an available-to-promise production-inventory system with ... · an available-to-promise...

IntroductionModels

Numerical Results

An Available-to-Promise Production-InventorySystem with Pseudo Orders

Long Gao

joint work with Susan Xu

University of DaytonPenn State University

MSOM Conference, June 6, 2008

Long Gao An ATP System with Pseudo Orders

IntroductionModels

Numerical Results

Outline

1 IntroductionMotivationResearch Questions

2 ModelsPseudo Order ModelOrder Promising Model

3 Numerical ResultsPolicy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy


IntroductionModels

Numerical Results

MotivationResearch Questions

Outline





IntroductionModels

Numerical Results


What is a pseudo order?

Pseudo OrdersIntended purchase orders, short-term forecasts, (imperfect)advance demand information in a B2B environmentSubject to change and cannot be enforcedContain information of the likelihood of becoming actualorders, due date, requested quantity, etc.Maintained and revised by sales personnelLumpy, nonstationary, volatile, highly uncertain

Trade-offThe presence of pseudo orders makes confirmed orders lesslikely to be accepted because it is more desirable to reservelimited resources for future higher value pseudo orders.


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A Pseudo Order Example

Siebel.com, (Forbes, January 21, 2002)Acting upon the information of the sudden cancellation ofhundreds of potential deals in February 2000, Thomas Siebel(CEO) anticipated the recession months ahead of rivals andeconomists. He realigned his sales force, readjusted resourceallocation decisions, and avoided the worst in 2001.

However,Pseudo order information is not well-integrated intobusiness planning and control systems.The cost of ignoring such information can be very high.


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Available-To-Promise System

What is an ATP system?A business function that matches incoming customerorders to planned resourcesDifferent from traditional planning, scheduling andinventory management processesOperate within a short-term operational environmentMost resources are considered fixed because ofprocurement leadtime limitationsDeal with multiple customer classes


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An ATP System with Pseudo Orders

ATP

System

Inventory

Mgmt

System

Production

Capacity

Component

Availability

Comitted

Orders

Pseudo

Orders

modify supplier orders reserve

comp avail

& prod cap

accept/process

orders

supplier order

lead time


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Numerical Results


ATP Examples: Dell and Toshiba

Dell two-stage order promising practice

Customer Differentiation: Home, Small Business, Medium& Large Business, Government, etc.Provide initial soft confirmation via emailGenerate hard confirmation after checking resourceavailability, based on batch ATP

Toshiba electronic product ATP systemOrders for several thousand models are collected andprocessed by a single central order processing systemATP execution every 1/4 ∼ 1/2 hourBook pseudo orders up to 10 weeks in advance of delivery


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Research Questions

Research QuestionsHow to model the lumpy, non-stationary and volatilenatures of pseudo order information?What is the optimal order promising policy in an ATPsystem with pseudo order information?How robust is the optimal policy?What are the costs of using suboptimal policies?What is the value of the pseudo order information?


IntroductionModels

Numerical Results

Pseudo Order ModelOrder Promising Model

Outline





IntroductionModels

Numerical Results


Pseudo Order Model

Three major characteristics of pseudo ordersLumpiness: non-negligible probability of cancellationNon-stationarity: demands are not identically distributedVolatility: attributes change before either confirmed orcancelled.

For each future pseudo order,

Random demand distribution: Ykt (ek) ∼ Fk

ek, where ek ∈ Ek

is a distribution state, evolving according to a Markovchain, qk

t (e′k|ek).

Random confirmation date: sk evolves according tohk

t (s′k|sk).


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An Example: Zero-Inflated Poisson Distribution

Two dist. states: E = { 0, 1 }, cancellation or PP(λ)

Time-homogeneous transition probabilities of distributionstates

[qk(·|·)] =

[1 0πk (1 − πk)

]However, if cancellation information is unknown, thedemand distribution is the mix of mass 0 and PP(λ),resulting in Zero-Inflated Poisson (ZIP) distribution

P(Yk = j) =

{πk + (1 − πk)e−λ, if j = 0,

(1 − πk)e−λλj/j!, if j > 0.

ConclusionInformation updates can remove one source of uncertainties.


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Order Aggregation Scheme

Less volatile, aid ATP decision making, increaseoperations and computation efficiencyAggregate demands with order confirmation dates s:

Xt,s =∑

{ k:sk=s }

Ykt (ek) ∼ ⊗k Fk

ek

Aggregated demands are temporally dependent, governedby

P { Et−1 | Et } =∏k∈Kt

hkt (s

′k|sk) · qk

t (e′k|ek)


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Order Aggregation Scheme: Poisson Example

Two dist. states E = {0, 1}, cancellation or PP(λ)

System state can be simplified to the total number ofuncancelled orders Et = (nt,1, . . . , nt,t−1).Aggregated demand Xt,s follows PP(nt,sλ).Transition probability of Pt {Et−1 | Et } is given by

Pt ((nt−1,1, nt−1,2, . . . , nt−1,t−1)|(nt,1, nt,2, . . . , nt,t−1))

=∑

n

∏t−1s=1

nt,s!n(0|s)!(nt,s−n(0|s))!π

n(0|s)(1 − π)nt,s−n(0|s)

×∏t−1

s=1

((nt,s−n(0|s))!

n(1|s)!···n(t−1|s)!∏t−1

s′=1(ht(s′|s))n(s′|s)) .

ConclusionOur Markov chain model completely describes the evolution ofpseudo orders at both the individual and the aggregate level.


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Assumptions for the Order Promising Model

T-period ATP system with MTO manufacturing strategyMultiple classes of orders bring in revenue r1 > · · · > rI

Each order consumes one unit of production capacity,takes a single production periodPseudo order forecast Et is updated by P(Et−1|Et)

Newly confirmed orders Nt

Accepted orders xt = {xit} must be fulfilled within L periods

Production: first-accepted, first-served


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Sequence of Events in Each Period t

Nt orders confirmedFuture pseudo orders updated to Et

Observe net capacity Qt

Planned capacity Kt becomes availableDecision: accept orders xt = {xi

t : i ∈ I}

ObjectiveMaximize the expected total profit over the ATP executionhorizon T


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A Markov Decision Process Formulation

Vt(Qt, Nt, Et) = maxxt∈At

r · xt − p(Qt + Kt − |xt|)++

∑Et−1

∑Nt−1

p(Nt−1|Et)P(Et−1|Et)

×Vt−1 (Qt−1, Nt−1, Et−1)

,(1)

The action space At is defined by

0 ≤ xt ≤ Nt, (2)|xt| ≤ Qt + [K]tt−L. (3)

Nonlinear system dynamics

Qt−1 = [Qt + Kt − |xt|] ∧ 0. (4)


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Characterization of the Optimal Policy

Optimal order acceptance policyAccept in an increasing order of the indexReject class i if class i − 1 are not fully acceptedAccept class i until

1 all N it are accepted (demand)

2 cumulative leadtime capacity for i is exhausted (supply)3 the net capacity rationing level is reached (rationing)

Formally, for class i ∈ I, the optimal acceptance is

x̂it = min

Ni

t ,[Qt + [K]tt−L − [N]i−1

1

]+,[

Qt + Kt − [N]i−11 − ηi

t−1(Et)]+

. (5)


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Contributions of the Characterization

Explicitly reveal the dependence on demand quantity, leadtime capacity, and capacity rationing level in a simple form

Result in (Qt, Nt)-state independent threshold ηit−1(Et),

depending on forecast only

Ease the “Curse of Dimensionality” for such multi-dim MDP

O(Q × E × I) v.s. O(Q × E × NI)

For example, if N = 100, I = 3, save 0.3 × 106 times incomputation efforts!


IntroductionModels

Numerical Results

Policy ComparisonValue of Pseudo Order UpdatingRobustness of the Optimal Policy

Outline





IntroductionModels

Numerical Results


Numerical Results

When are rationing and pseudo order informationnecessary?What are the costs of using suboptimal policies?Is it beneficial to use short term volatile forecast, or justuse long term forecast?How robust is the optimal policy?


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Experiment Design

Two-class inventory ATP system, horizon T = 10

4 demand settings: { SL, SH, NL, NH }3 resource availability ρ levels: scarce, intermediate, ampleρ = S/[EX1

t + EX2t ]

3 profit ratio γ = r1/r2 levels, r1 + r2 = 10

Holding cost: h = 0.5

2 lead time levels: L = { 0, 2 }72 scenarios, each generates 100 instances, total 7200instances


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Policy Comparison

OPT: rationing with complete pseudo order informationMVE: rationing with mean demand, ignore stochasticityPRO: priority rule only, ignore pseudo order informationFS: fair share or first-come first-served, ignore bothprioritization and pseudo order information

Performance Gap: percentage difference of total profits,e.g., ∆VM = [V∗ − VM]/V∗ × 100%


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What are the costs of using suboptimal policies?

VFS ≤ VPRO ≤ VMVE ≤ V∗

Prioritization is effectiveregardless of the capacitylevelRationing with mean valueis necessary when thecapacity level is low tointermediateStochasticity of pseudoorders cannot be ignoredwhen capacity is atintermediate level


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Policy selection: partitioning of the parameter space

0.2 0.4 0.6 0.8 1.0 1.21.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

22

2

2

22

2

22

Resource Availability: ρ

Pro

fit R

atio

: γ

MVE

OPT

PRO / MVE

FS / PRO / MVE


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Are later due dates always beneficial?

Figure: Impact of Lead Time L OPT, MVE and FSbenefit fromincreased lead timeresource availabilityPRO may sufferfrom later due date!Customercannibalization:larger percentage ofclass-2 acceptancedue to increasedresource availability


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Value of Pseudo Order Information Updates

Volatility and dynamic information availability

Question: Given the volatile nature of pseudo orders, is itbeneficial to use the short term forecast, or just use longterm forecast?

The percentage difference of the systems with and withoutupdating quantifies the value of pseudo order updating


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Value of Pseudo Order Information Updates2

2

22

2

2

44

4

4

4

6

6

6

Resource Availability ρ

Pro

fit R

atio

γ

0 0.2 0.4 0.6 0.8 11.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

significant region: ∆ V*S > 2%

Updating is always beneficialSignificant region: ∆V ≥ 2%,scarce capacity,heterogenous customersIn this region, updating canfurther strengthen theeffectiveness of rationing by2% ∼ 7%


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How robust is the optimal policy?

RobustnessWhat if the the forecast is inaccurate?What if the underlying distributions changed?Is OPT still better than others, especially forecastindependent policies?


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Robustness Comparison

Table: Robustness for Forecast Errors over 4200 Instances

Forecast Errors Dominance over suboptimalsType (ε2, ε3) ∆µ1% ∆cv1% PMVE% PPRO % PFS%

I (+3, +3) 25.42 2.32 64.74 71.85 85.45II (−3, +3) −5.08 26.67 76.73 94.42 100.00III (−3,−3) −25.42 0.97 87.35 100.00 100.00IV (+3,−3) 5.08 −3.61 83.08 93.76 98.85

Overall (±3,±3) ±15.25 ±8.58 77.98 90.01 96.08

OPT is robust for small to moderate forecast errors.OPT should be implemented with forecast updatingmechanisms.


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Conclusions

We quantify lumpy volatile pseudo order information, andcharacterize the optimal order acceptance policy.Commonly used policies may suffer severe losses due toignoring pseudo order information and rationing.Prioritization without rationing may reduce the profitabilitywith extended due dates!OPT is fairly robust and should be implemented withpseudo order updating mechanisms.


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Future Research

Multiple components, class-specific lead timeImpact of pseudo order information on strategic or tacticalresources planningRandom supply and production processesOther Applications: Hub Group, Inc. intermodal shippingload acceptance


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Characterization of the Value Function: Proof

Sketch of the ProofDifficulty: nonlinear dynamics of capacityInduction for three cases: both positive, both negative, andone eachObserve that: r1 ≥ ∆Vt−1(Qt−1|Et) ≥ −p

Use complementary property of max{x, 0} and min{x, 0}:at least one of them is 0Regarding Nt, there is no lost sale penalty and the actionspace is convex


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Literature Review: Rationing Models

Continuous time rationing modelsHa. (1997)Benjaafar & ElHafsi (2006)

Discrete time rationing modelsTopkis. (1968)Only one nonperishable resource, available at thebeginning, no pseudo ordersWang and Gupta. (2007)Two classes, single resource, no pseudo ordersOur model deals with multi-period, both perishable andnonperishable resources, incorporating pseudo orders


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Characterization of the Optimal Value Function

Lemma(i) Vt(Qt, Nt, Et) is concave in net capacity Qt.(ii) Vt(Qt, Nt, Et) is increasing concave in realized demand Nt.

Managerial InsightsMarginal value of unit capacity diminishes when capacityincreases.Carefully plan and allocate capacity over time [T, 1], usingpseudo order information.Marketing activities on demand management, such asorder expedition and postpone, need to be coordinatedwith the planned resources.


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