Demand Point Aggregation for Location Models

1
Demand Point Aggregation for Location Models

• R. L. Francis University of Florida
• francis_at_ise.ufl.edu
• T. J. Lowe University of Iowa
• tlowe_at_blue.weeg.uiowa.edu

2
Acknowledgement

• We are happy to thank Hulya Emir-Farinas, and
  Brenda Rayco, for their help, particularly
  computational work summarized in many of the
  figures and tables.
• This presentation is a shortened version of a
  tutorial given at the Istanbul EURO-INFORMS
  meeting in July, 2003.

3
Outline

• Location problems
• Location analysis
• Location model typology
• References
• MIP location models, MP aggregation
• Demand point modeling, aggregation
• Common DP aggregation approach

• Law of diminishing returns
• Aggregation error measures
• SAND location models
• Error bounds
• Paradox of aggregation
• Overview some aggregation algorithms
• Example aggregations
• Conclusions

4
Quote (Frank Plastria, 2002)

• A location problem arises whenever a question is
  raised like
• Where are we going to put the thing(s)?
• The next two questions then immediately follow
• Which places are available?
• On what basis do we choose?

5
Example location problems

• house or apartment
• branch banks
• automobile dealerships
• ATMs
• tax offices
• grocery stores

• schools
• lock boxes for periodic payments
• warehouses/
• distribution centers
• factories

This is a problem you have solved. Tradeoffs
include 1) rent 2) travel time to UF.
6
Distances (or times) matter

• distance to work
• distance to bank
• distance to shopping

• time check is in mail
• transportation distance to/from factory
• transportation distance to/from warehouse/DC

7
Location models

• Location models often try to capture some of the
  above distance aspects, as well as fixed costs in
  many cases.
• We seek a location, or locations, from either a
  finite or infinite set, with distances
  appropriately defined, to minimize some cost
  expression.

8
Purpose of location analysis

• Suggest and identify options for
• Number of facilities (servers)
• Locations of facilities
• Size of facilities
• Allocation of demands (supplies) to facilities

9
Types of location models tradeoffs

• Discrete/MIP Models most accurate, least
  computationally tractable, better for numbers
  than insights
• Continuous Planar Models least accurate, often
  very computationally tractable, better for
  insight than accuracy
• Network Models a compromise between MIP
  Planar Models, use shortest-path distances,
  require network data base may discretize to solve

10
Strategic, tactical, operational management
courtesy of Stefan Nickel (Bender et al., 2001)

• Strategic Management
• Long planning horizon, high aggregation level,
  planar location models.
• Tactical Management
• Medium planning horizon, medium aggregation
  level, network location models
• Operational Management
• Short planning horizon, low aggregation level,
  scheduling routing models.

little data
more data
much data
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References

• Discrete Location Theory, Mirchandani and
  Francis, eds., Wiley 1990
• Facility Location, Z. Drezner ed., Springer, 1995
• Network and Discrete Location, M. Daskin, Wiley,
  1995.
• Facility Location Applications and Theory, Z.
  Drezner and H. Hamacher, eds, Springer 2002
• Various published papers (see handout)
• Papers from ISOLDE IX, June 2002
• SOLA e-mail sola_at_bobcat.ent.ohiou.edu
• EURO Working Group on Locational Analysis
• http//www.vub.ac.be/EWGLA/homepage.htm

Includes chapter on demand point aggregation.
12
Outline

• Location problems
• Location analysis
• Location model typology
• References
• MIP location models, MP aggregation
• Demand point modeling, aggregation
• Common DP aggregation approach

• Law of diminishing returns
• Aggregation error measures
• SAND location models
• Error bounds
• Paradox of aggregation
• Overview some aggregation algorithms
• Example aggregations
• Conclusions

13
Demand point (DP) modeling background

• Many location problems deal with locating
  facilities with respect to demand points.
• In urban settings, there can be more than 100,000
  demand points.
• Demand point data is often readily available
  CD-ROM phone books, GIS address matching, U.S.
  post Office Delivery Point Validation (DPV) data
  base with 145 million addresses many commercial
  suppliers.

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DP aggregation benefits costs

• aggregation reduces
• data collection cost
• statistical uncertainty
• modeling cost
• computing cost
• confidentiality concerns

• aggregation increases
• modeling error

our focus
15
DP aggregation Basic question

• How do we aggregate DPs to keep the modeling
  error low, yet have a tractable model?

16
Common aggregation modeling approach used in
practice

• Replace every DP in each postal code region/zip
  code area by the centroid of the region.
• This is inexpensive, but may cause a large
  aggregation modeling error.

17
Basic aggregation idea

• p1 p2
  p1001
• p3 p1002
  
• 
  
• c1
  c2
• 
  p2000
• 
  p1999
• p999 p1000

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Basic aggregation idea

• Choose some single point in each region,
  aggregate every demand point in the region into
  this single point replace each pi by pi'.
• Example
• pi' c1, i 1, , 1000
• pi' c2, i 1001, , 2000.
• c1, c2 are aggregate demand points (ADPs).

Note the pi are distinct, but the pi' are not.
19
Fundamental Aggregation Insight Law of
Diminishing Returns (LDR)

• some aggregation
• error measure
• 
  aggregate DPs

costly choice
bad choice
better choice
20
Law of diminishing returns

• Our experience with DP aggregation is that this
  well-known law usually applies, and is
  practically important.
• Too few ADPs give a high error many ADPs may not
  accomplish much more than somewhat less.

21
Numerical example LDR for covering problem,
using RC-Cen (explained later)
22
Aggregation decisions to make

• (D-1) The number of ADPs
• (D-2) The locations of the ADPs
• (D-3) The replacement rule replace each pi by
  some pi'.
• Choosing ADPs is itself a location problem. DP
  aggregation is a kind of second-order location
  problem.

23
Location model notation

• d(x,y) some distance/metric (e.g.,shortest path,
  Euclidean, rectilinear)
• X x1, , xn collection of n facilities to
  locate
• P (p1, , pm) vector of DPs
• P' (p1', , pm') vector of ADPs
• M 1, , m DP index set
• Each pi is aggregated into (replaced by) pi'.

Commonly p is used here, but we use p for a
demand point.
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Notation comment

• We have
• P (p1, , pm) vector of DPs
• P' (p1', , pm') vector of ADPs
• Before solving an aggregated problem, we use the
  fact that the ADPs are not distinct by combining
  terms and doing away with redundancies. For
  analytical purposes, it is useful to refer to ADP
  pi' for each DP pi.

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Location model notation

• D(X,pi) distance between DP pi and a closest new
  facility in X.
• D(X,pi') distance between ADP pi' and a closest
  new facility in X.
• D(X,P) (D(X,pi)), D(X,P') (D(X,pi '))
  corresponding m-vectors of all closest distances

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Location model notation

• g a costing function that maps each of D(X,P),
  D(X,P') into a cost
• f(XP) g(D(X,P)) original model
• f(XP') g(D(X,P')) aggregated model
• Because the ADPs are not distinct, f(XP') has
  less distinct DPs than f(XP), and is a smaller
  model. Some algebraic steps are typically
  necessary to simplify f(XP').

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Examples of f(XP) g(D(X,P))

• D(X,P) (D(X,p1), , D(X,pm))
• g(Y) w1 y1 wm ym or
• maxw1 y1, , wm ym
• f(X) g(D(X,P)) w1 D(X,p1) wm D(X,pm)
• n-median model, X n
• f(X) g(D(X,P)) maxw1 D(X,p1), , wm
  D(X,pm) n-center model , X n

Use this as the Y vector in g(Y).
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Simplifying the agg. model f(XP')

• w1 D(X,p1') wm D(X,pm') aggregated
  n-median model
• maxw1 D(X,p1'), , wm D(X,pm') aggregated
  n-center model
• Extreme Examples if all pi' c,
  1) becomes
• W D(X,c), with W w1 wm.
• 2) becomes W D(X,c) with
• W maxw1, , wm.

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Outline

• Location problems
• Location analysis
• Location model typology
• References
• MIP location models, MP aggregation
• Demand point modeling, aggregation
• Common DP aggregation approach

• Law of diminishing returns
• Aggregation error measures
• SAND location models
• Error bounds
• Paradox of aggregation
• Overview some aggregation algorithms
• Example aggregations
• Conclusions

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Aggregation error

• There is no generally accepted way of measuring
  aggregation error.
• Note, however, that we have objective function
  value error unless f(XP) f(XP') for all X.

Here is an error, but not the kind we are
interested in.
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Various error measures

• ABC Error (Hillsman Rhoda, 1978) for n-median
  model.
• D(X,pi) D(X,pi') three cases depending on X,
  pi, pi'.
• This is a myopic error measure. The model
  objective structure is ignored.

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More error measures n-median model

• DP i Error
• ei(X) wi D(X,pi) wi D(X,pi'), i ? M.
• These errors can be negative or positive.
• Total Error
• e(X) e1(X) em(X) f(XP) f(X,P')
• Self-Canceling Error Because each DP i error
  ei(X) can be negative or positive, e(X) can be
  nearly zero.

33
Self-canceling error

• Very useful for n-median, and similar models.
  There is some experimental and theoretical
  evidence that centroids work well for such models
  if there are enough centroids.
• For other types of models, such as center and
  covering models, there is no self-canceling
  error.

34
More error measures any location model

• Absolute Error
• ae(X) f(XP) f(XP') for all X.
• The closer ae(X) is to zero, the better.
• Relative Error
• rel(X) ae(X)/f(XP), all X.
• Maximum Absolute Error
• mae maxae(X) X

This is difficult to compute if minimizing f(XP)
is NP-hard.
35
Logical difficulty computing error measures

• Error measures logically involve both f(XP) and
  f(XP'). However, we have to aggregate P into P'
  because f(XP) is difficult to compute for many
  choices of X.
• Thus it may be difficult to compute error measure
  values for many choices of X.

36
Error bounds (EBs)

• Recall
• mae maxae(X) X
• maxf(XP) f(XP') X.
• An error bound is a number eb for which mae
  eb.
• That is,
• f(XP) f(XP') eb for all X.
• A small eb value gives a small mae value.

37
Mathematical programming results due to Geoffrion
(1977)

• Suppose f(XP) f(XP') eb for all X ? S.
  Let X solve minf(XP)X ? S
• let X' solve minf(XP')X ? S. We have
• f(XP) f(X'P') eb
• f(X'P) f(XP) 2 eb.

38
Zemel early asymptotic work

• Zemel (1985) gives error bounds for planar
  n-median and n-center problems with Euclidean
  distances. He was not interested in aggregation,
  but in finding asymptotically optimal solutions.
  
• For large n, and many DPs approximated by a
  planar set of area A, each minimal objective
  function is of the form k v(A/n), for a given
  positive constant k.

We call this a square root formula.
39
Square root formulas, theoretical basis for
LDR Zemel, Francis Rayco (96)
40
Later error bound work
Closely related work Carrizosa, E., H. W.
Hamacher, S. Nickel and R. Klein, 2000

• Francis and Lowe, 1992. Explicit aggregation
  error bounds for n-median, n-center, covering
  problems, with any distance. (At the time, they
  were unaware of Zemels work.)
• Francis, Lowe and Tamir, 2000. Error bounds for
  any location model of the form
• f(XP) g(D(X,P)), when g is subadditive and
  nondecreasing, i.e., SAND.

vector of closest distances
41
SAND model error bounds (g is SAND)

• f(XP) g(D(X,P)), f(XP') g(D(X,P')).
• T(P,P') (d(p1,p1'), , d(pm,pm')), the vector
  of distances between DPs and ADPs.
• eb g(T(P,P')). f(XP) f(XP') eb for all
  X.
• This error bound is nonnegative, is nondecreasing
  in the distances between each pi and pi', and
  (when g(0) 0) is zero if and only if there is
  no aggregation (pi pi' for all i). It applies
  to a large class of models.

42
Underlying error bound basis
triangle inequality for distance

• Assume the distance d satisfies the triangle
  inequality and symmetry
• d(x,p) d(x,p') d(p',p)
• d(x,p') d(x,p) d(p,p').
• Thus
• d(p,p') d(x,p') d(x,p) d(p,p') ?
• d(x,p') d(x,p) d(p',p).

an eb
43
Example error bounds

• n-center model maxwi d(pi,pi') i ? M
• n-median model ?wi d(pi,pi') i ? M
• Basic DP aggregation insight each DP should have
  a nearby ADP.

44
Covering example error bounds can apply to
constraints also

• Original problem
• min X s. to f(XP) 1, X ? S
• f(XP) maxD(X,pi)/ri i ? M.
• Aggregated problem
• min X s. to f(XP') 1, X ? S
• f(XP') maxD(X,pi')/ri i ? M.
• We know f(XP) f(XP') eb for all X ? S,
  with

n-center structure

• eb maxd(pi,pi')/ri i ? M.

45
Reformulated MIP versions of models, error bounds

• Many of the models of interest can be
  reformulated as MIPs, using various finite
  dominating set principles, to obtain numerical
  solutions.
• If an error bound applies to the original model,
  it also applies to the reformulated MIP model.

46
Using error bounds for aggregation

• Basic Idea Aggregate to make the error bound
  small.

We want a small error.
47
Ideal aggregation approach center problem

• Minimize the error bound
• Find Q to min Q, Q q eb(QP)
• i.e., min Q, Q q maxD(Q,pi) i 1, ,
  m.
• Think of Q as the set of q (distinct) ADPs pi'
  is the closest ADP in Q to pi for each DP i.
• Paradox of aggregation the latter problem is
  an NP-Hard q-center problem.

48
Avoiding the paradox

• Use a low-order heuristic algorithm to
• min Q maxD(Q,pi) i 1, , m approximately.
• For network problems, instead of using
  shortest-path/network distances, use simpler
  Euclidean or rectilinear distances.
• A similar paradox, and way around it, occur with
  the n-median, and the covering location models.

49
Conceptual aggregation algorithm

• Problem Inputs
• Location Model
• Aggregation Algorithm
• Outputs
• Solution to Aggregated Model

error feedback loop
50
Evaluation criteria DP aggregation algorithms

• EC-1 aggregation error
• EC-2 cost to a) get DP data, b) develop, run
  algorithm, c) solve aggregated model
• EC-3 ease of explanation
• EC-4 problem structure exploitation
• EC-5 robustness/flexibility
• EC-6 GIS implementability
• Tradeoffs EC-1 2 EC-1 3 EC-4 5

51
Reminder planar distances

• Y
• c
  X (x1,x2), Y (y1,y2)
• X b
• a
• a x1-y1, b x2-y2, c v(a2
  b2)
• Euclidean d(X,Y) c,
• Rectilinear d(X,Y) a b
• Tchebyshev d(X,Y) maxa, b

52
Assumed underlying problem structure, aggregation
algorithms

• Large metropolitan areas with a rectilinear
  street structure.

53
Some row-column aggregation algorithms

• RC-Med planar n-median problem, rectilinear
  distances. Francis, Lowe and Rayco (1996)
• RC-Cen planar n-center problem, rectilinear
  distances. Rayco, Francis and Lowe (1997)
• RC-Cov planar covering location problem,
  rectilinear distances. Emir-Farinas and Francis
  (2002)

54
Basic idea row-column aggregation

• Imagine we impose a grid on the DP data.
• Project data onto x-axis, solve a related
  location problem on x-axis to adjust column
  spacing.
• Project data onto y-axis, solve a related
  location problem on y-axis to adjust row spacing.
• Solve a related location problem for each cell
  and use solution as the ADP for the cell.

55
Basic idea row-column aggregation

• Imagine we impose a grid on the DP data.
• Project data onto x-axis, solve a related
  location problem on x-axis to adjust column
  spacing.
• Project data onto y-axis, solve a related
  location problem on y-axis to adjust row spacing.
• Solve a related location problem for each cell
  and use solution as the ADP for the cell.

56
Basic idea row-column aggregation

• Imagine we impose a grid on the DP data.
• Project data onto x-axis, solve a related
  location problem on x-axis to adjust column
  spacing.
• Project data onto y-axis, solve a related
  location problem on y-axis to adjust row spacing.
• Solve a related location problem for each cell
  and use solution as the ADP for the cell.

57
Basic idea row-column aggregation

• Imagine we impose a grid on the DP data.
• Project data onto x-axis, solve a related
  location problem on x-axis to adjust column
  spacing.
• Project data onto y-axis, solve a related
  location problem on y-axis to adjust row spacing.
• Solve a related location problem for each cell
  and use solution as the ADP for the cell.

58
Basic idea row-column aggregation

• Imagine we impose a grid on the DP data.
• Project data onto x-axis, solve a related
  location problem on x-axis to adjust column
  spacing.
• Project data onto y-axis, solve a related
  location problem on y-axis to adjust row spacing.
• Solve a related location problem for each cell
  and use solution as the ADP for the cell.

59
RC-Med for planar n-median problem, with r rows,
c columns

• Project all DP data onto x-axis.
• Solve c-median problem on x-axis to adjust
  column spacing.
• Project all DP data onto y-axis.
• Solve r-median problem on y-axis to adjust row
  spacing.
• Use rows, columns to create a grid.
• Choose as ADP in each cell the 1-median for that
  cell.
• Aggregate all DPs in each cell to the 1-median.

60
RC-Cen for planar n-center problem, with r rows,
c columns

• Apply a 45 degree rotation to all the DP data.
• Project DP data onto x-axis.
• Solve c-center problem on x-axis to adjust
  column spacing.
• Project DP data onto y-axis.
• Solve r-center problem on y-axis to adjust row
  spacing.
• Use rows, columns to create a grid.
• Choose as ADP in each cell the (Tchebyshev)
  1-center for that cell. Aggregate all DPs in
  each cell to the 1-center.
• Apply the inverse 45-degree rotation to the grid
  to get the aggregation.

Given a 45 degree rotation, the rectilinear
distance between any two points equals the
Tchebyshev distance between the transformed
points.
61
RC-Cov for planar covering problem

• Apply a 45 degree rotation to all the DP data.
• Project DP data onto x-axis.
• Pick a small covering radius rs and solve a
  covering problem on the x-axis to get c centers.
  Use the c centers to adjust" the grid column
  spacing.
• Project DP data onto y-axis.
• Use the small covering radius rs and solve a
  covering problem on the y-axis to get r centers.
  Use the r centers to adjust the grid row
  spacing.
• Use rows, columns to create a grid.
• Choose as an ADP in each cell the (Tchebyshev)
  1-center for that cell. Aggregate all DPs in
  each cell to the 1-center.
• Apply the inverse 45-degree rotation to the grid
  to get the aggregation.

62
Computational Order RC Methods

• With m DPs, ordering the projected DPs on each
  axis takes O(m log m).
• With t r or c, a t-median, t-center, or
  covering problem - on the line - is solved on
  each axis. The order depends on the method used,
  but O(m log m) or less is typical.
• There are r c cells, with one ADP per (nonempty)
  cell. Solving the location problem in each cell
  is linear order in the number of DPs in the cell.
  This is basically O(r c).
• Typically O(m log m) is dominant, since m gtgt r c.

63
D-F Another aggregation method center, covering
problems

• D-F For q ADPs, use the Dyer-Frieze q-center
  algorithm, Dyer Frieze, 1985, with the m
  original DPs.
• It works nicely for network or for planar
  problems, but is O(q m) O(q2).

64
Outline

• Location problems
• Location analysis
• Location model typology
• References
• MIP location models, MP aggregation
• Demand point modeling, aggregation
• Common DP aggregation approach

• Law of diminishing returns
• Aggregation error measures
• SAND location models
• Error bounds
• Paradox of aggregation
• Overview some aggregation algorithms
• Example aggregations
• Conclusions

65
Application of D-F 6600 DPs, 198 ADPs,
aggregation for rectilinear distance covering
location problem Alachua Co., Florida
66
8-cover of DPs using ADPs of last slide,
rectilinear distances
A 3-center based on this approach was implemented.
67
Application of RC-Med LDR with sample average
absolute error
68
Application of RC-Med LDR with sample average
relative error
69
Example LDR for covering problem, using RC-Cen
70
Example aggregation for a covering problem

• The next slide shows an aggregation produced with
  RC-Cov. A restriction of the aggregated covering
  problem provided a provably optimal solution to
  the original problem with 69,960 DPs. The DP
  data is from Palm Beach County, Florida.

71
Application of RC-Cov 69,960 DPs (blue), 703
ADPs (yellow)
Lake Okeechobee
Atlantic Ocean
72
From RC-Cov lower, upper bounds on covering ofv
for 69,960 DP problem
73
14-cover of aggregated covering problem with
smallest upper bound
74
From RC-Cov actual aggregation eb versus square
root model prediction
75
Summary

• Location problems
• Location analysis
• Location model typology
• References
• MIP location models, MP aggregation
• Demand point modeling, aggregation
• Common DP aggregation approach

• Law of diminishing returns
• Aggregation error measures
• SAND location models
• Error bounds
• Paradox of aggregation
• Overview some aggregation algorithms
• Example aggregations

76
Questions?