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 interarrival 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
Work-in-process Production Finished goods
system
inventory
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Example 2: A Series System
Customer
demand
Stage 1
Stage N-1
Stage N
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External supply
Example 3: An Assembly System
Customer
demand
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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 1
Stage N-1
Stage N
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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
Total WIP  K
 Limits on WIP at individual stages (or groups of stages)
WIP1  k1
WIPN-1  kN-1
WIPN  kN
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MTO/MTS Systems
Customer
demand
Stage 1
Stage 2
Make-to-stock segment
Stage 3
Stage 4
Stage 5
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
s1
sN-1
sN
Demand signal
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Base-Stock Systems
 Each stage manages an output buffer according to a basestock 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 workin-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 capped.
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Implementation
 One card systems
 Two card systems
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One-Card Kanban
Outbound
stockpoint
Production
cards
Completed parts with cards
enter outbound stockpoint.
When stock is
removed, place
production card
in hold box.
Outbound
stockpoint
Production
card authorizes
start of work.
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Two-Card Kanban
Inbound
stockpoint
Outbound
stockpoint
Move stock to
inbound stock point.
Move card
authorizes
pickup of parts.
When stock is
Remove move
removed, place
card and place
production card
in hold box.
Production
Production in hold box.
Move
card authorizes
cards
cards
start of work.
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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 Work-In-Process (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 workcontent rather than work-in-process).
 Bottleneck starvation due to upstream 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-bufferrope, 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
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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 > 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38PFB
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
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