Control of ProductionInventory Systems with Multiple Echelons

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Control of Production-Inventory Systems with
Multiple Echelons
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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

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Example 1 A Single Stage Production-Inventory
System
Customer demand
Raw material
Finished goods inventory
Production system
Work-in-process
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Example 2 A Series System
Customer demand
Stage N-1
Stage N
Stage 1
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Example 3 An Assembly System
Customer demand
External supply
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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.

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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

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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.

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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.

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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)

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

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

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

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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

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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.

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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.

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Kanban Systems

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

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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 maintained constant.

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Implementation

• One card systems
• Two card systems

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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.
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Two-Card Kanban
Outbound stockpoint
Inbound 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
Production card authorizes start of work.
Move cards
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Signaling

• Cards
• Lights sounds
• Electronic messages
• Automation

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The Main Design Issue

• How many Kanbans should we have at each
  stage of the process and for each product?

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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.

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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

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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)

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Constant WIP (CONWIP) System
Customer demand
Total WIP ? K

• Basic CONWIP
• Multi-loop CONWIP
• Kanban

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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

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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

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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 downstream failures.
• Premature production due to early release.
• Lack of coordination within the CONWIP loop.

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Other Systems

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

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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

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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 gt 0, ki 8, for all i ? Base-stock
• si 0, ki 8, for all i ? MTO
• sN gt 0, kNlt 8 si 0, ki 8, for i ? N ?
  CONWIP
• sbottleneck gt 0, si 0 for i ? bottleneck, ki
  8 for all i? PFB

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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.

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Push or Pull?

• MTO
• Base-stock
• Kanban
• CONWIP
• PFB