Demand Point Aggregation for a Class of Minimax Location Models

1
Demand Point Aggregation for a Class of Minimax
Location Models

• R. L. Francis University of Florida
• T. J. Lowe Iowa University
• A. Tamir Tel-Aviv University
• H. Emir-Farinas University of Florida

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Overview ( 1) 6) background )

• Some minimax location models
• Demand point aggregation
• Aggregating for small error
• Error bounds
• Covering location model, error bound
• Ideal aggregation, related paradox
• Two idealized, related aggregation models
• Partitioning demand points by communities

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Overview Continued

• Error bound threshold, ß
• ß-separate communities
• Double-duty ADPs for communities
• Two square root models for initial first-cut
  approximate solutions
• Optimal aggregations, examples, law of
  diminishing returns (LDR).
• Using models for aggregation decomposition.

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Example N-center Location Model

• d(x,y) any metric (shortest path,
  Euclidean, rectilinear)
• M 1, , m DP index set
• P pi i ? M DP collection
• X x1, , xN an N-center
• D(X,pi) mind(xj,pi) j 1, , N
• f(XP) maxD(X,pi) i ? M
• N-center problem min X f(XP)

y
x
v
Demand point
NP-hard
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Idea Demand Pt. Aggregation Replace each DP pi
by some pi '

• pi ' c, i 1, , 50
• p1 p20
• p21 c p38
• p39 p50

Example we replace all the households in a
postal code area by the area centroid.
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Notation Aggregate DPs

• P' (p1', , pm') list or vector of aggregate
  demand points. For each i, pi' replaces pi. The
  pi are distinct, but the pi' are not distinct.
• f(XP') or f '(X) resulting aggregated location
  model (of reduced size), when for each i, pi'
  replaces pi.

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Aggregated N-center Model

• f(XP ') maxD(X,pi') i ? M

Nearest distance

• We can delete duplicate nearest-distances in the
  max expression obtain a smaller, more tractable
  model.
• This model is also less accurate than the
  original one.

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Motivating Research Question

• How do we aggregate DPs to get a smaller, more
  tractable problem, while at the same time keeping
  the resulting error small?
• Alternatively, how much error is acceptable in an
  aggregation?

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Location Model Error Bound

• The absolute error at X for location model f(XP)
  and aggregate model f(XP') is defined as
  f(XP) f (XP').
• An error bound for the model is a number, say
  eb(P'P), so that
  
• 
  f(XP) f(XP') eb(P'P) for
  all X.

P' is a good aggregation if this error bound is
small.
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Known N-center Error Bound

• eb(P' P) maxd(pi',pi) i ? M
• Recall the ADPs are not distinct. Let Q denote
  the set of all distinct ADPs.
• If pi' is a closest ADP in Q to pi, then
• eb(P' P) maxD(Q,pi) i ? M eb(QP)
• eb(QP) maxD(Q,pi) i ? M.

Nearest distance
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Other Models, Same Error Bound
Closely related work Carrizosa, E., H. W.
Hamacher, S. Nickel and R. Klein, 2000

• Conditional N-center model
• Obnoxious facility model
• Multifacility minimax model
• Multistop model with probability parameter
• Covering location model (constraint error bound)
• (Francis, R. L., T. J. Lowe and A. Tamir, On
  Aggregation Error Bounds for a Class of Location
  Models, Operations Research, 48, 2, 294-307,
  2000.)
• It can be shown that twice the N-center error
  bound also applies to the unweighted p-center hub
  location problem.

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Covering Location Model Constraint Aggregation

• Min X s. to D(X,pi) r, i ? M
• Or, if
• f(XP) (1/r)maxD(X,pi)i ? M
• Min X s. to f(XP) 1.

A scaled N-center model
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Aggregated Covering Model
Many redundant constraints we can omit.

• min X s. to D(X,pi') r, i ? M
• Or, with
  f(XP') (1/r) maxD(X,pi') i ? M
• min X s. to f(XP') 1.
• Note f(XP') is a (scaled) N-center model.
• Hence f(XP) f(XP') eb(P 'P), all X with
  
  eb(P 'P)
  (1/r) maxd(pi',pi)i ? M

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Covering Error Bound

• Conclusion
• eb(P 'P) (1/r) maxd(pi',pi) i ? M
• is an error bound for constraint aggregation
  for the covering model. Except for the scaling
  factor, 1/r, its the same eb as for the N-center
  problem.

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Covering EB A Constraint Penalty Viewpoint

• For the covering constraints, define
• Pen(X) (1/r) max maxD(X,pi) 1, 0 i ?
  M,
• Pen '(X) (1/r) max maxD(X,pi ') 1, 0 i
  ? M.
• It can be shown that
• Pen '(X) Pen(X) eb(P 'P) for
  all X.

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Ideal Aggregation Approach Find n ADPs to
minimize the error bound

• Find Q to min Q, Q n eb(QP).
• That is, solve min Q, Q n maxD(Q,pi) i ?
  M.
• Paradox of Aggregation (Francis Lowe, 92) the
  latter problem is an n-center problem, with all
  the structure of the original N-center model. If
  we must aggregate to solve the N-center model,
  then we cant minimize eb(QP) !

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Getting Around the Paradox

• Use a low-order heuristic algorithm to min
  Q maxD(Q,pi) i ? M approximately.
• Instead of using shortest-path/network distances,
  use simpler Euclidean or rectilinear distances,
  and consider approximate solution methods.

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Two Closely Related Models (New Material Now
Begins)

• First Idealized Model (n-Center)
• (Pcen) min Q eb(QP) s. to Q n
• Second Idealized Model (Covering)
• (Pcov) Min Q Q s. to eb(QP) b

find n ADPs to minimize the eb
minimize ADPs needed for an eb of at most b
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Research Question

• How can we exploit geographic structure to
  decompose the above n-center and covering
  aggregation problems?

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Assumption There are K DP Communities

• (P1, ,Pk, , PK) is a partition of P, with Pk
  denoting the DPs in community k.
• (P1 ', ,Pk ', , PK ') is the corresponding
  partition of P'.
• If eb(Pk'Pk) maxd(pi',pi) pi ? Pk, then
  eb(P'P) maxeb(Pk'Pk)k.

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Error Bound Threshold, ß

• We are given some positive number, ß. Any
  aggregation with an error bound value greater
  than ß is not acceptable.
• For example, maybe ß 10 km. Any aggregation
  with an error bound value of more than 10 km is
  not acceptable.

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Error Bound Threshold, ß

• The analyst must choose ß. This may require some
  thought. What is the maximum acceptable error?

23
ß-Separate Communities

• Define any 2 communities, say Ps and Pt, to be
  ß-separate if for every DP ps ? Ps and pt ?
  Pt, d(ps,pt) gt 2 ß.
• For example, if ß 10 km, then Kaiserslautern
  and Frankfurt are 10-separate, since the distance
  between them exceeds 20 km.

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Double-duty ADPs

• An ADP does double duty if it is an ADP for DPs
  in two distinct communities.
• For example, suppose two counties are adjacent.
  Each county defines a community. Two demand
  points, one in each county, close to each other,
  might be aggregated into one ADP. This ADP does
  double duty.

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Double Duty, ß-Separateness

• Suppose 2 communities, Ps and Pt, are ß-separate.
  If an ADP p' does double duty for any DP ps ? Ps
  and any DP pt ? Pt, then
• maxd(ps,p'), d(pt,p') gt ß
• gt2 ß
• ps p'
  pt
• gtß

use triangle inequality
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Conclusions Double-Duty ADPs

• If any two communities are ß-separate, and some
  ADP in P' does double duty for both, then
  eb(P'P) gt ß.
• We will not accept any aggregation with error
  bound more than ß.
• To get an acceptable aggregation we cannot have
  any ADP doing double duty for any two ß-separate
  communities.

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Conclusions Double-Duty ADPs

• To get an acceptable aggregation we cannot have
  any ADP doing double duty for any two ß-separate
  communities.
• Assuming every pair of communities is ß-separate,
  then no ADP can do double duty.
• Thus if we add the numbers of ADPs for all
  communities we get the total number.

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ß Separateness Assumption

• For the following two models, every pair of
  communities is ß-separate. Also, b ß.
• (Pcen) min Q eb(QP) s. to Q n
• (Pcov) min Q Q s. to eb(QP) b

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Exploiting ß-Separateness Model Decomposition

• Let Qk denote the set of all distinct ADPs for
  community k, nk Qk, n Q.
• With eb(QkPk) maxD(Qk,pi) pi ? Pk,
• eb(QP) maxeb(QkPk) k
• n Q means S k nk n

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Exploiting ß-separateness

• We rewrite min Q eb(QP) s. to Q n
• (Pcen)
• min max k maxD(Qk,pi) pi ? Pk
• s. to
• S k nk n, nk 0, all k

nk Qk n Q
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Exploiting ß-separateness

• We rewrite min Q Q s. to eb(QP) b
• (Pcov)
• min S k nk
• s. to
• max k maxD(Qk,pi) pi ? Pk b
• nk 0, all k

This problem now decomposes into K independent
problems.
nk Qk n Q
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Key Insight, and Simplification

• The expression
• maxD(Qk,pi) pi ? Pk
• is an nk-center model, Qk nk.
• Assumption. Each community k has area Ak, we
  use Euclidean or rectilinear distances, we
  replace the above expression by a known simple
  asymptotic approximation (accurate for large
  nk) for the n-center model.

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Approximation, maxD(Qk,pi) pi ? Pk

• Euclidean distances (due to Zemel, 85)
• Ck v(Ak/nk) with Ck 0.6204
• Rectilinear distances (Francis and Rayco, 96)
• Ck v(Ak/nk) with Ck 0.7071 1/v2

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Actual Aggregation EB versus Square Root
Approximation
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Actual Aggregation EB versus Square Root
Approximation
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Actual Aggregation EB versus Square Root
Approximation
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Resulting Square Root Models - for initial
first-cut approximate solutions
An alternative to using professional judgement.

• (Pcen)
• min zcen max k Ck v(Ak/nk)
• s. to
• S k nk n, nk 0, all k

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Resulting Square Root Models

• (Pcov)
• min zcov S k nk
• s. to
• max k Ck v(Ak/nk) b
• nk 0, all k

Equivalent to simple LP lower bounds on variables.
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Optimal (non-integer) Solutions

• (Pcen) Define A S k Ck2 Ak.
• Take
• nk (Ck2 Ak/A) n, all k,
• zcen v(A/n)

Obeys Law of Diminishing Returns in n
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Optimal (non-integer) Solutions

• (Pcov) Define A S k Ck2 Ak.
• Take
• nk (Ck/b)2 Ak, all k,
• zcov A/b2

Obeys Law of Diminishing Returns in b
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Conclusions Square Root Models

• Models of this form are very well solved (in
  closed form).
• Omitting the integrality condition has little
  effect, since the nk are not small integers.
• Optimal integer solutions are easily available if
  really necessary.
• Optimal solutions give all communities the same
  error bound value (equity).
• We believe these models can be useful in the
  initial stages of an aggregation study.

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Law of Diminishing Returns

• (Pcov) obeys the law of diminishing returns in b.
  We have zcov A/b2.
• (Pcen) obeys the law of diminishing returns in n.
  We have zcen v(A/n).

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Law of Diminishing Returns (LDR)

• error bound zcen
• 
  aggregate DPs

costly choice
bad choice
better choice
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Example (hh means household)
1 sq. mile 2.60 sq. km.
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Combining Two Counties

• We take ß 1 mile (1.61 km).
• Johnson and Linn counties are adjacent, and not
  ß-separate. We treat the two counties as a
  single community, of area
  1,331 614 717 square miles.
• We treat Dubuque and Polk Counties as the other
  two (ß-separate) communities.

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ß-separate communities
1 sq. mile 2.60 sq. km.
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Solution to (Pcen) n 5,000 ADPs

• (n1,n2,n3) (1212, 2654, 1134)
• Zcen 0.50 miles (with rectilinear distances)
• Zcen 0.44 miles (with Euclidean distances)

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Solution to (Pcov) b 0.5 miles

• (n1,n2,n3) (1216, 2662, 1138)
• Zcov 5016 ADPs
• (with rectilinear distances)

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Example LDR for (Pcov)

• b 10 miles, Zcov 14 ADPs
• b 5 miles, Zcov 52 ADPs
• b 1 mile, Zcov 1,255 ADPs
• b 0.5 miles, Zcov 5,016 ADPs
• b 0.1 miles, Zcov 125,400 ADPs

WOW!
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Problem Decomposition

• Either model helps with decomposition of the
  aggregation problem. This allows us to aggregate
  larger problems than otherwise.
• We can estimate the number of ADPs for each
  community, then use a DP aggregation algorithm to
  aggregate for each.
• The resulting ADPs can then all be put into the
  aggregated location model.

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Any Questions?
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53
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