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Control of Production-Inventory Systems with Multiple Echelons

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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

Characteristics

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Example 1: A Single Stage Production-Inventory System

Finished goods inventory

Customer demand

Production system

Work-in-process

Raw materi

al

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Example 2: A Series System

Customer demand

Stage 1 Stage N-1 Stage N

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Example 3: An Assembly System

Customer demand

External supply

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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.

The State of the System

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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

Costs, Decisions, and Objectives

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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.

The Optimal Production Policy

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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.

Challenges

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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)

Heuristic (but Common) Policies

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MTO Systems

Customer demand

Stage 1 Stage N-1 Stage N

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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

MTO Systems

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MTO Systems with Limits on WIP

Total WIP K

WIP1 k1WIPN-1 kN-1 WIPN kN

Limits on total WIP

Limits on WIP at individual stages (or groups of

stages)

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MTO/MTS Systems

Customer demand

Stage 1 Stage 2 Stage 4Stage 3 Stage 5

Make-to-stock segment Make-to-order segment

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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)

MTO/MTS Systems (Continued…)

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Base-Stock Systems

Customer demand

sN sN-1s1

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

Kanban Systems

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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

Outbound stockpoint

Productioncards

Completed parts with cards enter outbound stockpoint.

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

Production cards

Move stock to inbound stock point.

When stock is removed, place production card in hold box. Production

card authorizes start of work.

Move card authorizes pickup of parts.

Remove move card and place in hold box.

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 Work-In-Process (CONWIP) System

Customer demand

Basic CONWIP Multi-loop CONWIP Kanban

Total WIP K

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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

CONWIP Mechanics

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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 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-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

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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 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