is introduced whenever one completes • The next job is selected from a dispatching list based on current demand • The mix of jobs is not fixed • Priorities can be assigned to jobs in the dispatching list • WIP level can be dynamically adjusted

33. Advantages of CONWIP Systems • Accommodates multiple products and low production volumes • Protects throughput and prevents bottleneck starvation • Less vulnerable to demand and process variability • Allows expediting and infrequent orders • Less vulnerable to breakdowns

34. Challenges • Difficulties in setting WIP limits and adjusting WIP levels with changes in product mix (a possible fix is to limit work-content rather than work-in-process). • Bottleneck starvation due to upstream failures. • Premature production due to early release. • Lack of coordination within the CONWIP loop.

35. Other Systems • Pull from the bottleneck systems (e.g., drum-buffer-rope, DBR) • Generalized Kanban Systems

36. Generalized Kanban System • Each stage has two parameters, si and ki • si: maximum inventory level (Ii) that stage i can keep in its output buffer of stage i • ki: maximum of number production orders (IOi) that stage i can place

37. Generalized Kanban System • Each stage has two parameters, si and ki • si: maximum inventory level (Ii) that stage i can keep in its output buffer of stage i • ki: maximum of number production orders (IOi) that stage i can place

38. Generalized Kanban System • Each stage has two parameters, si and ki • si: maximum inventory level (Ii) that stage i can keep in its output buffer of stage i • ki: maximum of number production orders (IOi) that stage i can place • si = ki , for all i Kanban • si > 0, ki = ∞, for all i  Base-stock • si = 0, ki = ∞, for all i MTO • sN> 0, kN< ∞; si = 0, ki = ∞, for i  N CONWIP • sbottleneck> 0, si = 0 for i  bottleneck,ki = ∞ for all i PFB

39. Push versus Pull Many competing definitions, including the following: • Definition 1: A pull system is a one where production is driven by actual inventory consumption (or immediate need for consumption). • Definition 2: A pull system is one where WIP is kept fixed or bounded by a finite (usually small) upper limit.

40. Push or Pull? • MTO • Base-stock • Kanban • CONWIP • PFB