Robert Savit

1
Agent Based Models of Supply Chains

• Robert Savit
• University of Michigan
• Program for the Study of Complex Systems

2
Credits and Contacts
Van Parunak and Steve Clark Center for
Electronic Commerce, ERIMAnn Arbor, MI
48105-2467vparunak_at_erim.org, 734-769-4049 Bob
Savit and Rick Riolo Program for the Study of
Complex Systemsand Department of
PhysicsUniversity of MichiganAnn Arbor, MI
48109savit_at_umich.edu, 734-764-3426 Supported by
DARPA under contract F33615-96-C-5511 to ITI,
administered by WPAFB, Mr. James Poindexter, COTR
3
Program for the Study of Complex Systems atthe
University of Michigan

• Further information
• http//www.pscs.umich.edu
• ERIMs CEC
• http//www.erim.org/cec/projects/dasch.htm

4
Overview

• Nature of supply chains
• Brief comparison of methods
• The model
• Results
• Analyses
• Conclusions and outlook

5
What are the features that make supply chains so
problematic?
6
The Seven Habitsof Highly Ineffective Supply
Chains

• Information that flows through a supply chain is
  often late and wrong.
• Effective planning is difficult or impossible in
  a distributed multi-firm enterprise.
• Supply chain dynamics impact both delivery and
  cost.
• Suppliers can't trust forecasts and don't trust
  forecasters.
• Suppliers lack technical sophistication.
• Support technologies are inadequate.
• Standards (de facto and de jure) are lacking for
  Data Interchange and Enterprise Control

7
7
These features may be stated more generally

• 1. Supply chains consist of heterogeneous sets
  of agents, each acting on the basis (typically)
  of incomplete information.
• 2. Motivations are generally local--eg.
  Maximizing local profit or local efficacy.
  Efficacy of other elements of chain are typically
  discounted.
• 3. Supply chains are intrinsically dynamical.
• 4. Agents are adaptive.
• 5. There are inherent nonlinearities and
  feedback effects in dynamics and structure of
  supply chain.

8
These are all hallmarks of a complex adaptive
system

• Behaviors of the system can be highly
  counter-intuitive.
• Need methods of modeling and analysis that are
  appropriate to nonlinear, dynamical and adaptive
  systems--particularly distributed systems.

9
What are some of the specific ways in which
supply chains manifest characteristics of complex
systems?
10
Supply Chains as Dynamical Systems Structural
Issues

• Asynchronous decentralized network
• Structural irregularities

5
11
Supply Chains as Dynamical Systems Intrinsic
Nonlinearities and Adaptivity

• Batching of orders into "economic order
  quantities"
• Maintenance of "safety stocks" in excess of
  actual production requirements
• Capacity limitations on production equipment
• Time delays in movement of information and
  material
• Algorithms (PPIC) used at each company to
  translate demand from customers into orders to
  suppliers
• Agents in supply chains are adaptive

5
12
An Additional Problem Data Analysis

• Common time series methods best suited to
• long time series (103 or more observations)
• stationarity
• few degrees of freedom
• Available supply chain data offer
• short time series (typically, weekly data over a
  few years 102 observations)
• highly nonstationary
• many available variables
• Solution Model-driven approach

7
13
Modeling of complex systemsLightning discussion

• Standard analytic or semi-analytic methods are
  often of limited use for distributed, nonlinear,
  adaptive systems.
• Requires abstraction
• Translation of distributed actions into analytic,
  equation based forms
• Often requires introduction of lumped or average
  variables
• Can drastically alter the nature and effect of
  fluctuations.

14
Better approach Direct emulation of the
distributed agents in the system

• Some advantages
• Dont introduce lumped variables
• more direct representation of systems under study
• easier to control and sample relevant parameters
• can more reliably do what-if experiments

15
Down side

• Disadvantages
• computationally intensive
• different set of validation issues
• Need to give up epistemology of exactness

16
The Model
17
The Supply Chain Hourglass
Consumers
Distribution
Distributors
Manufacturer
Subassy Suppliers
Input
Part Suppliers
18
Entities
Product Flow
4
19
Some Agent Behaviors and Independent Variables
Site
PPIC
Shipper, Mailer
3
20
Operating Assumptions

• Sites fill orders from inventory
• An order ships only when complete
• Incoming raw material is
• held in WIP for delayMean to model processing
  time
• gated through site capacity
• added to finished goods inventory, available for
  shipment immediately

6
21
PPIC Algorithm

• R(t) is order placed at time t by site k to site
  k1.
• A is safety stock.
• I(t) is inventory at time t.
• P(t) (Q(t)) is what is in the pipeline at site k
  (not yet at site k),available for inventory at
  time t.
• F(t) is the forecast of what is expected to be
  ordered by site k-1 from k at time t.

22
Forecasting Methods

• Constant ( Gaussian noise of specified variance)
• Pass-through from customer ( Gaussian noise of
  specified variance)
• Weighted average of past demand
• n forecastWindow
• wi n 1 - i

23
Typical Forecasting Methods
7
24
Results

• Examples two sets of results.
• Strictly linear phenomena (but with feedback
  effects)
• variation amplification
• variation correlation
• variation persistence
• Nonlinear phenomenon
• inventory oscillations

25
Some quantities we study

• Orders issued
• Shipments
• Inventory (finished goods)
• Work-in-Process (raw materials)
• Throughput (running average over 5 ticks)
• Average Time to Fill at Consumer
• Average Overdue Time at Consumer

7
26
Variation Amplification

• Subtier suppliers see a higher range of variation
  in their incoming orders than the OEM generates
  in its orders to the first tier.
• Configuration
• All batch sizes at 1
• No capacity limitations (capacity mean 10000)
• Consumer orders 100/week, variance 10
• Linear phenomenon

6
27
Variation Amplification Mechanisms

• (Stermans Beer Game People forget items already
  in the pipeline--not active in DASCh)
• Lee, Padmanabhan, and Whang, The Bullwhip Effect
• è Demand signal processing changing forecast
  based on past observations
• Rationing by suppliers
• Order bunching across consumers
• Adjusting purchase volume to price

28
Variation Modification in Weighted Forecasting
Order variation to supplier scales as VL2/f2
4
29
Amplification Raw Data
30
Variation Amplificationin DASCh Orders at Each
Level
3
31
Variation Amplification in Automotive Orders
3
32
Order Volatility, 1971-1995(Anderson, Fine, and
Parker, MIT, 1996)
Annual Volatility (?)
GDP
2
12
37
4
33
Variation Modification in Weighted Forecasting
Theory

• Parameters
• L Order lead time
• f historical period used for forecasting
• ?2 variance in customer demand
• S(t) ? ?(t)
• ?(t) is IID by construction
• ?2 variance in Site 2 orders to Site 3
• R(t) ? ?(t)
• Preliminary analysis suggests
• lt ?(t)2gt ? L2lt ?(t)2gt
• ?(t) is linearly correlated over range of order f
  (see below)

8
34
Confirmation of lt ?(t)2gt ? L2lt ?(t)2gt Site2 ?
Site3 Variance
450
400
Lead
350
Time
300
3
250
Secondary Order Variance
10
200
20
150
30
100
50
0
0
10
20
30
40
50
60
70
Primary Order Variance
35
Confirmation of lt ?(t)2gt ? L2lt ?(t)2gt Site2 ?
Site3 Variance
450
400
Primary
Order
350
Variance
300
5
250
Secondary Order Variance
10
200
30
150
60
100
50
0
0
500
1000
(Lead Time)2
36
Confirmation of lt ?(t)2gt ? L2lt ?(t)2gt/f2Site 2 ?
Site3 Variance
Primary Order Variance
37
Variation Modification in Pass-Through Forecasting
Order variation at bottom of chain scales as sum
of (Noise L) from each level!
4
38
Pass-Through Forecast Fit183.7 3.1(ConsVar
N2L2 N3L3)
39
Implications of Variation Amplification

• Long lead times are bad.
• Result in variance amplification, but of a
  different nature depending on the forecasting
  algorithm.
• Distorting your customers forecast is bad.
• Its especially bad when both of these are
  excessive at the same site.
• In weighted forecasting, longer forecast windows
  reduce amplification (but cause other problems)

4
40
Variation Modification Correlation
Time-Delay Plots detect Variation Correlation
(One-Step Time-Delay Plots)
3
41
Variation Amplification Correlation in Real Data
3
42
Linear correlation due to linear feedback
generated by the supply chain--Eg. Orders from
site 2 to site 3

• r(t)R(t)-m
• d(t)F(t)- m
• r(t) and d(t) are not IID
• (but, actual variation in REAL customer
  demand,?(t), is IID)
• Yet more complicated further down the chain

43
Implications of Variation Correlation

• PPIC algorithms can impose structure on the
  demand stream that does not reflect top-level
  requirements.
• Suppliers need ways to distinguish between this
  spurious structure and real variation that
  demands their attention.

44
Variation Persistence

• Step function demand with weighted average
  forecast
• Anomalous peaks and dips in demand
• Settling time is forecastWindow
• Other manifestations discussed later
• Still linear

5
45
Variation Persistence
5
46
Trade-Off in Forecast Window Length
47
Implications of Variation Persistence

• Supply chains have memories that retain the
  state of the chain
• Forecasting windows
• Backlogged orders
• High WIP levels
• These memories must be shortened to improve
  agility.

5
48
A nonlinear phenomenon Generation of
inventory oscillationsEven with constant
supply and demand, internal dynamics of the
supply chain can generate oscillations.
49
Oscillations Description

• A chain with stable boundary conditions can
  generate variance internally.
• Even with constant top-level demand and unlimited
  bottom-level supply, intermediate sites see
  oscillating activity.
• Configuration
• Batch sizes all 1
• Sites 2 and 3 have capacity of 100, no variance
  (threshold nonlinearity)
• Consumer demand constant at varying levels 95,
  150, 200, ...

33
50
The DASCh Oscillator (Phenomena)

• Demand gt Capacity inventory oscillates _at_ sites 2
  3.
• Negative ramps (sawtooth)
• Period is Demand/(Demand - Capacity)
• Amplitude is max(Capacity, Demand - Capacity)
• Subsequent drop of Demand lt Capacity continues
  oscillation until WIP drops
• Positive ramps
• Period is Demand/(Capacity - Demand)
• Amplitude is Demand of oldest fillable order
• Low excesses (lt 100)
• Single period oscillations
• Sites 2 and 3 lock in phase at about 150
• High excesses (gt100)
• Multiperiodic oscillations
• Sites 2 and 3 lock out of phase

51
Oscillation Generation
Demand/Capacity
3
52
Characteristics of the Inventory Oscillator

• Let D(emand)/P(roduction) have no common factors
• Represent I(nventory)(t) in the same units.
• Let H ? Min(P, D-P). Then
• P ? I lt DP is an attractor
• For I(t) in the attractor, I(t) I(tD)
  (periodicity)
• Between t and tD, I(t) assumes every value in P
  ? I lt DP .
• There are H intermediate maxima between maxima of
  the same size (counting one of the ends).
• If H ? P, these intermediate maxima are of two
  periods, differing by 1, with one period more
  common than the other.
• No two maxima of the less common period are
  adjacent.

9
53
The DASCh Oscillator (Mechanism)

• Defect (order - output) erodes safety stock
• Infinite capacity PPIC builds WIP
• When safety stock 0,
• backlog that periods order
• production provides new safety stock
• Backlogged orders can delay impact of changed
  demand (fill oldest orders first)
• When demand drops below capacity, excess (output
  - order) accumulates until it can satisfy a
  backlogged order

7
54
Geometrical interpretation

• The complicated multi-periodic structure can be
  understood in the context of non-linear dynamics
  as flow on a 2-torus.
• Take a periodic segment of the graph of
  inventory. Identify I0 and ID (vertical axis).
  Wrap it around to form a cylinder.
• Further, identify the beginning and end of a
  period to form a doughnut.
• I(t) is then a flow on the doughnut (2-torus).
  The more complicated the ratio D/C, the more
  complicated the flow on the doughnut.

55
Implications for nonlinear dynamics

• If D/C were replaced by an irrational number, we
  would have a truly quasiperiodic structure, and
  the function I(t) would not repeat.
• Because this dynamics is restricted to a 2-torus,
  it cannot become chaotic (flows), without the
  introduction of other dynamical effects

56
Implications of Oscillation Generation

• A bottleneck in the supply chain
• not only limits Capacity
• but also introduces Variation.
• The nonlinear nature (periodic or quasiperiodic)
  may have implications for predictability in
  supply chains.

57
Summary

• Supply chains are complex adaptive systems with
  intrinsic nonlinearities capable of very rich
  counter-intuitive behaviors.
• Emulation of such systems are important tools to
  gain the best understanding of such complex
  systems-both their linear and non-linear
  properties.
• Studied very simple model and saw a number of
  interesting features.
• Linear
• variation amplification
• variation persistence
• variation correlation

58
Summary, cont.

• Nonlinear oscillation generation. Which
  oscillations could approximate quasiperiodicity
• Some features we did not consider
• more complicated topologies
• structural feedbacks
• competitions and induced frustrations
• nonstationarity
• decision making under incomplete information
• adaptivity

59
Overarching conclusions

• The best way to approach the complicated and
  important problem of supply chains is as a CAS
  using agent based modeling techniques and
  insights from nonlinear dynamics and adaptive and
  evolutionary computation.
• Effective and robust strategies for supply chain
  design and control require the fundamental
  insight into the dynamics and nonlinear structure
  of these systems that this approach can help
  provide.

60
CEC Approach Philosophy (1)
Supply Chains are Complex Systems
Trade-offs Are Difficult To Understand and
Control
Firm Performance
Supply Chain Performance
The True Costs and the Benefits of Change Are
Often Hidden