Control of High Volume AssembletoOrder Systems

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Control of High Volume Assemble-to-Order Systems
Amy R. Ward (USC) ANS Lecture, Monday, Aug. 18
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Control of High Volume Assemble-to-Order Systems
Amy R. Ward (USC) ANS Lecture, Monday, Aug. 18

• Plambeck and Ward (2008)
• Optimal Control of a High-Volume
  Assemble-to-Order System with
• Maximum Leadtime Quotation and Expediting
• --- and Ward (2006)
• Note A Separation Principle for a Class of
  Assemble-to-Order
• Systems with Expediting
• --- and Ward (2006)
• Optimal Control of a High-Volume
  Assemble-to-Order System

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The Basic Assemble-to-Order System
K products
price p
?(p) arrival rate akj
requirements c component unit
production cost
assembly
sequencing
capacity ?
J components
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The Basic Problem

• How should an Assemble-to-Order Manufacturer
• price products
• set component production capacity
  
• sequence orders for assembly
• to maximize expected infinite horizon discounted
  profit?

Component arrival process
Assembly process
Inventory process
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Our Approach
1. Assume the system experiences a high volume
of demand. 2. Show an asymptotic reduction in
problem dimensionality. 3. Solve an
approximating Brownian control problem. 4. Show
asymptotic optimality. 5. The same methodology
applies for systems with maximum leadtime
quotation and expediting and salvaging.
(Though we can only show asymptotic optimality
when expediting is expensive and there is
no salvaging.) 6. We can also allow for
product demand rates and component
production costs that change at certain points in
time.
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Literature Review
Extensive review of the operations management
literature on assemble-to-order systems in Song
and Zipkin (2002).

• Exogenous demand.
• Given leadtime distributions.
• Assumed sequencing rule.
• Goal optimize basestock levels.

• Agrawal and Cohen (2001)
• Zhang (1997)
• Ackay and Xu (2004)

Brownian Approximation for ATO systems Kushner
(1999).
Expediting Control Lawson and Porteus
(2000) Sethi, Yan, and Zhang (2003) Cho and Meyn
(2003, 2005)
Single-item inventory management
Power networks
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Our Approach
1. Assume the system experiences a high volume
of demand. 2. Show an asymptotic reduction in
problem dimensionality. (Set prices, component
production rates, and assembly policy.) 3. Solve
an approximating Brownian control problem. 4.
Show asymptotic optimality. 5. The same
methodology applies for systems with maximum
leadtime quotation and expediting and
salvaging. (Though we can only show
asymptotic optimality when expediting is
expensive and there is no salvaging.) 6. We
can also allow for product demand rates and
component production costs that change at
certain points in time.
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Static Planning Problem (SPP)
Prices and component production rates should be
close to the SPP solution.
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The Profit Function
Order arrival process
Queue-lengths
Inventory levels
There is a tension between the loss of revenue
associated with having order queues and the loss
of revenue associated with having inventory.
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The Key Insight
The Shortage Process
is not affected by the assembly policy.
However, queues and inventory levels are, and the
shortage process can also be written as
FCLT implies S behaves as a J dimensional
Brownian motion. For a given S, there is an
optimum arrangement of Q and I. We would like an
assembly policy under which Q and I track those
levels.
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The Ideal Assembly Policy
Can we implement this assembly policy?
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The Proposed Assembly Policy
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Reduction in Problem Dimensionality
High volume assumption Consider a sequence of
systems, indexed by n. Demand in the nth system
is n?k(p), k1,,K.
The choice of review period length depends on the
number of moments assumed on the inter-arrival
time random variables See Ata and Kumar (2004).
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Our Approach
1. Assume the system experiences a high volume
of demand. 2. Show an asymptotic reduction in
problem dimensionality. 3. Solve an
approximating Brownian control problem. 4. Show
asymptotic optimality. 5. The same methodology
applies for systems with maximum leadtime
quotation and expediting and salvaging.
(Though we can only show asymptotic optimality
when expediting is expensive and there is
no salvaging.) 6. We can also allow for
product demand rates and component
production costs that change at certain points in
time.
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Asymptotic Behavior of the System
Brownian motion with zero drift.
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Adjusting Prices and Component Production Rates
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Brownian motion with drift.
Observe the cost trade-off between queues and
inventory that mimics what we saw in the profit
function.
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An Asymptotically Optimal Policy
Implication Capacity utilization near 100
is economically optimal.
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Our Approach
1. Assume the system experiences a high volume
of demand. 2. Show an asymptotic reduction in
problem dimensionality. 3. Solve an
approximating Brownian control problem. 4. Show
asymptotic optimality. 5. The same methodology
applies for systems with maximum leadtime
quotation and expediting and salvaging.
(Though we can only show asymptotic optimality
when expediting is expensive and there is
no salvaging.) 6. We can also allow for
product demand rates and component
production costs that change at certain points in
time.
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ATO System with Maximum Leadtime
K products
price p
?(p) arrival rate l maximum
leadtime akj requirements c
component unit production cost x
expediting cost
assembly
sequencing, expediting
capacity ?
J components
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Problem

• How should an Assemble-to-Order Manufacturer
• price products
• set component production capacity
  
• sequence orders for assembly
• expedite (costly) components
• to maximize expected infinite horizon discounted
  profit,
• subject to satisfying maximum leadtime quotations?

Assembly Process Expediting Process
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Static Planning Problem (SPP)
Proposition Any asymptotically optimal policy
has
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The SPP Solution
Customers are averse to waiting and instantaneous
delivery is free in the SPP (because it ignores
stochastic variability). Does the SPP solution
always have l0?
Yes, whenever customers have homogeneous delay
tolerance.

• Example
• Customers arrive according to a Poisson process.
• Each customer has product valuation (v1, v2, ,
  vK),
• drawn from a general joint distribution, with a
  pdf.
• Each customer has delay cost function f(l),
  regardless
• of product, and purchases product
• if the optimal objective value is non-negative,
  and
• otherwise does not make a purchase.

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The SPP Solution
However, when customers may differ in their delay
tolerances, as in Afeche (2004), the SPP solution
may have a strict subset of products for which
the optimum maximum leadtime strictly exceeds 0.
The key characteristic is a positive correlation
between impatience and willingness to pay a high
price. Then, the system manager may want to
offer a discount to customers willing to accept a
long leadtime.
For convenience in presentation, we will assume
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The Brownian Approximation
The capacity imbalance rate is
Proposition Any asymptotically optimal policy
has
Hence suppose
Maximum leadtime quotations imply shortages are
not too big
The shortage process can be approximated by a
reflected Brownian motion restricted to the
region
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The Proposed Prices and Capacity
Reflected Brownian motion having state space S.
Pushing process
Solution to a perturbed SPP.
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Perturbed SPP
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The Proposed Assembly Policy
Penalty for large order queues.
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The Proposed Expediting Policy
Expedite the minimum amount of components
required to assemble orders in the next review
period.
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An Asymptotically Optimal Policy
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The Case of Heterogeneous Customers

• The shortage process approximation

• The implications are
• Expediting is not necessary for any component
  used
• in a product for patient customers.
• Actual leadtimes for products targeted to
  patient
• customers are consistently lower than maximum
  leadtimes.

Why do many firms have maximum leadtime quotation?
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Exact Leadtime Quotation
The effect of giving customers information about
actual leadtimes would, in equilibrium, be
similar to that of requiring exact leadtime
quotation.
Discounted profit
Static Planning Problem
The loss in profit is of order n1/2 when l0,
and is of order n otherwise. There is a strong
financial incentive for having the greater
flexibility inherent in maximum leadtime
quotation, and preventing customers from knowing
the actual leadtime distributions.
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Our Approach
1. Assume the system experiences a high volume
of demand. 2. Show an asymptotic reduction in
problem dimensionality. 3. Solve an
approximating Brownian control problem. 4. Show
asymptotic optimality. 5. The same methodology
applies for systems with maximum leadtime
quotation and expediting and salvaging.
(Though we can only show asymptotic optimality
when expediting is expensive and there is
no salvaging.) 6. We can also allow for
product demand rates and component
production costs that change at certain points in
time.
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Systems with Salvagingand a General Cost of
Expediting

• How should an Assemble-to-Order Manufacturer
• price products
• set component production capacity
  
• sequence orders for assembly
• expedite and salvage components
• to maximize expected infinite horizon discounted
  profit,
• subject to satisfying maximum leadtime
  quotations?

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An Approximating Diffusion Control Problem
To determine price and capacity.
Numeric solution using Kumar and Muthurman
(2004) Also, Kushner and Dupuis (1992), Kushner
and Martins (1991)
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Our Approach
1. Assume the system experiences a high volume
of demand. 2. Show an asymptotic reduction in
problem dimensionality. 3. Solve an
approximating Brownian control problem. 4. Show
asymptotic optimality. 5. The same methodology
applies for systems with maximum leadtime
quotation and expediting and salvaging.
(Though we can only show asymptotic optimality
when expediting is expensive and there is
no salvaging.) 6. We can also allow for
product demand rates and component
production costs that change at certain points in
time.
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Adapting to Shifts in Demand and Supply Conditions
It is common to change prices, maximum
leadtimes and component production rates in
response to shocks in the business environment.
This leads us to consider piecewise
constant policies for setting prices, maximum
leadtimes, and Component production rates.
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An Approximating Diffusion Control Problem
Transition probability
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Conclusions and Future Research

• Realistically complex models of
  assemble-to-order
• systems are tractable under the assumption that
  a
• high volume of customers arrive per unit time.
• Under the high volume assumption, we derived a
• policy for setting prices, maximum leadtimes,
• component production rates, and managing
  component
• inventory through expediting and salvaging.
• Our policy is provably asymptotically optimal
  when
• expediting is expensive and there is no
  salvaging.
• The key insight is that there is a reduction in
  
• dimensionality to the number of components,
  J.
• This is important because typical systems
  have
• many more products than components.
• Interesting future research would allow for
  dynamic
• leadtime quotation.