Abstract Location problems occurring in urban or regional settings may involve many tensof thousands of “demand points,” usually individual private residences. In modeling suchproblems it is common to aggregate demand points to obtain tractable models. We surveyaggregation approaches to a large class of location models, consider and compare variousaggregation error measures, identify some effective (and ineffective) aggregation error mea-sures, and discuss some open research areas.
Many location problems involve locating services (called servers) with respect to customersof some sort (called demand points, and abbreviated as DPs). Usually there is travel betweenservers and DPs, so that travel distance, or (more generally) travel cost, is of interest. Loca-tion models represent this travel cost, and solutions to the models provide locations of theservers of (nearly) minimal cost. For books on location models and modeling, see Daskin
R.L. Francis (�)Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USAe-mail: [email protected]
T.J. LoweTippie College of Business, University of Iowa, Iowa City, IA 52242, USAe-mail: [email protected]
M.B. RaycoModular Mining Systems, Inc., 3289 E. Hemisphere Loop, Tucson, AZ 85706, USAe-mail: [email protected]
A. TamirDepartment of Statistics & Operations Research, School of Mathematical Sciences, Tel-AvivUniversity, Tel-Aviv 69978, Israele-mail: [email protected]
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(1995), Drezner (1995a), Drezner and Hamacher (2002), Francis et al. (1992), Handler andMirchandani (1979), Love et al. (1988), Mirchandani and Francis (1990), and Nickel andPuerto (2005).
A common difficulty with modeling location problems that occur in urban or regionalareas is that the number of DPs may be quite large, since each private residence might bea DP. In this case it may be impossible, and also unnecessary, to include every DP in thecorresponding model. Further, the models may be NP-hard to optimize (Kariv and Hakimi1979). For problems as diverse as ones including the location of branch banks (Chelst et al.1988), tax offices (Domich et al. 1991), network traffic flow (Sheffi 1985), and vehicle ex-haust emission inspection stations (Francis and Lowe 1992) a popular aggregation approachis used: to suppose every DP in each postal code area or zone of the larger urban area isat the centroid of the postal code area or zone, and to compute distances accordingly. Theresult is a smaller model to deal with, but one with an intrinsic error. If the modeler wishes toaggregate to have both a small number of aggregate demand points (abbreviated as ADPs),and also a small error, then how to aggregate becomes a nontrivial matter.
For earlier reviews of aggregation for location models, see Francis et al. (1999) and Fran-cis et al. (2002b). For a general review of aggregation and disaggregation in optimization,see Rogers et al. (1991). For Ph. D. dissertations on DP aggregation, see Rayco (1996), Zhao(1996), and Emir-Farinas (2002). For an overview of location theory research see Hale andMoberg (2003). The literature on DP aggregation has been growing more rapidly in recentyears. Of the DP aggregation journal publications we reference, about 65% have appearedafter 1995.
It is important to note that DP data is becoming widely available (although at some cost).For example, a well-known CD-ROM telephone book for the USA includes the latitude andlongitude for every street address having a street number. Latitudes and longitudes are foundusing the Geographic Information System capability called address matching.
It is tempting to ask the following question: How many ADPs are enough? There areno general answers to this question. This is because there are important tradeoffs in doingaggregation. Aggregation often decreases: (1) data collection cost, (2) modeling cost, (3)computing cost, (4) confidentiality concerns and (5) data statistical uncertainty. The firstfour items seem self-explanatory; item (5) occurs because aggregation leads to pooled data,which provides larger samples and thus smaller sample standard deviations. The price paidfor aggregation is the increase in model error: instead of working with the actual locationmodel we work with some approximating location model. How to trade off the benefits andcosts of aggregation is still an open question. The question is open in part because thereis no general agreement on how to measure error, and also because there is no acceptedway to attach a cost to model aggregation error. As best we know, professional judgment isgenerally used to do the tradeoffs.
One can categorize location models as strategic, tactical, or operational in scope. Aspointed out by Bender et al. (2001), planar distances are often used for strategic-level loca-tion models, and network distances for tactical-level location models. Such models are oftenconverted to equivalent mixed integer programming (MIP) models for solution purposes, us-ing some finite dominating set principle to reduce the set of possible locations of interest toa finite set (Hooker et al. 1991). Thus results to follow for these planar and network mod-els also apply to their MIP representations, including those for the covering, p-center, andp-median location models. Operational-level location models are not too common (mobileservers are one example), but for such models no aggregation may be best. Note that thescope of the location model may well indicate the degree of aggregation; a more detailedscope requires a more detailed aggregation.
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Consider now some terminology and several aggregation examples. We suppose thatservers and DPs are either all points in the plane, or on some travel network. In either case,there is some well-defined set, say S, and a distance d(x, y) between any two points x, y
in S. If S is a travel network (assumed undirected) then d(x, y) is usually the length ofa shortest path between x and y. For planar problems when S = R2, with x = (χ1, χ2),y = (ψ1,ψ2), d(x, y) is often the �p-distance, ‖x − y‖p = [|χ1 − ψ1|p + |χ2 − ψ2|p]1/p ,with p ≥ 1. Taking p = 1 or 2 gives the well-known rectilinear or Euclidean distance, re-spectively. The limiting case of the �p-distance as p goes to infinity, denoted by ‖x − y‖∞,‖x − y‖∞ = max{|χ1 − ψ1|, |χ2 − ψ2|}, is called the Tchebyshev distance. The Tchebyshevdistance in R2 is sometimes analytically convenient because it is known (Francis et al. 1992)to be equivalent to the planar rectilinear distance under a 45-degree rotation of the axes. Wedefine the diameter of S by diam(S) = sup{d(x, y) : x, y ∈ S}, with the understanding thatpossibly diam(S) = +∞. More generally, S can be a metric space (Goldberg 1976) withmetric d , but no loss of insight occurs by considering the network and planar cases for S.
Suppose we have n DPs, aj ∈ S, j = 1, . . . , n. Denote the list (or vector) of DPs by A =(a1, . . . , an). When we aggregate, we replace each DP aj by some ADP a′
j in S, obtainingan ADP list A′ = (a′
1, . . . , a′n). While the DPs are usually distinct, the ADPs are not, since
otherwise there is no computational advantage to the aggregation. When we wish to modelm distinct ADPs, we let Á (upper case Alpha) denote the set of m distinct ADPs, say Á ={α1, . . . , αm}. We use the former (latter) ADP notation when the correspondence betweenDPs and ADPs is (is not) of interest. Usually we have m � n.
For any positive integer p, let X = {xk : k = 1, . . . , p} denote any p-server, the set oflocations of the p servers, X ⊂ S. (This symbol p is a different symbol from the one definingthe �p-distance.) Denote the location model with the given original DPs by f (X : A), andthe one with the aggregate DPs by f (X : A′). (The notation f (X : A) and f (X : A′) capturesa key idea that an aggregation is a replacement of A by A′, with the entries of A′ not alldistinct.)
For the large class of location models with similar or indistinguishable servers, with onlythe closest one to each DP assumed to serve the DP, for any p-server X ⊂ S and DP a ∈ S
we denote by D(X,a) ≡ min{d(xk, a) : k = 1, . . . , p} the distance between a and a clos-est element in X. Then define closest-distance vectors D(X,A) ≡ (D(X,aj )),D(X,A′) ≡(D(X,a′
j )) ∈ Rn+. Suppose g is some “costing” function with domain Rn+ attaching a cost toD(X,A) and D(X,A′). This gives original and approximating location models f (X : A) ≡g(D(X,A)) and f (X : A′) ≡ g(D(X,A′)), respectively. Important and perhaps best-knowninstances of g are the p-median and p-center costing functions, g(U) = w1u1 +· · ·+wnun,and g(U) = max{w1u1, . . . ,wnun} respectively; the wj are positive constants, often called“weights”, and may be proportional to the number of trips between servers and DPs. Thusf (X : A) is either the p-median model, w1D(X,a1) + · · · + wnD(X,an), or the p-centermodel, max{w1D(X,a1), . . . ,wnD(X,an)}. These models originate with Hakimi (1965)(each is called unweighted if all wj = 1). They are perhaps the two best-known modelsin location theory. Subsequently, we refer to the p-center, p-median, and covering loca-tion model as PCM, PMM, and CLM, respectively. These models are NP-hard to minimize(Kariv and Hakimi 1979; Megiddo and Supowit 1984).
Consider several aggregation examples which serve to illustrate our notation and basicaggregation ideas. Let N = {1, . . . , n} denote the set of all DP indices. We suppose, forexample, that the DPs will be aggregated into two postal code area centroids. Let Ni denotethe subset of indices of the DPs in postal area i = 1,2. Let αi denote the centroid of postalarea i = 1,2. Clearly then, the Ni form a partition of N . To aggregate the DPs in the postalcode areas into the centroids we replace by αi each aj with j ∈ Ni , for i = 1,2. Thus a′
j = αi
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for j ∈ Ni and i = 1,2. Hence A′ is now the n-vector of ADPs, and Á = {α1, α2} is the ADPset.
Example aggregation (PMM) f (X : A) = ∑{wjD(X,aj ) : j ∈ N}. Let ω1 = ∑{wj :j ∈ N1}, ω2 = ∑{wj :j ∈ N2}. We then have f (X : A′) = ∑{wjD(X,a′
j ) : j ∈ N} =∑{wjD(X, α1) : j ∈ N1} + ∑{wjD(X, α2) : j ∈ N2} = ω1D(X, α1) + ω2D(X, α2). Thisexample illustrates how aggregation error can occur. If only p-servers are of interest (withp ≥ 2), then taking X to be {α1, α2} minimizes f (X : A′) with minimal value of 0, giving auseless underestimation of min{f (X : A) : X}.
If there is only one server, X = {x}, and the �p-distance is used, then it is known thatthis 1-median under-approximation can hold for all x. If we let ω = ∑{wj : j ∈ N}, andlet α = ∑{(wj/ω)aj : j ∈ N} be the centroid of the DPs, so that f (x : A′) = ω‖x − α‖p ,it is known (Francis and White 1974) that f (x : A) ≥ f (x : A′) for all x. This is an impor-tant reason why underestimation can occur for PMM aggregation when few centroid ADPsare used. It is also known that for �p distances Plastria (2001) the difference f (x : A) −f (x : A′) goes to zero as x gets farther from α along an infinite ray with one end point at α.There are good theoretical reasons due to self-canceling error (Plastria 2000, 2001; Franciset al. 2003) for using centroids as ADPs for the PMM, but none that we know of for the PCMand CLM. Indeed, better choices than centroids are available for the latter two models.
Example aggregation (PCM) f (X : A) = max{wjD(X,aj ) : j ∈ N}. Let w+1 = max{wj :
j ∈ N1},w+2 = max{wj : j ∈ N2}. We then have f (X : A′) = max{wjD(X,a′
p-servers (p ≥ 2) are of interest, then taking X to be {α1, α2} minimizes f (X : A′) withminimal value of 0, giving an underestimate of f (X : A).
Example aggregation (CLM) minimize |X| subject to D(X,aj ) ≤ rj , j ∈ N,X ⊂ S. Allbut two covering constraints for the aggregated model are redundant. Define ρ1 = min{rj :j ∈ N1}, ρ2 = min{rj : j ∈ N2}. Thus the aggregated model has constraints D(X, α1) ≤ρ1,D(X, α2) ≤ ρ2,X ⊂ S. This means it takes at most two servers to solve the aggregatedmodel. CLMs and PCMs are known to be closely related (Kolen and Tamir 1990). We shallsee that aggregation results developed for one model often also apply to the other.
These examples illustrate two equivalent approaches for representing n DPs with anaggregation of m ADPs. Either we have a partition of the DP index set N into m setsN1, . . . ,Nm with one ADP per set, or for each aj there is a replacing ADP a′
j , with each
a′j in the set Á of m distinct ADPs. In either case, three aggregation decisions (Francis et al.
1999) must be made: (1) the number of ADPs, (2) the location of ADPs, (3) the replacementrule: for each aj , what is a′
j ? The (reasonable) replacement rule often used is to replace eachDP by a closest ADP. Further, for the aggregation to be computationally useful we requirethe number of ADPs, m, be less (usually much less) than the number of DPs, n; also itis reasonable to have p � m. These authors note that versions of these three aggregationdecisions occur in location modeling. Hence results in location theory help in doing DP ag-gregation, so DP aggregation is a sort of “second-order” location problem to solve prior tosolving the original or “first-order” problem.
These three examples may possibly also suggest that as more ADPs are used the aggre-gation error decreases—ideally, if we could use m = n ADPs, then we need have no DPaggregation error at all. In fact there are classes of location models where the law of dimin-ishing returns applies: aggregation error decreases at a decreasing rate as m increases. Thus
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a very small value of m may cause a very high aggregation error, while a large value of m
might give little less error than an appreciably smaller value of m.It is the purpose of this paper to survey aggregation approaches to a large class of lo-
cation models, consider and compare various error measures, identify some effective (andineffective) aggregation error measures, and discuss some areas where more work is neededto achieve agreement in order to further research in the area.
We now give an overview of our paper. In Sect. 2 we consider various approaches toaggregation error measurement. In Sect. 3 we provide a general literature discussion ofDP aggregation. In Sect. 4 we consider several specific aggregation topics in more detail,including limitations and interrelationships of various aggregation error measurements, andpresent some new results related to measuring location error.
2 Aggregation error measurements
While there can be other types of error in location models, the one we focus on is demandpoint aggregation error, which comes by replacing DPs by ADPs. Thus, instead of actualdistances we obtain approximating ones. The use of these approximating ADPs creates er-ror. It is thus important for the location modeler who does the aggregation to be aware ofthe aggregation error being created. The modeler who does DP aggregation intentionallyintroduces error into the model. The use of ADPs is the cause of the aggregation error, butthere are error effects—including inaccurate values of the objective function and of serverlocations, due to using the approximating distances. It is important to consider both causeand effects in order to get the whole picture. There are a number of ways to measure erroreffects; further, the magnitude of aggregation effects can depend on model structure—forthe same aggregation, some models can have more error than others. What is clear, in anycase, is that the way to minimize DP aggregation error is to not aggregate DPs—certainlythis is what we recommend when it is feasible. The ideal way to aggregate DP data is notto aggregate it.
If DP data must be aggregated, then we need to consider aggregation error measures. Welist and summarize ten such measures in Table 1. All these error measures have an ideal valueof zero. One simple way to measure aggregation error is to consider ADP-DP distances. Ifthese distance values are all zero then ADPs and DPs are identical, so there is no error. Laterwe establish a relationship between ADP-DP distances and other error measures, includingthe distance difference error. For the PMM, this distance difference error leads to an errorwe call DP error. Like the difference error, the DP error can be negative or positive. Stillconsidering the PMM, note that total DP error e(X) in Table 1 satisfies e(X) = f (X : A′)−f (X : A), the difference between the aggregated PMM and the original model. Even thoughno DP error is zero, total DP error can be zero or nearly zero, since negative errors cancancel out positive errors—this is self-canceling error. Unfortunately self-canceling erroronly applies to models with an additive cost structure.
Next, consider ABC errors for the PMM, due to Hillsman and Rhoda (1978). Note thatABC errors are sums of the DP errors which are organized by the ADPs. Suppose werepresent an aggregation by a partition of N = {1, . . . , n}, say N1, . . . ,Nm, such that fori = 1, . . . ,m, every DP aj with j ∈ Ni is aggregated into the ADP αi; that is, a′
j = αi for j ∈Ni . Thus
∑{wjD(X,aj ) : j ∈ Ni} is replaced in the aggregate model by∑{wjD(X, αi) :
j ∈ Ni} = ωiD(X, αi), where ωi ≡ ∑{wj : j ∈ Ni}. In the parlance of Hillsman and Rhoda,the ABC error illustrates their Source A error, which they define actually as ωiD(X, αi).Using ωiD(X, αi) instead of
∑{wjD(X,aj ) : j ∈ Ni} is a source of error. The special
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Table 1 Various demand point aggregation error measures for a location model f (X : A). Ideal error mea-sures have value zero for all j and all X
No. Error name Error definition
1 ADP-DP distances d(a′j, aj ), j ∈ N
2 Distance difference error D(X,a′j) − D(X,aj ), j ∈ N , all X
3 DP error, PMM ej (X) = wj [D(X,a′j) − D(X,aj )], j ∈ N , all X
4 Total DP error, PMM e(X) = ∑{ej (X) : j ∈ N}, all X
5 ABC error for PMM: N1, . . . ,Nm is abci (X) = ωiD(X, αi ) − ∑{wj D(X,aj ) : j ∈ Ni}a partition of N = {1, . . . , n}; for all X
ωi ≡ ∑{wj : j ∈ Ni } for i = 1, . . . ,m
6 Absolute error, any location model ae(X) = |f (X : A′) − f (X : A)|, all X
7 Relative error, for all X with f (X : A) > 0 rel(X) = ae(X)/f (X : A), all X
8 Maximum absolute error mae(f ′, f ) = max{ae(X) : X,X ⊂ S, |X| = p}9 Error bound eb a number eb with ae(X) ≤ eb for all X
Ratio error bounds |f (X : A′)/f (X : A) − 1| ≤ eb/f (X : A),
(when f (X : A),f (X : A′) > 0) |f (X : A)/f (X : A′) − 1| ≤ eb/f (X : A′) for all X
10 Location error some measure, diff(X′,X∗), of the
“difference” between p-servers X′ and X∗
case of Source A error when αi ∈ X, so that ωiD(X, αi) = 0, is their Source B error. IfωiD(X, αi) = 0, then it is useless as an estimate of
∑{wjD(X,aj ) : j ∈ Ni}. Source C er-ror is a related sort of allocation error involving closest-distance definitions. Suppose xk ∈ X
and is closest in X to αi; we might then assume that every aj ∈ Ni will be closest to xk .However, in reality, some aj ∈ Ni may be closer to another element of X than xk . In effect,we would allocate some DPs to a wrong server location that is not closest to them. Noteabci (X) = ∑{ej (X) : j ∈ Ni} for all i, so total ABC error is e(X) = f (X : A′) − f (X : A).ABC error can be negative or positive, again resulting in possible self-cancellation effects.Hillsman and Rhoda recognize and discuss both total error and error self-cancellation.
Now consider any location model f (X : A) with p-server X and its approximationf (X : A′). A difficulty with error measures that can be negative or positive is that a smallererror (e.g., −3000) can be worse than a bigger error (e.g., +3). We can avoid this difficultyby using the (nonnegative) absolute error, ae(X) ≡ |e(X)| = |f (X : A′) − f (X : A)| de-fined for all X. This measure is familiar from Calculus for measuring how well one functionapproximates another. Related to ae(X) is the idea of an error bound: a number eb for whichae(X) ≤ eb for all X. An equivalent way to define an error bound, using f ′ and f to denotethe functions f (X : A′) and f (X : A) respectively, is based on the maximum absolute error,mae(f ′, f ), a number which may very well be quite difficult to compute. Any error bound isthen an upper bound on the maximum absolute error. Good error bounds may be much easierto compute than the maximum absolute error. Relative error can be defined when f (X : A)
is always positive: rel(X) ≡ ae(X)/f (X : A), perhaps converted to percent. Depending onmodel structure, ae(X) may be large but rel(X) may still be small due to the magnitudeof f (X : A). Relative error is not affected by the measurement scale chosen, whereas thepreceding error measures are.
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Table 2 Various types of optimality errors for any location model f (X : A). Ideal error measures are zero
No. Error name Error definition
1 Total error at X′ e(X′) = f (X′ : A′) − f (X′ : A)
2 Opportunity cost error f (X∗ : A) − f (X′ : A′)3 Optimality error f (X∗ : A) − f (X′ : A)
Assuming f (X : A) > 0 and f (X : A′) > 0 for all X ⊂ S, the relative error idea suggestsother equivalent ways of expressing the error bound:∣∣∣∣f (X : A′)f (X : A)
− 1
∣∣∣∣ ≤ eb
f (X : A)∀X ⊂ S ⇔
∣∣∣∣f (X : A)
f (X : A′)− 1
∣∣∣∣ ≤ eb
f (X : A′)∀X ⊂ S.
If the model f (X : A) is on a national scale, but aggregation is done on a city/town scale(e.g., eb = 10 miles, f (X : A) = 500 miles), we could have relatively small ratios eb/
f (X : A) and eb/f (X : A′), in which case the model ratios would be nearly one and wewould have a good aggregation. By contrast, if the model is on a city/town scale and theaggregation is also on a city/town scale, we might have a poor aggregation. We need theaggregation scale to be substantially smaller than the model scale in order to have a goodaggregation. This is one reason that aggregation may be of more interest for problems ofcity/town/regional scope than those of national or international scope.
There is another way to view the use of an aggregation error bound. The error boundallows us to draw conclusions about a family of original models, instead of just one. If theactual location model is F(X : A) instead of f (X : A), but the error bound applies to both,that is
|f (X : A′) − F(X : A)| ≤ eb and |f (X : A′) − f (X : A)| ≤ eb for all X,
then whatever conclusions we draw about the function f using the error bound inequalityalso apply to the function F . While we lose accuracy when we aggregate, we gain the abilityto draw approximate conclusions about a family of original functions. As a general exampleof the function F , suppose instead of DP set {aj : j ∈ N} we have a different DP set, say{bj : j ∈ N}, defining F , while all other model data is the same as for f (X : A). If each DPbj is aggregated into a′
j , then each of the functions F and f will be aggregated into the sameapproximating model, denoted by f ′. Further, if also d(aj , a
′j ) = d(bj , a
′j ) for j ∈ N , then
the methods we present later would provide both F and f ′, and f and f ′, with the sameerror bound. The data for F and f differ, but are sufficiently similar that the aggregationdoes not detect the differences.
Denote (globally) minimizing solutions to any original and approximating location mod-els f (X : A) and f (X : A′) by X∗ and X′ respectively. While we usually cannot expect tofind X∗ if we must aggregate DPs, we can still obtain some information about X∗ if weknow an error bound eb and X′. Geoffrion (1977) proves that |f (X′ : A′) − f (X∗ : A)| ≤eb, |f (X′ : A)−f (X∗ : A)| ≤ 2eb. Supposing f (X′ : A) > 0, we thus have |1 −f (X∗ : A)/
f (X′ : A)| ≤ 2eb/f (X′ : A). Hence, if 2eb is small relative to f (X′ : A), we may reason-ably accept X′ as a good substitute for X∗. We assume henceforth that we can compute X′but not X∗. Note that if we wish to use X′ to approximate X∗, then it makes no sense toallow p ≥ m, for then we can place a new facility at every ADP and may achieve a minimalapproximating function value of f (X′ : A′) = 0. Certainly it is desirable to have p � m.
Various authors, cited in Sect. 3, have proposed different types of optimality errors whichwe list in Table 2. The first error can be computed, and indicates how well the approximating
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function estimates the original function at X′. For large models, the second two errors cannotbe computed without knowing X∗. They can be computed for smaller models where X∗ canbe found without the need to aggregate, or for larger models if one assumes the algorithmused to solve the original problem provides X∗. Unless one can be certain that X∗ is known,or that some properties of X∗ are known, the latter two measures do not seem useful.
Although it is reasonable to use some measure of the difference between f (X : A′) andf (X : A) to represent aggregation error, doing so results in what may well be called theparadox of aggregation (Francis and Lowe 1992). Often our principal reason to aggregate isbecause we cannot afford, computationally, to make many function evaluations of f (X : A).We want to aggregate to make the error small; however, algorithms to do this typicallyrequire numerous function evaluations of f (X : A) and thus cannot be used for this purpose.Usually it is practical, however, to compute error measures for at least a few X, and wecertainly recommend doing so whenever possible. For example, given we know A and A′,we can use a sampling approach to compute a random sample of size K of p-servers, sayX1, . . . ,XK , compute f (Xk : A′) and f (Xk : A) for each sample element Xk , and thencompute a sample error estimate of any error measure of interest.
Location error (Casillas 1987; Daskin et al. 1989) involves some comparison of the p-server locations X∗ and X′. There are several difficulties with using this concept. First, ifwe really knew X∗ we would not need to do the aggregation. Second, when |X∗| ≥ 2, thereappears to be no accepted way to define the difference between X∗ and X′. Third (assumingwe do know X∗), the function f (X : A), particularly if it is the PMM function, may well berelatively flat in the neighborhood of X∗, as pointed out by Erkut and Bozkaya (1999). Thismeans we could have some X′ with f (X′ : A) only a little larger than f (X∗ : A), but X′ is“far” from X∗. Fourth, X′ and X∗ may not be unique global minima. Why are comparisonsmade of X′ with X∗? We speculate they are made in part due to unstated subjective evalu-ation criteria, or known but unstated supplementary evaluation criteria. As another possibleexample of the use of location error, we might solve the approximating model with threedifferent levels of aggregation (numbers of ADPs), obtaining three corresponding optimalp-servers say X′,X′′ and X′′′. In this case, differences between successive pairs of these p-servers might be of interest; we might want to know how stable the optimal server locationsare as we change the level of aggregation (Murray and Gottsegen 1997).
Subjective or unstated aggregation error criteria may well be important, but are not well-defined. Thus two analysts can study the same DP aggregation and not agree on whether itis good or not. Further, if a subjective evaluation derives from some visual representation ofDPs and ADPs, such an analysis may single out some relatively simple visual error featurethat is inappropriate for the actual model structure. For example, a visual analysis could notevaluate the (computationally intense) absolute error for the PMM. Some generally acceptedway to measure location error is desirable.
How should we measure the location error diff(X,Y ), the “difference” between any twop-servers X and Y ? The answer is not simple, because the numbering of the elements ofX and of Y is arbitrary, and we must find a way to match up corresponding elements.Further, X and Y are not vectors, but sets. We propose the use of a method discussed byFrancis and Lowe (1992). For motivation, consider the case where for each element xk
of X there is only one “nearby” element of Y , say yk∧ . In this case we might use eithermax{d(xk, yk∧) : k = 1, . . . , p} or
∑{d(xk, yk∧) : k = 1, . . . , p} as diff(X,Y ). More gener-ally, define the p × p matrix C = (cij ) with cij = d(xi, yj ). Define an assignment (permu-tation matrix) to be any 0/1 p × p matrix Z = (zij ) having a single nonzero entry of one ineach row, and a single nonzero entry of one in each column, and let P denote the set of allsuch p! assignments (permutation matrices). Define the objective function value v(Z) for
Ann Oper Res
every assignment Z ∈ P by v(Z) ≡ max{cij zij : Z ∈ P }, so that v(Z) is the largest entryin C for which the corresponding entry in Z is one. Define �(X,Y ) = min{v(Z) : Z ∈ P },so that �(X,Y ) is the minimal objective function value of the min-max assignment prob-lem with cost matrix C. We propose using �(X,Y ) for diff(X,Y ). There are severalgood reasons for using �(X,Y ). One reason is that it has all the properties of a dis-tance (see Goldberg 1976): symmetry: �(X,Y ) = �(Y,X); nonnegativity: �(X,Y ) ≥ 0and �(X,Y ) = 0 ⇔ X = Y ; triangle inequality: �(X,Y ) ≤ �(X,Z) + �(Z,Y ) for anyp-servers X,Y and Z. Another reason, further explored in Sect. 4, is that it is related toabsolute error. (We could also use the optimal value of the conventional min-sum assign-ment model for diff(X,Y ). This optimal value also has all the properties of a distance, butwe know of no useful relationship between it and absolute error.) We call the distance �
the min-max distance. Note, for any two p-servers X,Y ⊂ S,�(X,Y ) ≤ diam(S). Further,when p = 1 the min-max distance is just the usual distance, d(x1, y1).
Both min-max and min-sum assignment models are well-studied and are efficiently solv-able in low polynomial order for any set of real coefficients (Ahuja et al. 1993). In the as-signment models we study, the coefficients typically correspond to distances between pointsin some geometric spaces, e.g., planar Euclidean or rectilinear cases. For these geometricmodels significantly more efficient algorithms have become available (Agarwal et al. 1999;Agarwal and Varadarajan 1999; Efrat et al. 2001 and Varadarajan 1998).
There are a number of relationships between the error measures of Table 1. These re-lationships, some of which may not be obvious, will be a subject of discussion of Sect. 4,where we also give numerical examples of many of the error measures. It also seems worthpointing out that error measures 2 through 7 of Table 1 are local error measures, since theydepend on X. By contrast, measures 1, 8 and 9 may be considered global error measures.
There is no general agreement on which aggregation error measure is best. Until the re-search community agrees on one or more error measures, progress in comparing variousaggregation approaches, and in building a cumulative body of knowledge, will necessar-ily be limited. The lack of agreement on error measures also limits progress in trading offaggregation advantages and disadvantages. Further, because comparisons of various aggre-gation algorithm results should all be based on the same error measures, there is currentlylittle point in developing for the profession a test data base of DPs to aggregate. For rea-sons discussed further in Sect. 4, we personally recommend the uses of relative error basedon absolute error and/or error bounds, together with ADP-DP distances. The bound in theinequality |1 − f (X∗ : A)/f (X′ : A)| ≤ 2eb/f (X′ : A) seems particularly promising.
An alternative to using some low computational order approach to aggregate the originaldemand point set, and then solving the resulting aggregated location model to optimality, isto use some low computational order metaheuristic approach (Pardalos and Resende 2002;Reeves 1993; Resende and de Sousa 2004) to approximately minimize the original, unaggre-gated location model. The first approach gives bounds on optimality to the original model.The second approach introduces an additional source of error, since a heuristic is used, butmay possibly result in a better solution. Given the current state of the art, which approach isbest is not known. Indeed, “best” may not even be well-defined, since there is no generallyaccepted measure of aggregation error.
3 Literature discussion
We now provide an overview of some of the principal papers dealing with aggregation error,although we discuss some papers in other sections. The review is organized in two main
Ann Oper Res
Tabl
e3
p-m
edia
nag
greg
atio
nlit
erat
ure
clas
sific
atio
n
Yea
rR
efer
ence
Em
phas
isSe
tting
Agg
mea
s.R
ange
ofR
ange
ofSe
rver
sD
ata
Dis
tanc
eC
ompu
teC
ompu
teC
ompu
te
used
DPs
AD
Ps(R
ange
Rea
l?m
easu
reX
∗X
′di
ff(X
∗ ,X
′ )of
p)
Y/N
1978
Hill
sman
and
Rho
da(1
978)
CP
#4,5
Tabl
e1
Uni
form
ly1
to50
per
N/A
NE
uclid
ean
N/A
N/A
N/A
dist
ribu
ted
serv
er
1979
Goo
dchi
ld(1
979)
CP
and
N#1
0Ta
ble
190
0at
1to
900
1to
4Y
&N
Euc
lidea
nY
YG
raph
ical
mos
t(h
euri
stic
s)(h
euri
stic
s)co
mpa
riso
n
1981
Bac
h(1
981)
CD
#8Ta
ble
128
6(1
2,62
,171
)1
to12
YE
uclid
ean,
YY
No,
but
netw
ork
(heu
rist
ics)
(heu
rist
ics)
solu
tions
are
liste
d
1986
Mir
chan
dani
and
Rei
lly(1
986)
TP
#2Ta
ble
1N
A1
N/A
NE
uclid
ean,
N/A
N/A
N
rect
iline
ar
1987
Cur
rent
and
Schi
lling
(198
7)C
D#1
,#3
Tabl
e2
681
(30,
70)
5,7,
9Y
Euc
lidea
nY
YN
(heu
rist
ics)
(heu
rist
ics)
1987
Cas
illas
(198
7)C
D#4
Tabl
e1
500
50-2
001,
2,4
&6
NE
uclid
ean
YY
Gra
phic
al
(heu
rist
ics)
(heu
rist
ics)
com
pari
son
1991
Ohs
awa
etal
.(19
91)
TP
#2Ta
ble
2;N
/AN
/A1
NE
uclid
ean
Y(c
lose
dY
(clo
sed
Y
#10
Tabl
e1
(rea
llin
e)fo
rm)
form
)
1992
Fran
cis
and
Low
e(1
992)
TN
#9Ta
ble
1N
/AN
/AN
/AN
Net
wor
kN
/AN
/AN
Dis
tanc
e
1993
Hod
gson
and
Neu
man
(199
3)C
D#1
,#3
Tabl
e2
276
to25
3Y
Euc
lidea
nY
YN
1994
Bal
lou
(199
4)C
P#1
Tabl
e2
900
25-9
001-
100
YE
uclid
ean
YY
N
(heu
rist
ics)
(heu
rist
ics)
1995
Foth
erin
gham
etal
.(19
95)
CD
#4Ta
ble
187
125
-800
10Y
Euc
lidea
nY
YN
(heu
rist
ics)
(heu
rist
ics)
Ann Oper Res
Tabl
e3
(Con
tinu
ed)
Yea
rR
efer
ence
Em
phas
isSe
tting
Agg
mea
s.R
ange
ofR
ange
ofSe
rver
sD
ata
Dis
tanc
eC
ompu
teC
ompu
teC
ompu
te
used
DPs
AD
Ps(R
ange
Rea
l?m
easu
reX
∗X
′di
ff(X
∗ ,X
′ )of
p)
Y/N
1996
Fran
cis
etal
.(19
96)
C,T
P#9
Tabl
e1,
5500
–25
0-12
251,
3,5,
7Y
Rec
tilin
ear
NN
N
sam
ple
erro
r69
960
mea
sure
s
1997
Hod
gson
etal
.(19
97)
CD
Scal
edva
lues
668
158
only
1-25
YE
uclid
ean
YY
N
of#1
,#3
(heu
rist
ics)
(heu
rist
ics)
Tabl
e2
1997
Mur
ray
and
Got
tseg
en(1
997)
CD
#3Ta
ble
291
310
0-80
010
YE
uclid
ean
YY
Gra
phic
al
(heu
rist
ics)
(heu
rist
ics)
com
pari
son
1998
And
erss
onet
al.(
1998
)C
Nan
dP
#9Ta
ble
1,49
Kto
1(2
16,3
76,
5,7
N,b
utR
ectil
inea
rN
NN
com
bine
dsa
mpl
eer
ror
mill
ion
832)
real
and
mea
sure
s(u
nifo
rm)
netw
ork
netw
ork
1999
Bow
erm
anet
al.(
1999
)C
N,D
#1,#
3Ta
ble
226
3020
95-
50Y
Net
wor
kY
YN
Dis
tanc
e(h
euri
stic
s)(h
euri
stic
s)
1999
Erk
utan
dB
ozka
ya(1
999)
CD
#4Ta
ble
166
8(5
0,20
0,2-
6Y
Euc
lidea
nY
YN
300)
(heu
rist
ics)
(heu
rist
ics)
1999
Zha
oan
dB
atta
(199
9)C
,TD
#1,#
3Ta
ble
250
3250
1,5
YE
uclid
ean
YY
Gra
phic
al
com
pari
son
2000
Fran
cis
etal
.(20
00)
TG
ener
al#6
Tabl
e1
N/A
N/A
N/A
Nar
bitr
ary
N/A
N/A
N
2000
Plas
tria
(200
0)T
P#4
Tabl
e1
arbi
trar
y1
1N
/AV
ario
usN
/AN
/AN
norm
s
2000
Zha
oan
dB
atta
(200
0)T
N“O
ppor
tuni
tyN
/AN
/AN
/AN
Net
wor
kN
/AN
/AN
cost
”,#3
Tabl
e2
Dis
tanc
e
Ann Oper Res
Tabl
e3
(Con
tinu
ed)
Yea
rR
efer
ence
Em
phas
isSe
tting
Agg
mea
s.R
ange
ofR
ange
ofSe
rver
sD
ata
Dis
tanc
eC
ompu
teC
ompu
teC
ompu
te
used
DPs
AD
Ps(R
ange
Rea
l?m
easu
reX
∗X
′di
ff(X
∗ ,X
′ )of
p)
Y/N
2001
Plas
tria
(200
1)C
,TP
#5Ta
ble
120
0040
01,
2,5
NE
uclid
ean
YY
N
and
Rec
tilin
ear
2002
Hod
gson
(200
2)C
D“S
urro
gatio
n”N
/AN
/A5-
50Y
Euc
lidea
nN
/AN
/AN
/A
erro
r
2003
Hod
gson
and
Hew
ko(2
003)
CD
“Sur
roga
tion”
722
N/A
1-25
YE
uclid
ean
N/A
N/A
N/A
erro
r&
#5
Tabl
e1
2003
Fran
cis
etal
.(20
03)
T,C
P#9
Tabl
e1
5000
–25
-900
1,3,
5Y
(2R
ectil
inea
rN
NN
2500
00da
ta
(com
pute
rse
ts)
gene
rate
d);
1193
&
6996
0fo
r
real
data
2004
Har
-Pel
edan
dM
azum
dar
(200
4)T
P#6
,#7
Tabl
e1
N/A
N/A
N/A
NE
uclid
ean
N/A
N/A
N
2005
Har
-Pel
edan
dK
usha
l(20
07)
TP
#6,#
7Ta
ble
1N
/AN
/AN
/AN
Euc
lidea
nN
/AN
/AN
Ann Oper Res
Tabl
e4
p-c
ente
ran
dco
veri
ngag
greg
atio
nlit
erat
ure
clas
sific
atio
n
Yea
rR
efer
ence
Em
phas
isSe
tting
Agg
mea
s.R
ange
ofR
ange
ofSe
rver
sD
ata
Dis
tanc
eC
ompu
teC
ompu
teC
ompu
teO
bjec
tive.
used
DPs
AD
Ps(R
ange
Rea
l?m
easu
reX
∗X
′di
ff(X
∗ ,X
′ )Fu
nctto
n.
ofp
)Y
/N
1989
Das
kin
etal
.(19
89)
CD
#1,#
3Ta
ble
235
5(6
7,20
1)4
YE
uclid
ean
YY
YM
ax.
Cov
erin
g
1990
Cur
rent
and
Schi
lling
(199
0)C
,TD
#1,#
3Ta
ble
268
1(1
85,4
15)
30,7
0Y
Euc
lidea
nY
YN
Max
.
(heu
rist
ics)
(heu
rist
ics)
Cov
erin
g
1996
Fran
cis
and
Ray
co(1
996)
TP
#8Ta
ble
1N
/AN
/AN
/AN
Euc
lidea
nN
/AN
/AN
Cen
ter
1996
Ray
co(1
996)
T,C
P#9
Tabl
e1
5000
–N
otkn
own
1,3,
5,7
Y(2
data
Rec
tilin
ear
NN
NC
ente
r
1000
0;a-
prio
ri;2
5–se
ts)
(Med
ian
1199
3an
d25
00by
for
one
6996
0fo
rco
nstr
uctio
nca
se)
real
data
sets
1999
Ray
coet
al.(
1999
)T,
CP
#9Ta
ble
150
00–
Not
know
n1,
3,5,
7Y
(oth
erR
ectil
inea
rN
NN
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ter
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prio
ri;2
5–da
ta
6996
0fo
r25
00by
com
pute
r
real
data
cons
truc
tion
gene
rate
d)
set
2000
Fort
ney
etal
.(20
00)
CP
N/A
435
N/A
4000
YE
uclid
ean
N/A
N/A
NC
ente
r,
Cov
er
2002
Aga
rwal
and
Proc
opiu
c(2
002)
TP
#6,#
7Ta
ble
1N
/AN
/AN
/AN
Euc
lidea
nN
/AN
/AN
N
2002
Aga
rwal
etal
.(20
02)
TP
#6,#
7Ta
ble
1N
/AN
/AN
/AN
Euc
lidea
nN
/AN
/AN
N
2004
Fran
cis
etal
.(20
04a)
TG
ener
al#1
Tabl
e1
N/A
N/A
N/A
NG
ener
alN
/AN
/AN
/AC
ente
r,
Cov
er
2004
aH
ar-P
eled
(200
4a)
TP
#6,#
7Ta
ble
1N
/AN
/AN
/AN
Euc
lidea
nN
/AN
/AN
N
2004
bH
ar-P
eled
(200
4b)
TP
#6,#
7Ta
ble
1N
/AN
/AN
/AN
Euc
lidea
nN
/AN
/AN
N
Ann Oper Res
categories, median problems and center/covering problems, but there is some overlap sincesome authors considered both types of problems. Also, unless it is clear, we indicate whetherthe study was for planar data (P), data on a network (N), or discrete data (D). Some of thesereferences have been mentioned earlier in the paper but are discussed here for completeness.Refer to Tables 3 and 4, and Sect. 3.3, for a summary of our findings. The tables also givesupplementary information about the papers, including whether the papers are primarilycomputational (C), or theoretical (T).
3.1 Median problems
Most authors agree that Hillsman and Rhoda (1978) proposed the first formal method forclassifying and measuring aggregation error for the median problem. As discussed in Sect. 2,they identified three error sources labeled A, B and C. Source A essentially involves theinaccuracy of measuring average distance from a server location to a set of DPs via distanceto a single ADP. Source B, a special case of A, occurs when a server coincides with an ADP.Source C occurs when there is more than one server and some of the represented pointsare closer to one server while others are closer to another server. To investigate aggregationerror, they utilized a grid of regular polygons (hexagons, squares, or triangles) over a planardistribution of DPs, and aggregated the total demand in each polygon to its centroid. Theythen placed servers at regularly spaced points on a lattice. By varying the spacing betweenservers they were able to vary the number of ADPs assigned to each server. They concludedthat aggregation error was larger when few ADPs were assigned to each server.
Goodchild (1979), in a planar setting, studied the problem of aggregating a zone intoone ADP so that the average distance to any point in the zone from any arbitrary facilitylocation was equal to the distance to the ADP. He concluded that this is not possible. Hethen conducted an empirical study using 900 uniformly distributed DPs in the plane byaggregating them into different numbers of ADPs. He also conducted an empirical studyof aggregation effects on a network. The primary focus was to study location errors due toaggregation. He compared solutions to resulting problems graphically and concluded that“. . . the effects of aggregation error on median problems are substantial.” He also noted that“. . . aggregation tends to produce much more dramatic effects on location than on the valuesof the objective function.”
Bach (1981) studied aggregation effects for the discrete PMM, PCM and CLM usingdata sets for the cities of Dortmund, Kleve and Emmerich, Germany. Using these data sets,Bach studied both location error as well as objective function error by examining severallevels of aggregation and different distance measures, e.g., Euclidean distance in the plane,travel time by car on a road network, etc. Bach states, “Thus it is possible to conclude thatthe level of aggregation exerts a strong influence on the optimal locational patterns as wellas on the values of the locational criteria.” Similar conclusions were stated for variations indistance measures.
Mirchandani and Reilly (1986) studied the problem of approximating distances to pointsin a region by distance to a single point representing the region. The paper has a goodreview of pre-1986 literature on aggregation. The authors give an algorithm that partiallycompensates for the distance error discussed above.
Current and Schilling (1987) proposed a method for eliminating error sources A andB , and tested it on real discrete data. Using n = 681 DPs, they solved problems forp = 5,7, and 10 and with 30 and 70 ADPs. Four different aggregation schemes were usedin their experiments. In addition to using
∑{wjD(X, αi) : j ∈ Ni} for aggregated mod-els, they experimented with using a modified distance matrix as follows. The cost of as-signing ADP αi to facility xk was taken to be
∑{wjd(aj , xk) : j ∈ Ni}, when xk �= αi ,
Ann Oper Res
and∑{wjd(aj , αi) : j ∈ Ni}, when xk = αi . For any given aggregation scheme, use of
the above modified distance matrix eliminated error sources A and B , but C still re-mained. Letting X∗
AB be the optimal solution to the aggregated problem using the mod-ified distance matrix, Current and Schilling reported the following results on their testproblems: (f (X′ : A′) − f (X′ : A))/f (X′ : A), (f (X∗
AB : A′) − f (X∗AB : A))/f (X∗
AB : A),(f (X′ : A) − f (X∗ : A))/f (X∗ : A) and (f (X∗
AB : A) − f (X∗ : A))/f (X∗ : A). Error wasreduced by using the modified distance, but still remained due to the presence of errorsource C.
Casillas (1987) proposed that error sources A, B and C can result in two types of error:(1) error as a result of wrong server locations (location error) and (2) error associated withan inaccurate objective function value (comparison of f (X∗ : A) and f (X′ : A′)). He stud-ied aggregation effects using randomly generated planar demand data. The DP data baseconsisted of 500 points generated using the negative exponential distribution, thus favoringthe placement of points near the origin. He then ran experiments using m = 200,150,100and 50 ADPs. For each level of aggregation m, he randomly selected m of the original DPsto be seeds. Then for each seed he created a zone by clustering each non-seed point with itsclosest seed. Next, the centroid of each of the m zones was chosen as an aggregate point. Hethen solved a PMM over these m ADPs for p = 1,2,4 and 6. He concluded that optimalityerror (the difference f (X∗ : A)−f (X′ : A)) was small for small p, but larger for higher lev-els of aggregation and larger values of p. Casillas plotted the locations of optimal facilitiesfor some of the problems solved.
Ohsawa et al. (1991) considered the continuous min-sum and min-max (1-median, 1-center) problems in one-dimensional space and the effect of aggregating DPs into the mid-points of intervals of equal width. Thus, the authors refer to aggregated data as “roundeddata.” As the aggregate data points are determined by an origin Z and the specified intervalwidth ω, the authors studied the case when Z is a random variable that is uniformly distrib-uted between −ω/2 and ω/2, and analyze the expected location error E{(X∗(Z)−X′(Z))2}and expected cost E{(f (X∗(Z) : A) − f (X′(Z) : A′))2} for each of the models. They con-cluded that: (1) rounding appears to exert more serious influence on the median problemthan on the center problem, and (2) for both models, the circumstance that maximizes costerror minimizes location error.
As discussed in Sects. 2 and 4, Francis and Lowe (1992) developed bounds on the ab-solute error due to aggregation of demand data. Error bounds were developed for both thePCM and PMM. For each problem, the minimal error bound is obtained by using the leastobjective function value obtained of a location problem that has the same structure as, butalso has more variables than, the original problem.
Hodgson and Neuman (1993) proposed a method for eliminating error source C by spa-tially disaggregating data “as needed” during the solution to a p-median problem. Theirmethod made use of Thiessen (Voronoi) overlay polygons, where every point within a givenpolygon is closer to that polygon’s centroid than the centroid of any other polygon. Thedisaggregation process was based on membership in a polygon. Their method made use ofa geographic information system (GIS) and appears to be computationally intensive.
Ballou (1994) studied costing error (f (X′ : A) − f (X′ : A′))/f (X′ : A) for the PMMusing U.S. population data with initial DPs being the centroids of 900 3-digit zip code areas.The weights corresponded to population. Transport cost was approximated by using the for-mula: rate = a+b× (Euclidean distance), with a and b obtained from freight rate tables. Thecomputed rate is multiplied by population to obtain the cost for serving a given DP. Differentrate functions were also tested. Ballou examined cost errors by varying m and p. Facilitieswere located using Cooper’s (1967) location/allocation heuristic. He found that his error
Ann Oper Res
measure increased as p increased (for fixed m), and that error decreased as m increased.This latter relationship displayed diminishing returns. Computational work indicated that tokeep the costing error below one percent, m should be approximately 5 to 10 times p, whileif costing errors in the five to ten percent range are acceptable, then m can be 3 to 5 times p.
Fotheringham et al. (1995) studied the sensitivity of both the objective function value andoptimal locations for the discrete problem to different aggregation schemes for a real dataset. They examined different “zone definitions” involving 871 Buffalo, New York censusblocks aggregating them into different sized aggregate units. Using p = 10, they consideredaggregation into 800, 400, 200, 100, 50 and 25 ADPs. For each aggregation level, theyrandomly selected m seeds from the 871 original units and then clustered the remaining(871 − m) units to the seeds creating m ADP-regions. The centroid of each ADP-regionwas then computed and a 10-median problem was solved. At each aggregation level thisprocess was repeated 20 times. The resulting objective function values were then plotted.To measure location error, they first found X∗ (the optimal solution to the original problem)and arbitrarily indexed the elements of X∗ as x∗
1 , x∗2 , . . . , x∗
10. Then, at a given aggregationlevel and from each of the 20 solutions, letting X′ be one of these solutions, they calculatedG = ∑
i=1,...,20 d(x∗i , x
′[i]), where x ′
[i] is the closest member of X′ to x∗i . In this procedure,
x ′[1] is found first and removed from the set of x ′ values, etc. Note this approach amounts
to a heuristic solution to a min-cost assignment problem, as discussed in Sect. 2. The 20 Gvalues for a given aggregation level were then plotted. The authors concluded that objectivefunction values did not seem to depend much on the level of aggregation, but the optimalfacility locations, i.e., location error, did seem to be quite sensitive to changes in the level ofaggregation.
Francis et al. (1996) considered DP aggregation for the planar PMM with rectilinear dis-tances, and weights normalized (scaled) to total 1. Their approach was to find an aggregationgiving a small value of the error bound of Francis and Lowe (1992). The authors showedthat the error bound they studied, while worst-case, is attainable, and gave necessary andsufficient conditions for attainability. They also gave a discussion of various error measures,and an argument, based on work of Geoffrion (1977), that their error bound was a “nat-ural” error measure to use. The authors considered a collection of row-column aggregationswith one ADP per cell. Their approach, which built on work of Hassin and Tamir (1991),minimized the error bound over a (restricted) class of row-column aggregations. Rows (andcolumns), need not have equal spacings. Adjusting the number of rows and columns adjuststhe total number of ADPs. For problems with real data, the authors found it helped to add anextra “touchup” step of low computational order to obtain a better bound. Their procedure,predominantly of O(n logn), was tested extensively using both randomly generated dataand real data. Error bounds, as well as various sample error estimates, were computed. Thelaw of diminishing returns occurred with all the testing; each error measure decreased at adecreasing rate as the number of ADPs increased.
Hodgson et al. (1997) studied different types of aggregation error using Edmonton,Canada census data. In addition to error sources A, B and C identified by Hillsman andRhoda (1978) they also defined and studied source D error, the error introduced by aggregat-ing potential server locations. That is, an error might possibly arise when a potential facilitysite is not available because it has been aggregated with one or more other potential sites.Computational work was performed on problems of sizes 668 × 668,158 × 158,158 × 668,and 668 × 158, where the first entry is m (smaller values are the result of aggregation)while the second entry is the number of potential facility locations. The problem was solvedon these data. The authors reported computational estimates of (f (X′ : A′) − f (X′ : A))/
f (X′ : A) (called measurement error), and (f (X′ : A) − f (X∗ : A))/f (X∗ : A) (called lo-cation error) for various values of p. They found that measurement error decreased with p,
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while location error increased with p. They also studied isolating the various error sourcesinto types A through D.
Murray and Gottsegen (1997) studied the stability of facility locations and objective func-tion values for the planar PMM with respect to various levels of demand data aggregationand various aggregation schemes for a fixed level of aggregation. They used U.S. Bureau ofCensus block group data from the 1990 census for the Buffalo, New York metropolitan area.The original DPs consisted of 913 block groups of elderly citizens totaling 138,515 peoplewith demands for senior services centers. A total of 10 servers were located using the DPsgiving f (X∗ : A). Then f (X′ : A′) was found for aggregation levels of m = 100,200,400and 800. Furthermore, for each level of aggregation, various clustering methods were used tocreate the aggregate sets. They found that although the server locations varied quite a bit de-pending on the level of aggregation and aggregation method used, for a fixed value of m, thecorresponding objective function values of the various location sets, when evaluated againstthe original data, i.e., f (X′ : A), did not seem to vary that much. Comparing f (X∗ : A) withf (X′ : A), they found that more aggregation (smaller m values) produced poorer results.They concluded that “. . .more intelligent and sophisticated methods of aggregating data doappear to be worth the additional effort in terms of reducing error.”
Andersson et al. (1998) studied DP aggregation methods for network versions of thePMM and PCM. A coarse aggregation structure was first obtained by partitioning the DPsaccording to a variable-spacing grid imposed over the demand region using “Row-Column”aggregation algorithms of Francis et al. (1996) or of Rayco et al. (1997). A second step lo-cated an ADP on each of the subnetworks induced by the cells of the grid; each ADP is eithera network 1-median or 1-center, depending on the problem. Optionally, the ADP set initial-ized an iterative network location-allocation method to find improved ADPs. The conceptof a network Voronoi diagram (see Hakimi et al. 1992) was used extensively to find closestADPs to DPs. The network data sets—not aggregated—were computerized maps from theTIGER/Line database of the U.S. Census Bureau. The maps were of large U.S. cities, in-cluding Jacksonville, Florida and Chicago, Illinois. Hypothetical DPs were spaced equallyalong network arcs. In addition to the four sample error measures discussed in Sect. 2, a “rel-ative error bound” eb/f (X : A), an upper bound on the relative error, was considered. Errormeasures obeyed the law of diminishing returns with increasing numbers of ADPs. For the5-median problem and the city of Jacksonville with n = 206,761, for example, the samplemaximum relative error was at most 1% with at least 500 ADPs. While the aggregationswere sensitive to the street network structure, the error estimates were not, and not too sen-sitive to the value of p. Computed error bounds largely overestimated the sample errors forboth models, but the sample errors closely tracked the error bound values. Error estimatesbased on rectilinear distances were consistently less than the more accurate estimates basedon network distances. Overall, the approach worked well for the PMM, while results weremixed for the PCM.
Bowerman et al. (1999) proposed a data partitioning method to eliminate sources A, Band C for the PMM. This iterative method essentially combined the approaches developedby Current and Schilling (1987) to eliminate source A and B, and of Hodgson and Neuman(1993) to eliminate source C. The method was tested using network data from the CentralValley of Costa Rica. The disaggregated data set had 2,630 demand nodes along with 731potential sites for facilities. They tested their method using 209 ADPs with p ranging from5 to 50. They reported results for error measures 1 and 3 in Table 2. They found that theirmethod was more effective than that of Current and Schilling (1987) in reducing error butthe computation times were higher.
Erkut and Bozkaya (1999) provided an excellent review of the pre-1995 literature onaggregation error analysis for the planar problems. In addition to the review, Erkut and
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Bozkaya asserted that source A–C errors result in three types of PMM output perturbations:(a) location error (see error 10, Table 1), (b) cost error, f (X∗ : A) versus f (X′ : A′), and(c) optimality error, i.e., f (X∗ : A) versus f (X′ : A). They made a case that some of theerror analysis reported in the literature was exaggerated because of selection of test data,e.g., using uniformly distributed data as opposed to a data distribution that is more realistic.They also gave “dos” and “don’ts” for spatial aggregation.
Zhao and Batta (1999) presented a theoretical analysis of aggregation error for the pla-nar PMM with Euclidean distance and centroid ADPs. They studied both worst-case andaverage-case error. Among other theoretical results they developed bounds on the total DPerror (error measure 4, Table 1) for the 1-median problem evaluated at any point X. Theyapplied their analytical results to housing location data from Buffalo, New York; and On-tario, California. The Buffalo problem involved 5032 housing locations and m = 50 ADPs.In their experiments they measured both total error and optimality error (error measure 3,Table 2). The Ontario experiments involved sort of a reverse procedure. They started with76 zip code areas and then randomly generated n > 76 data points in the zones, measuringthe same kind of errors as for the Buffalo problem.
We discuss material in Francis et al. (2000) in Sect. 4.Plastria (2000, 2001) studied aggregation error for the planar problem. The papers are
primarily theoretical, and have good literature discussions. He considered various distancemeasures derived from gauge functions. He made a strong case for aggregating data at thecentroid of the data set. An important finding of his is that aggregation error for the 1-medianproblem goes to zero as the distance between the centroid and the server increases as theserver location extends out on a ray originating at the centroid. Plastria (2001) contains someresults on computational experiments with a DP set consisting of 2000 randomly generatedpoints in the plane. He measured error sources A and B in his experiments, with m = 400ADPs and p = 1,3 and 5.
Zhao and Batta (2000) considered a seldom studied type of aggregation error. Perhapsdue to budgeting constraints it may be the case that only a subset of all possible feasiblesolutions to a location problem may be considered. They studied a problem of this typeon a network where demands could be on links of the network and could be discrete orcontinuous. They found that, by restricting the search for a solution to the nodes of thenetwork, the resulting error associated with this restriction involved only the demands ona single link of the network. Based on this result, they also suggested a “link aggregation”scheme to further reduce the error.
Hodgson (2002) defined data surrogation error for a PMM as the error which occurswhen an inappropriate parameter is used to stand in for a target population’s demand. Anexample of this error was illustrated for 25 Canadian cities where general population datais substituted for target data: either small children or else elderly citizens. In a follow-uppaper, Hodgson and Hewko (2003) investigated the relative sizes of total DP error (errormeasure 4, Table 1) and surrogation errors using Edmonton, Canada data. In their limitedtesting, they found that for the data they used, surrogation error was a much more seriousproblem than was ABC error. The paper also contains a disaggregation technique that isdesigned to reduce both error types.
Francis et al. (2003) presented theory and algorithms for aggregations for the 1-medianrectilinear distance problem on the plane. The authors first studied properties of an AlignedRow-Column Aggregation (ARC) algorithm that minimizes—over all aligned row-columnaggregations—the maximal absolute error for the 1-median rectilinear distance problem.The Centroid Row-Column (CRC) aggregation then is based on the partitioning defined bythe ARC aggregation using the centroids of the sets as the aggregate points. The row-column
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partitioning is itself determined by solving two contiguous DP aggregation problems on aline. Two algorithms were presented for solving these problems on a line: a bisection searchmethod, and a dynamic programming procedure. While the development of the CRC wasbased on the 1-median problem, the authors posited that the aggregate sets so obtained canbe quite effective for PMM aggregations, for p ≥ 2. In their computational testing theyused values of p = 1,3,5. They found that the error measures examined with the CRCaggregation could be well-modeled by a power function of the form a/mb, where m is thenumber of aggregate points, a is a positive constant, and b ≥ 1.
Finally we refer to the work of Har-Peled and Mazumdar (2004) and Har-Peled andKushal (2007) who studied the existence and construction of small ADP sets for weightedEuclidean p-median problems defined in Rk . Their main motivation was to improve on andobtain more efficient approximation algorithms for weighted Euclidean p-median problems.
They use the relative error measure, rel(X), defined in Sect. 2. Specifically, given aweighted set A of n DP’s in Rk , and a parameter ε, Har-Peled and Mazumdar (2004) provethat there is a weighted set A′, A′ ⊂ A, of size O((p/εk) logn), such that for the weightedp-median models, denoted by f (X|A);f (X|A′) respectively, we have
(1 − ε)f (X|A) ≤ f (X|A′) ≤ (1 + ε)f (X|A),
for any subset X of p points in Rk .Equivalently, with our terminology, when f (X|A) > 0, rel(X) ≤ ε.They call such a set A′ an ε-coreset for the p-median problem on A, and show that it
can be constructed in time O(n + poly(p, logn,1/ε)), where poly(·) is a polynomial.Har-Peled and Kushal (2007) improved the above result and showed that one can con-
struct a coreset whose size is independent of n, the size of A. In particular, they constructan ε-coreset A′ for the weighted Euclidean p-median problem of size O(p2/εk). However,the coreset A′ that they construct is not necessarily a subset of A.
We note that with the above results, if we let X∗ and X′ denote optimal solutions forf (X|A) and f (X|A′), respectively, then one can show that
0 ≤ f (X′|A) − f (X∗|A) ≤(
2ε
1 − ε
)
f (X∗|A).
3.2 Center and cover problems
Daskin et al. (1989) studied aggregation error for discrete planar maximum covering prob-lems. Using n = 355 DPs representing demand areas in the U.S., they measured three errortypes with three different aggregation schemes. In their experiments, they aggregated bothDPs as well as candidate facility site nodes. For each aggregation scheme, they measured avalue they call “optimality error” a number computed using (a) the optimal objective valueof the model using all 355 DPs, and (b) facility locations determined via an aggregatedmodel but evaluated using all DPs, i.e., error measure 3 in Table 2. A second error measurewhich they called “coverage error” used (b) above in comparison with the objective value ofthe aggregated problem, i.e., error measure 1 in Table 2. Also they measured location errorusing a rather interesting definition of this metric. To measure location error, for each aggre-gate candidate site αi they identified the set of DPs that are closest to αi . Then after solvingboth the aggregated and disaggregated versions of the problem they measured location erroras follows. If a node is chosen in the disaggregated solution and if it is associated with αi
chosen in the aggregated solution, then no location error occurs. However, if αi is not chosen
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in the aggregated solution, then a location error has occurred. They found that the percent-age of location error tended to decrease as m increased. They also found that location errorexceeded both optimality error and coverage error for all of the problems examined. Fromthis observation, they “qualitatively confirmed” Goodchild’s finding that “aggregation has agreater effect on location decisions than on the values of the objective function”.
For planar problems, Current and Schilling (1990) defined CLM counterparts to the Hills-man and Rhoda (HR) source A, B and C errors. They also presented three aggregation rules,which when applied during the data aggregation process, will reduce these errors. The ana-logue of the HR source A error applied to the covering problem is that once an aggregatedproblem is formulated and solved, an original DP may be reported as covered (because itsADP is covered), when in actuality it is not. Also, it may be that the original DP is covered,but the aggregate model indicates that it is not. Current and Schilling noted that there isno analogue of error source C in the covering problem. Their three aggregation rules weretested using an n = 681 node data set representing Baltimore City, Maryland. They foundthat their aggregation rules were effective in reducing problem size as well as aggregationerror.
Francis and Rayco (1996) considered aggregation schemes for the unweighted PCM inthe plane with rectilinear distances. They noted that the error bound function for the PCMdeveloped by Francis and Lowe (1992) might be difficult to minimize when the number ofaggregate points is large. Thus they developed an over-approximation to the error boundfunction and studied its behavior. They also developed a lower bound for the minimal valueof this error bound and gave necessary and sufficient conditions for the lower bound to beattained. They then gave a constructive aggregation algorithm that attains the lower boundasymptotically as the number of ADPs increases. Their results parallel the results derivedby Zemel (1985) for Euclidean distance PCMs and PMMs. However, Zemel did not focuson the implications of the results for designing good aggregation schemes.
Rayco et al. (1999) presented an aggregation algorithm for the rectilinear distance PCM.The aggregate set identified imposed a grid structure on the plane. The cells comprisingthe grid structure were diamond-shaped and all of the same user-specified dimensions. Thepositioning of the grid was determined by minimizing an upper bound eb on the objectivefunction error, where |f (X : A′) − f (X : A)| ≤ eb, for all X. Minimizing this error boundwas shown to decompose into 1-center problems on cycles. In their computational testing,the authors applied the algorithm to computer-generated data sets, as well as a real-worlddata set instance; they found a decreasing rate of improvement in the error measures as thenumber of ADPs increased.
In a study of access to health services, Fortney et al. (2000) examined metrics related toboth center and covering objectives for planar data from the state of Arkansas. They stud-ied aggregation of location data for 435 patients in need of care as well as aggregation ofover 4000 health care provider locations. Since all data was available beforehand, no opti-mization was involved. Aggregation was done at the zip code level with centroids taken aslocations. Results of coverage using aggregate data were then compared with coverage usingstreet address encoding. They found that for the center-type objectives (called accessibilityin their paper), when compared with zip code aggregation, a large increase in accuracy canbe obtained by using a GIS to encode both patient and provider locations at the street level.For the covering type objective (called availability) they found the street level geocoding didnot provide substantial accuracy.
Francis et al. (2004a) considered aggregation issues for location problems that involvenearest distances of DPs to servers where these distances are in the objective function as wellas in a set of constraints. They used and extended error bound results developed by Francis
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et al. (2000) for the constrained problems to be able to analyze the effect of aggregation onthe problem constraints. The paper made use of penalty functions as a method for collapsingthe constraints into the objective function. An application of the method was illustrated viathe CLM. (See Sect. 4 for more details.)
Finally we refer to the work by Agarwal and Procopiuc (2002), Agarwal et al. (2002),and Har-Peled (2004a, 2004b) and Agarwal et al. (2005), who studied the existence andconstruction of small ADP sets for unweighted Euclidean p-center problems defined in Rk .Using our terminology, they use the relative error measure, defined in Sect. 2. Specifically,given a set A of n DPs in Rk , and a parameter ε, they prove that there is a set A′,A′ ⊂ A, ofsize O(p/εk), such that for the unweighted p-center models, f (X|A);f (X|A′), we havef (X|A) ≤ (1 + ε)f (X|A′), for any subset X of p points in Rk . With our terminology,when f (X|A) is not zero, rel(X) ≤ ε. (Note that for this unweighted model we clearly havef (X|A′) ≤ f (X|A).)
They call such a set A′ an additive ε-coreset for the p-center problem on A, and showthat it can be constructed in time O(n + p/εk). (It should be pointed out that the additiveε-coreset that they define is more general. We use a simplified version sufficient for ourpresentation and discussion.)
We note that with the above results, if we let X∗ and X′ denote optimal solutions forf (X|A) and f (X|A′), respectively, then one can show that
0 ≤ f (X′|A) − f (X∗|A) ≤ εf (X∗|A).
3.3 Other related work, and literature conclusions
In this section we discuss a few papers that do not fit into our classification scheme. Alsowe draw some conclusions from our study of the literature.
Webber (1980) considered a spatial interaction model, which expressed the idea that theproportion of flows pij from origin i to destination j depends on the attractiveness of j ,typically measured as the “size” mj of j , and some function of the distance dij betweeni and j : m
γ
j f (dij ), where γ is a parameter reflecting the significance of the size of thedestination as an attractive force. The author argued that the use in spatial interaction modelsof parameter estimates that are derived using the assumption that all individuals in a zoneare located at the center of the zone is flawed. He indicated that computations that considerinterzonal distance to the distance between the zone centers give rise to biased estimators.Instead, Webber used maximum likelihood methodology to propose the use of estimatorsthat eliminate such bias.
Rodriguez-Bachiller (1983) looked into errors introduced with the use of intercentroiddistances for measuring separation between discrete zones, especially for spatial modelsinvolving a power or exponential function of distance. He proposed an error-correcting
approach which utilized the following formula: Iij =√
d2ij + d2
ii + d2jj , where dii (djj ) is
the average distance between all the points in zone i(j) to its centroid, and dij is the in-tercentroid distance between zones i and j . Modifications to this transformation formuladepending on whether the underlying network of interest exhibited low connectivity or con-tained so-called “privileged” links were also provided. The error correction transformationproposed was tested on one particular urban area, using three levels of aggregation (119, 29,and 4 zones), and reduction in the distortion was reported. Rodriguez-Bachiller character-ized his approach as “purely intuitive.”
Rushton (1989) focused on the degree to which locational complexity and geographicalcomplexity in a location model influence solution quality. The former refers to the iden-tification and evaluation of a large number of potential locations; the latter refers to the
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representation of the geography of an area. The level of spatial aggregation employed in amodel falls within the latter concern. In particular, Rushton considered the effect of employ-ing discrete structures to represent continuously distributed data. He questioned Goodchild’s(1979) statement that “aggregation tends to produce much more dramatic effects on locationthan on the values of the objective function,” and asserted that optimal locations are sensitiveat some level of data aggregation. He identified three ways to avoid error due to aggrega-tion, namely, (1) data disaggregation, (2) the removal of Type A and B errors (as outlinedby Current and Schilling 1987), and (3) modeling and solving the problem in continuousspace. Still, these suggestions do not come with clear cut methodologies, as for instance,the disadvantages of having large data sets with disaggregated data remain. Other geograph-ical representation issues Rushton analyzed were the representation of distance measures,the effect of representing boundaries to solution quality and the accuracy of modeling in-teraction patterns. In the end, Rushton argued for more explicit attention to the tradeoffsin the level of detail employed for capturing complexity—locational and geographical—inlocation problems.
Drezner (1995b) and Drezner and Drezner (1997), consider a market share location prob-lem for a single server. There are competitors with known locations competing for marketshare, and demand points providing the market. While the approach is not demand pointaggregation, it is related to it. It is assumed effectively that there are an infinite number ofDPs, so that the original (continuous) model, say f (X), is one involving integrals (instead ofsums) and the location X of a single server. The continuous model is compared to a discreteone involving sums, say f ′(X). The discrete model is constructed using what amounts toDP aggregation to obtain a finite number of ADPs. The authors compare graphs of surfacesof f (X) and f ′(X), and find f ′(X) has local optima that f (X) does not have. With the dis-crete model they also consider location error graphically, and find as well that “calculationof market share is inaccurate when two facilities are located close to one another and oneis located near a (aggregate) DP whereas the other is not.” Thus errors similar to Type A/Berrors for the PMM can occur. All the computational experience is for the case when theDPs are uniformly distributed over a given planar set of known area. Also for this uniformcase, the authors reduce A/B errors by using a modified distance measure in f ′(X) based ona “distance correction” approach.
Hale and Hale (2000) considered solving a single facility planar location problem, withone-dimensional barriers to travel. The motivating problem was to locate a facility in a citydivided by a river spanned by bridges. Travel from one side of the city to the other was thusby a bridge that yielded a shortest travel distance among all the bridges. The river representeda barrier to travel that could be “penetrated” at the bridge points. The bridges effected asimplification of the problem of computing the average travel distance between a point onone side of a river and any subregion on the opposite river side. Connections were madeto a substantial literature involving the use of integrals to compute average travel distancesbetween regions of various shapes, and there was a good literature discussion. The use ofintegrals can be viewed as a sort of DP aggregation. There was no computational experience,and no aggregation error measure was used. One integral was evaluated to illustrate theaverage travel distance idea assuming uniformly distributed DPs.
Hale et al. (2000) considered a planar region with DPs for which a PMM is to be solved.They assumed the region was partitioned in some way into N∧ subregions. Assuming a DPdensity function was given and making an independence assumption, they computed theaverage distance between each pair of subregions using an integral. They then constructeda network with N∧ nodes, where the length of arc (i, j) was the average distance betweensubregions i and j . Each node i representing a subregion of positive area had a self-loop
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representing the average distance between any two points in the subregion. They then usedthe network information to construct an uncapacitated facility network location model torepresent an aggregated approximation to the original PMM for the planar region. One nu-merical example was given of a network with 5 nodes assuming uniformly distributed DPsand rectilinear distances. The resulting uncapacitated facility location model was solved bycomplete enumeration for p = 2. One facility location, instead of being found, was local-ized to a subregion. There was no computational experimentation, and no error measure wasreported.
We draw the following conclusions from our literature survey:
1. There is much more aggregation literature for median than for center, covering and othermodels;
2. The work of Hillsman and Rhoda is widely recognized and influential; in particular,self-canceling error is a helpful concept for models with additive structure;
3. Justifying theory for using centroids as ADPs is limited to median models;4. Aggregation error bounds can be useful, particularly for center and covering models;5. Location modelers should be aware of the law of diminishing returns for aggregation
error;6. Although there are clear tradeoffs in doing aggregation, there are no analytical models
for addressing them;7. There is little average-case analysis of aggregation error;8. Progress is definitely being made in understanding aggregation error;9. DP data for aggregation algorithm testing is often computer-generated instead of being
real data;10. Sometimes the number of DPs used in testing aggregation methods is so small that
aggregation is not really necessary, although this can depend on the computational stateof the art at the time the work was done;
11. There appears to be little or no theoretical basis for the sometimes implicit assumptionthat conclusions for aggregation of “small” problems scale up to similar conclusions for“large” problems;
12. The variety of findings for location error, and the use of heuristics to compute X∗and X′, support statements in Sect. 2 that this error is difficult to deal with;
13. Aggregation error measures used vary greatly, and there is no agreement on how tomeasure error; hence it is pointless to ask which aggregation algorithm is best, since“best” is not defined.
4 ADP-DP distances, SAND functions and aggregation error bounds, constraintaggregation, big problems versus small ones
This section deals with specific aggregation concepts in more detail. In Sect. 4.1 we pointout worst-case failings of some aggregation error measures and suggest a repair based onADP-DP distance bounds. In Sect. 4.2 we consider the prevalent case when the real-valuedcosting function g with domain Rn+ has special structure. We say that g is subadditive (SA)if for every U,V ∈ Rn+, we have g(U + V ) ≤ g(U) + g(V ). We say that g is nondecreasing(ND) if for every U,V ∈ Rn+ with U ≤ V we have g(U) ≤ g(V ). We say that g is SANDif it is both subadditive and nondecreasing. When g is SAND, there are easily computederror bounds, as well as a Lipschitz-like property for the location function f (X : A) (withpossible implications for location error) that we believe to be new. In Sect. 4.3 we consideraggregation for constraints of location models, concentrating on the CLM. In Sect. 4.4 we
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Table 5 Example of aggregation and aggregation errors for unweighted planar 2-median X = {(0,4), (0,2)}using closest rectilinear distances, with N1 = {1,2},N2 = {3,4,5}
j aj a′j
D(X,a′j) D(X,aj ) ej (X) abci (X), i = 1,2 d(a′
j, aj )
1 (−r,1) (r,2) = α1 r r + 1 −1 2r + 1
2 (−r,2) (r,2) = α1 r r 0 abc1(X) = −1 2r
3 (−r,3) (r,4) = α2 r r + 1 −1 2r + 1
4 (−r,4) (r,4) = α2 r r 0 2r
5 (−r,5) (r,4) = α2 r r + 1 −1 abc2(X) = −2 2r + 1
f (X : A′) = 5r f (X : A) = 5r + 3 e(X) = −3, eb = 10r + 3
ae(X) = 3
discuss DP aggregation for PCMs and CLMs where big models are not like small ones,thus raising the question of the validity of drawing conclusions about aggregation for largemodels based on the study of aggregation for small ones.
4.1 Error measure comparisons
Table 5 presents a small aggregation example. The example serves to illustrate some basicaggregation ideas, as well as worst-case limitations of some aggregation error measures.There are five DPs, N = {1, . . . ,5};N1 = {1,2},N2 = {3,4,5} is the partition of N definingthe aggregation. DPs are aggregated into two ADPs α1 and α2, as shown; each ADP is aclosest one to its DP. For the given X, closest distances D(X,aj ) and D(X,a′
j ) are alsoshown. Since the example model is unweighted, ej (X) = D(X,a′
j ) − D(X,aj ) for j ∈ N ,so abc1(X) = e1(X) + e2(X) = −1,abc2(X) = e3(X) + e4(X) + e5(X) = −2. Distancedifference errors are identical to DP errors, and are all nearly zero. Total DP error e(X) isalso nearly zero so absolute error is nearly zero, as are the ABC errors. We have rel(X) =|f ′(X)−f (X)|/f (X) = 3/(5r + 3), which is small for large r . Other choices of 2-mediansplaced on the y-axis would lead to similar conclusions.
Note the value of r is arbitrary, so that (last column) the ADPs can be arbitrarily farfrom the DPs. Hence the error measures of the last paragraph, which have small values,are misleading for this X. This example demonstrates that local error measures may bemisleading, and that we really need the error measures to be nearly zero for all X. Thisneed for all the error measures to be small will lead (below) to a relationship between theADP-DP distance error measures and the others in Table 5.
If we use Table 5 to illustrate the unweighted 2-center model, then f (X : A) =max{D(X,aj ) : j ∈ N} = r + 1, f (X : A′) = r,ae(X) = 1, and rel(X) = 1/(r + 1) which issmall for large r . Except for the error bound measure, eb = 2r + 1, the error measures thatare defined for this model are unreliable for the given X.
The above example is a small one, but it is possible to construct examples with manyDPs where the error measures still fail in a similar manner.
The following observation is a basis for finding an aggregation criterion that holds forall X.
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ADP-DP distance bounds
(a) For any DP aj and its ADP a′j , j ∈ N , and any p-server X ⊂ S, the ADP-DP distance
differences satisfy
−d(a′j , aj ) ≤ D(X,a′
j ) − D(X,aj ) ≤ d(a′j , aj ) ⇔
|D(X,a′j ) − D(X,aj )| ≤ d(a′
j , aj ).
(b) We have max{|D(X,a′j ) − D(X,aj )| : X a p-server, X ⊂ S} = d(a′
j , aj ).
Part (a) above is presented in Francis and Lowe (1992), and is due to the triangle inequalityfor distances. Part (b) follows from part (a) and the following:
ADP-DP observation If aj ∈ X and is the closest point in X to a′j , or a′
j ∈ X and is theclosest point in X to aj , then |D(X,a′
j ) − D(X,aj )| = d(a′j , aj ).
Let ε be any small positive number. We now relate ADP-DP distances to distance differ-ence errors.
ADP-DP distance bound claim For any ε > 0, the following are equivalent:
−ε ≤ D(X,a′j ) − D(X,aj ) ≤ ε for all X ⊂ S, j ∈ N,
d(a′j , aj ) ≤ ε for all j ∈ N,
max{d(a′j , aj ) : j ∈ N} ≤ ε.
Proof The following are all clearly equivalent:
−ε ≤ D(X,a′j ) − D(X,aj ) ≤ ε for all X ⊂ S, j ∈ N,
|D(X,a′j ) − D(X,aj )| ≤ ε for all X ⊂ S, j ∈ N,
max{|D(X,a′j ) − D(X,aj )| : X ⊂ S} ≤ ε, j ∈ N,
d(a′j , aj ) ≤ ε, j ∈ N, (using (b) above)
max{d(a′j , aj ) : j ∈ N} ≤ ε. �
The above claim leads to a rationale to consider ADP-DP distances when doing aggre-gation; having them all small guarantees that other error measurements are also small forall X. Note, since max{d(a′
j , aj ) : j ∈ N} ≤ ε, the smallest that ε can be for the given ADPsis max{d(a′
j , aj ) : j ∈ N}.
ADP-DP error measure guarantee Consider the PMM, and let W = w1 + · · · + wn. Ifd(a′
j , aj ) ≤ ε for all j ∈ N , then
(0) (a) whenever D(X,a′j ) ≤ rj , then D(X,aj ) ≤ rj + ε, j ∈ N ,
(b) whenever D(X,a′j ) > rj , then D(X,aj ) > rj − ε, j ∈ N ,
(1) |ej (X)| ≤ wjε for j ∈ N and all X ⊂ S,(2) |e(X)| ≤ |e1(X)| + · · · + |en(X)| ≤ Wε and all X ⊂ S,(3) |abci (X)| ≤ ωiε for all X ⊂ S, i = 1, . . . ,m, (ωi ≡ ∑{wj : j ∈ Ni}),(4) ae(X) ≤ Wε for all X ⊂ S,(5) mae(f ′, f ) ≤ Wε,
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(6) eb = Wε is a valid error bound.
The conclusion we draw from the example of Table 5, and the above guarantee, is that ifwe keep all ADP-DP distances small in doing a PMM aggregation then other aggregationmeasures will also be small. If we do not, then Table 5 illustrates that other aggregationmeasures can be small but we can have a bad aggregation. In effect, if we omit from our setof aggregation error measures the condition that all ADP-DP distances are small, then theremaining error measures (2 through 8 of Table 1) can be incomplete and unreliable.
A similar observation and guarantee apply to the PCM, f (X : A) = max{wjD(X,aj ) :j ∈ N}. Define Wmax = max{wj : j ∈ N}. To obtain the observation from the one above,replace W by Wmax in items (4) through (6) above; omit items (2) and (3) (which do notapply).
4.2 SAND costing functions, error bounds, location error, and the Lipschitz condition
The topic of using ADP-DP distances for aggregation evaluation is closely related to aggre-gation error bounds. These bounds avoid the limitations of some aggregation error measuresby using ADP-DP distances to define the error bounds, thus implicitly using the ADP-DPdistance aggregation error measure. They also incorporate the model cost structure. It isworth considering these error bounds since the bounds are useful, easy to compute, and ap-ply to a large class of models (see Francis et al. 2000, 2004a, 2004c). The error bounds leadnaturally to aggregation algorithms and have been found to be computationally useful forPMMs (Francis et al. 1996), PCMs (Rayco et al. 1997, 1999), and CLMs (Emir-Farinas andFrancis 2005).
Our location model of interest is given by f (X : A) = g(D(X,A)), with D(X,A) =(D(X,aj )) ∈ Rn+ the n-vector of closest distances, and g a costing function. From Sect. 1,taking g to be the sum-function or the max-function gives the PMM or PCM respec-tively. Each of these g functions is an instance of a larger class of SAND functions con-sidered. These SAND costing functions provide a unifying aggregation theory as wellas a sort of Lipschitz condition relating location error �(X′,X∗) and optimality error|f (X′ : A) − f (X∗ : A)|.
Carrizosa et al. (2000) studied a model related to the one we discuss. They consideronly single-facility models. They assume g is nondecreasing and quasiconcave. Their work,like ours, involves using approximating distance functions, but these are defined by generalgauges. Also their work, like ours, involves Lipschitz conditions, which we consider in moredetail in the Appendix. There is a substantial location literature involving Lipschitz costingfunctions, for a review see Hansen et al. (1995); for specific uses see Plastria (1992) andRomero-Morales et al. (1997).
Let g be a real-valued function defined on Rn+. We assume that g is SAND. It is knownthat g being SAND implies 0 ≤ g(U) for all U ∈ Rn+. Often it is also reasonable to assumeg(0) = 0, giving f (X : A) = 0 if D(X,A) = 0. If g is SAND with g(0) = 0 and g(U) > 0for every nonzero U ∈ Rn+, we say that g is super-SAND, and write g is SSAND. The maxfunction and the sum function are SSAND.
Basic SAND inequalities Suppose g is SAND, with domain Rn+. For any U1,U2,U3 ∈ Rn+with U1 ≤ U2 + U3 and U2 ≤ U1 + U3, we have g(U1) ≤ g(U2) + g(U3) and g(U2) ≤g(U1) + g(U3) using the ND and SA properties in turn. Thus |g(U2) − g(U1)| ≤ g(U3).
We now use the fact that closest-distances satisfy several triangle inequalities (Zemel1985; Francis and Lowe 1992) and the SAND inequalities, to get several useful bounds.
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Closest-distance DP triangle inequalities Given the DPs aj and ADPs a′j , j ∈ N , de-
fine the vector of ADP-DP distances δ(A′,A) = δ(A,A′) = (d(a′j , aj )) ∈ Rn+. We have
D(X : A) ≤ D(X : A′) + δ(A′,A),D(X : A′) ≤ D(X : A) + δ(A,A′) for every p-serverX ⊂ S.
Basic SAND error bound Applying the basic SAND inequalities to the above triangleinequalities, with eb ≡ g(δ(A′,A)), gives |g(D(X,A′)) − g(D(X,A))| ≤ eb for every p-server X ⊂ S.
For examples of error bounds when either g(U) = w1u1 + · · · + wnun or g(U) =max{w1u1, . . . ,wnun}, we have either eb = w1d(a′
1, a1) + · · · + wnd(a′n, an) or eb =
max{w1d(a′1, a1), . . . ,wnd(a′
n, an)}, giving error bounds for either the PMM or PCM re-spectively. For the 2-median example of Table 5 with all wj = 1, eb = 10r +3 which is largeif r is large. If r = 0, then eb = 3 = ae(X), illustrating the error bound can be tight. For therelated 2-center example also based on the data of Table 5, f (X : A) = r +1, f (X : A′) = r ,ae(X) = 1, eb = 2r + 1 and suggests a bad aggregation. If r = 0 then ae(X) = eb, so theerror bound is tight. Note the error bound g(δ(A′,A)) is necessarily worst-case, since it isvalid for all X ⊂ S. The PCM error bound does not capture the self-canceling error con-cept that occurs with additive models like the PMM. When g is SSAND it is known thatg(δ(A′,A)) defines a distance on the set of all possible pairs of n-tuples of DPs.
There are several costing functions g determined to be SAND (Francis et al. 2000), whichinclude many others as special cases. The convex ordered median costing function discussedbelow unifies the PMM, PCM, p-centrum and p-centdian location models. This functionwas introduced independently by Nickel and Puerto (1999), Rodriguez-Chia et al. (2000),and Puerto and Fernandez (2000). A second instance of g of interest is the �p-norm costingfunction (Francis et al. 2000) studied by Shier and Dearing (1983) and Zemel (1985). Weconsider these two functions after stating the following property.
Basic SAND scaling property Suppose g is SAND with domain Rn+, and B is any n × n
matrix with all nonnegative entries. Then the function gB(U) ≡ g(BU) with domain Rn+ isSAND.
The property gives an easy way to build weights into a SAND costing function. Forexample, suppose B = (bij ) is a diagonal weight matrix, defined as a diagonal matrix withbjj = wj , all j ∈ N . The �p-norm function g(U) = ((u1)
p +· · ·+ (un)p)(1/p) is known to be
SAND for p ≥ 1. Therefore gB(U) = [(w1u1)p + · · · + (wnun)
p](1/p) is also SAND. Hencethe �p-norm location model gB(D(X,A)) is a SAND location model.
Now consider the convex ordered median model. Given any U ∈ Rn+, denote its compo-nents in decreasing order by u[1] ≥ u[2] ≥ · · · ≥ u[n]. Let λ1 ≥ · · · ≥ λn ≥ 0 be any sequenceof real constants. The function g(U) = ∑
j=1,...,n λju[j ] is known to be a SAND functionwith domain Rn+. Suppose again B is a diagonal weight matrix, and we modify g to ob-tain gB . Then gB(D(X : A)) is a SAND location model. If λ1 = 1 and λj = 0 otherwise,the result is the weighted PCM. If λj = 1 for all j ∈ N , the result is the weighted PMM. Ifλ1 = 1 and λj = α otherwise for some α with 0 < α < 1, we get the p-centdian model. Thecase λ1 = · · · = λk = 1 and λj = 0 otherwise gives the k-centrum model.
Consider now a Lipschitz-like result for the SAND location function. Francis and Lowe(1992) prove the following result.
Closest-distance and min-max distance p-server triangle inequalities Let e be the n-vector of ones, and A the DP vector. With the definition of �(X,Y ) from Sect. 2, for anytwo p-servers X and Y ⊂ S we have
Applying the basic SAND inequalities to (1) and (2), we obtain the following:
Lipschitz-like bound For any two p-servers X and Y ⊂ S we have
|f (X : A) − f (Y : A)| ≤ g(�(X,Y )e).
If g(0) = 0 we note the bound is tight with X = Y .We remark that the bound also applies to the approximating model f (X : A′).Using the Lipschitz-like bound, we conclude that |f (X′ : A) − f (X∗ : A)| ≤
g(�(X′,X∗)e), thus obtaining a new relationship between absolute optimality error andlocation error. If an aggregation algorithm can compute X′ satisfying �(X′,X∗) ≤ r , thenwe conclude (since g is ND) that |f (X′ : A) − f (X∗ : A)| ≤ g(re).
For our purposes, an actual Lipschitz inequality is of the form |f (X : A) − f (Y : A)| ≤L�(X,Y ) for some positive Lipschitz constant L, and holds for all p-servers X and Y .Consider now some conditions on the function g for which we obtain this inequality orsomething quite similar.
Comment
(a) If g is also convex, then (Rockafellar 1970, Thm. 4.7) it is known to be positively ho-mogeneous of degree 1. This means g(�(X,Y )e) ≤ �(X,Y )g(e), which gives the Lip-schitz condition with Lipschitz constant L = g(e).
(b) If g is SAND and also positively homogeneous of degree t, t ≥ 1 (Rockafellar 1970p. 135), then g(�(X,Y )e) ≤ (�(X,Y ))tg(e).
We next note that if g is SSAND then the function g(�(X,Y )e) can be interpreted as adistance.
Lipschitz bound distance property For any two p-servers X and Y ⊂ S, define β(X,Y ) =g(�(X,Y )e). If g is SSAND, then β(X,Y ) is a distance defined on the set of all pairs ofp-servers.
Examples with β(X,Y ) = L�(X,Y ) for some Lipschitz constant L For each example, g
is homogeneous of degree 1.
(1) gB is the �p-norm costing function with diagonal weight matrix B for p ≥ 1: we haveL = (w
p1 + · · · + w
pn )(1/p).
(2) gB is the convex ordered median costing function with diagonal weight matrix B , andwith ordered nonnegative DP weights w[1] ≥ w[2] ≥ · · · ≥ w[n]: we have L = λ1w[1] +· · · + λnw[n]. Note an appropriate choice of the λ′s from {0,1} gives either L = W orL = Wmax for the PMM and PCM respectively.
Following Francis et al. (2004c), the Lipschitz results may be useful if we wish to aggregatesome large set of feasible solutions into a smaller set. Let FS be some set of potential p-servers and denote by FSagg some aggregation of FS, possibly finite, with the property thatFSagg ⊂ FS. We say that FSagg is r-close to FS if for each X ∈ FS there exists a YX in FSagg
with �(X,YX) ≤ r . Suppose X∗ and Y ∗ are global minimizers of f (X : A) over FS and
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Table 6 Alternative relaxations and restrictions of both the original and aggregated covering location modelsassuming all δj < rj : approach I is tighter than approach II
I. Individual constraint approach II. Combined constraint approach
1 Definitions α1, . . . , αm: the m distinct ADPs f (X) ≡ max{D(X,aj )/rj : j ∈ N}δj ≡ d(a′
j, aj ), j ∈ N; δj < rj , j ∈ N f ′(X) ≡ max{D(X,a′
2 Original D(X,aj ) ≤ rj , j ∈ N , all X f (X) ≤ 1, all X
covering
constraints
3 Aggregate D(X,a′j) ≤ rj , j ∈ N , all X f ′(X) ≤ 1, all X
constraints
4 Restrictions of D(X,a′j) ≤ rj − δj , j ∈ N , all X ⇔ f ′(X) ≤ 1 − eb, all X ⇔
both original D(X, αi ) ≤ βi , i = 1, . . . ,m, all X D(X,a′j) ≤ rj (1 − eb), all X,j ∈ N
and aggregate
constraints
5 Relaxations D(X,a′j) ≤ rj + δj , j ∈ N , all X ⇔ f ′(X) ≤ 1 + eb, all X ⇔
of both original D(X, αi ) ≤ γi , i = 1, . . . ,m, all X D(X,a′j) ≤ rj (1 + eb), all X,j ∈ N
and aggregate
constraints
FSagg respectively. It is direct to use the Lipschitz inequality, with the definitions of X∗ andY ∗, to conclude
f (Y ∗ : A) − g(re) ≤ f (X∗ : A) ≤ f (Y ∗ : A) ⇔1 − g(re)/f (Y ∗ : A) ≤ f (X∗ : A)/f (Y ∗ : A) ≤ 1.
For example, FS might consist of all p-servers contained in a network, while FSagg mightbe the set of all p-servers with each server a vertex. If 2r is the length of the longest arc inthe network, then FSagg is r-close to FS and is contained in FS. Thus if g(re)/f (Y ∗ : A) issmall, then Y ∗ may be an acceptable approximation to X∗.
4.3 Constraint aggregation
Aggregation can occur in constraints (Francis et al. 2004a, 2004b), instead of the objective.For example, consider the CLM with n constraints whose aggregation was illustrated inSect. 1. If only m of the ADPs in the aggregated CLM are distinct, then n − m of theaggregated CLM constraints are redundant, and may be deleted as in the example of Sect. 1or as shown in Table 6.
Let us now develop a basic error bound idea for constraints. Generally, we have loca-tion constraints of the form fj (X) ≤ rj , j ∈ N,X ⊂ S. Suppose each function fj (X) isreplaced by some approximating function, say f ′
j (X), giving non-distinct constraints forthe aggregated model of f ′
j (X) ≤ rj , j ∈ N,X ⊂ S. If we now define functions f (X) andf ′(X) by f (X) ≡ max{(1/rj )fj (X) : j ∈ N}, f ′(X) ≡ max{(1/rj )f
′j (X) : j ∈ N}, then
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the constraints for the two models are equivalent to f (X) ≤ 1 and f ′(X) ≤ 1 respec-tively. Hence we can view f ′(X) as an aggregated version of the function f (X), and ap-ply whatever function error measures are of interest. It is known (Francis et al. 2004a),for example, that if f ′
j (X) and fj (X) have error bound bj (= d(a′j , aj ) for the CLM) for
j ∈ N , then f (X) and f ′(X) have the (unitless) error bound eb = max{bj/rj : j ∈ N}.For the CLM, the resulting error bound is identical in form to that for the PCM; henceaggregation methods providing small PCM error bounds also can provide small CLM er-ror bounds, and vice-versa. Note also, since the constraints of the CLM may be written asf (X : A) ≡ max{(1/rj )D(X,aj ) : j ∈ N} ≤ 1, that a Lipschitz bound applies to this func-tion f .
When f (X) and f ′(X) are any original and aggregated functions with some error boundeb, it follows directly that f ′(X) ≤ 1 − eb ⇒ f (X) ≤ 1;f (X) ≤ 1 ⇒ f ′(X) ≤ 1 + eb. Thusthe constraint f ′(X) ≤ 1 − eb gives a restriction of the original constraint, while f ′(X) ≤1 + eb gives a relaxation. Each can be easier to deal with than the original constraint andmay be used to compute lower and upper bounds on the optimal objective function value ofthe original model. Supposing eb � 1 (which is clearly desirable), feasibility conclusionsabout one model thus allow us to draw feasibility or “near-feasibility” conclusions about theother model.
Following Francis et al. (2004c), Table 6 illustrates the use of error bounds as discussedto obtain relaxations and restrictions of the aggregated CLM that are also relaxations andrestrictions of the original model. We can aggregate the individual constraints I (see Table 6),or deal with the combined constraints II. There is a resulting interesting modeling insight:Even though I and II give equivalent formulations of the original CLM, I gives at least astight a relaxation and at least as tight a restriction as II. This is because we have rj + δj ≤rj (1+ eb), rj (1− eb) ≤ rj − δj , for all j ∈ N . This insight is very much in the same spirit asthat of Cornuejols et al. (1977) for alternative formulations of the PMM as a mathematicalprogram.
Francis et al. (2004c) used approach I of Table 6. They solved to optimality a CLM withalmost 70,000 original CLM constraints by solving several aggregated CLMs each withless than 1,000 covering constraints. Their computational experience was usually that theminimal objective function value of the original model was underestimated when solvingthe approximating model without enough ADPs, which is consistent with the discussion inSect. 1. For a successful use of this aggregation approach to a CLM application see Dekleet al. (2005).
The error bound max{wjd(a′j , aj ) : j ∈ N} for the PCM and CLM for some choice of the
wj including wj = 1/rj is quite robust. It applies to an obnoxious facility location model(Francis et al. 2000; Erkut and Neuman 1989) and, when doubled, to a p-center hub locationmodel (Gavriliouk 2003; Ernst et al. 2002a, 2002b). It also occurs in the ADP-DP DistanceBound Claim and in a penalty function approach we next consider.
Emir-Farinas and Francis (2005) considered maximum violation error for the CLM,a special type of constraint penalty function. For constraint j,D(X,aj ) ≤ rj , (D(X,aj ) −rj )
+ ≡ max{D(X,aj ) − rj ,0} is the amount by which the constraint is violated, and canbe scaled by dividing by rj . Let mve(X) = max{(D(X,aj ) − rj )
+/rj : j ∈ N},mve′(X) =max{(D(X,a′
j ) − rj )+/rj : j ∈ N} denote the maximum (scaled) constraint violations for
the original and aggregated CLMs. Each of the penalty functions, mve(X) and mve′(X), isalways nonnegative and is zero if and only if X is feasible to the underlying problem. Ide-ally, we want each error measure to be zero if and only if the other is zero. More realistically,if one measure is zero we guarantee the other is small by having a small error bound on thequantity |mve′(X) − mve(X)|. We summarize this discussion in column 2 of Table 7. Theresult of Emir-Farinas and Francis appears in the last row of column 2 of Table 7.
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Table 7 Illustration of constraint penalty approach to covering constraints, and to general SAND constraintswith any SSAND penalty costing function g
Item name Covering example penalty analysis General SAND constraints
penalty analysis
Original constraint j D(X,aj ) ≤ rj , j ∈ N cj (X) ≤ rj , j ∈ N
Constraint j in standard D(X,aj )/rj ≤ 1, j ∈ N fj (X) ≡ cj (X)/rj ≤ 1, j ∈ N
form
Aggregated constraint j , D(X,a′j)/rj ≤ 1, j ∈ N f ′
Maximum violation mve′(X) ≡ max{ve′j(X) : j ∈ N} π ′(X) ≡ g(P (X : F ′))
error/composite constraint
penalty, aggregate
constraints
Composite error bound on ceb ≡ max{d(a′j, aj )/rj : j ∈ N} ceb ≡ g(EB)
absolute error of
maximum violation
error/composite
constraint penalty
Error bound inequality |mve′(X) − mve(X)| ≤ ceb, all X. |π ′(X) − π(X)| ≤ ceb, all X
Emir-Farinas and Francis (2005) presented several algorithms for aggregating DPs forCLMs assuming rectilinear distances. They considered four different infeasibility measures,including maximum violation error and related average violation error. They found boththeoretically and computationally that the choice of an infeasibility measure had a quitesubstantial effect on the best choice of an aggregation method. Their algorithms exploitCLM structure and allow the user to control the aggregation error by specifying a maximum
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allowable error bound value as an input to the algorithm. Their computational experienceindicated that the error bound was a good surrogate for the maximum absolute error.
Following Francis et al. (2004a), consider a generalization of the CLM and of the abovepenalty approach to constraint aggregation. Denote the original constrained location modelas follows: (Prorig) : minf0(X) s. to F(X) ≤ e,X ⊂ S. Here e is a vector of ones, eachentry fj (X) in F(X) ∈ Rn+ is SAND with n ≡ |N |. The functions fj (X) are distinct. Themodel is in standard form, in the sense that any constraint of the form cj (X) ≤ rj withcj (X) a SAND function, is rewritten equivalently as fj (X) = (1/rj )cj (X) ≤ 1. Column3 of Table 7 illustrates how to extend the penalty function approach of Emir-Farinas andFrancis to the more general context of any SAND constraints and any SSAND function g,including the max-function and the sum-function. Results in column 2 are a special case ofthose in column 3. The principal conclusion from Table 7 is in the bottom row of column 3.Note, if X is feasible to the aggregation of (Prorig), then |g(P (X,F )) − g(P (X,F ′))| ≤ cebbecomes g(P (X,F )) ≤ ceb thus giving a bound on how “infeasible” X can be to (Prorig).
There is a direct generalization of this penalty approach if the SAND objective func-tion f0(X) is also replaced by some approximating SAND objective function f ′
0(X) andeach penalty function pen(X,fj ) is multiplied by some appropriate positive weighting con-stant kj .
4.4 Big aggregation problems may not be like small ones
DP aggregation may be needed only when there are many DPs. While we may obtain in-sight by aggregating small models, as Table 5 illustrates, there is some evidence that bigaggregation models may not be like small ones. For some large aggregation models we canobtain simple closed-form “square root” formulas for minimal aggregation error bound val-ues, whereas this has not been the case for small models. This finding necessarily raisessubstantial questions about the validity of extrapolating from aggregation analysis of smallmodels to analysis of big models.
Frieze (1980), Marchetti-Spaccamela and Talamo (1983) and Zemel (1985) consideredthe case when the n DPs are uniform and iid random variables distributed on some planarset T of area a. The set T must satisfy a (weak) boundary regularity condition. Theseauthors studied the PMM with each weight 1/n and/or the unweighted PCM with Euclideandistances. Given some rather technical assumptions, they derived asymptotically accurate(for large values of p and n) “square root” approximation formulas of the form c
√(a/p)
for the model minimal objective function values. The formulas are based on tiling T with p
regular hexagons. The constant c is given by c = 0.3772 and c = 0.6204 for the PMM andPCM respectively.
For the rectilinear distance unweighted PCM, Francis and Rayco (1996) derived an ap-proximating minimal objective function value of c
√(a/p) with c = 0.7071. Again the set
T must satisfy a weak boundary regularity condition (different from Zemel’s). This value isasymptotically accurate as p and n increase and is a valid upper bound on the actual minimalobjective function value even for smaller p values. It does not require DPs to be modeled asuniform iid random variables but substitutes instead the assumption that every point in T isa DP. The authors observed, since the error bound for a PCM with m ADPs is an m-centermodel, that 0.7071√
(a/m) gives an approximate value for the error bound. The formula isbased on tiling T with m “diamonds” (squares rotated 45-degrees relative to the axes).
Francis et al. (2004b) used the square root formulas for aggregation error bound analy-sis of a family of both the PCM and CLM that decompose in a natural way into smaller“separate community” models. They also provide computational evidence, assuming rec-tilinear distance, that the formula is a good approximation to the error bound value for an
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aggregation algorithm presented by Emir-Farinas and Francis (2005) for m ≥ 900 (and over-estimates it for smaller m). It is critically important in using the square root formula to havean accurate estimate of the set T and its area a. For a DP set with about n = 70,000 fromPalm Beach County, Florida, their best choice of the area a of the set T was about 31% ofthe land area of the entire county. Major portions of the county have no DPs, and must beexcluded from consideration to obtain an accurate estimate. This exclusion can be viewedas beginning with a DP data set where the DPs are far from being uniformly distributed.Then the data set is reformulated to one where the uniform approximation is reasonable,thus improving the accuracy of the approximating model. If no such exclusion is made, thenwhat one obtains is likely to be an overestimate of the error bound.
How many ADPs are enough? The square root formulas imply that the foregoing questionis meaningless, since we must know not only m but also the area a in order to use theformulas. Even if we know both a and m, the aggregation tradeoffs discussed in Sect. 2should be carefully considered in order to provide a reasoned response to the question. Ifinstead, we can answer the question “How much error is acceptable?” then the square rootformulas can answer the original question. Given a and an acceptable error (bound) valueof ea , then the inequality ea ≥ c
√(a/m) gives m ≥� ((c/ea)
2a)�, where �Z�is the smallestinteger greater than or equal to Z.
The square root formulas provide some theoretical basis for the law of diminishing re-turns for DP aggregation for a class of PCMs and CLMs: DP aggregation error decreases at adecreasing rate as the number of ADPs increases. While the formulas are only approximate,using them may be better than relying on professional judgment.
Unfortunately the law of diminishing returns for error bounds as the number of ADPS m
increases does not always hold. See Francis et al. (2002a) for details for the PCM and PMM.
Acknowledgements We are happy to thank Xiuli Chao for an insightful discussion of ABC error. Wethank the operators of the EWGLA and SOLA web sites, Frank Plastria and Trevor Hale respectively, fordistributing our reference queries, as well as the colleagues who responded to these queries. In particular,we thank Emilio Carrizosa, Sanjay Melkote, Stefan Nickel and Frank Plastria for their help with referencesinvolving the Lipschitz condition. We thank a referee for careful editing. A portion of this material was givenin invited presentations at the Seville ISOLDE X meeting in June, 2005.
Appendix: Lipschitz conditions, SAND, and convexity
We need some general definitions.Let g be a real function defined on some nonempty set S ⊂ Rn. We say that g is positively
homogeneous (homogeneous of degree 1) if for any U ∈ S, and a nonnegative r , such thatrU ∈ S,g(rU) = rg(U). g is subhomogeneous if for any U ∈ S, and a nonnegative r , suchthat rU ∈ S,g(rU) ≤ rg(U). g is quasi-homogeneous if for any U ∈ S, and a real r ≥ 1,such that rU ∈ S,g(rU) ≤ rg(U). g is Lipschitzian relative to S if there exists a finite con-stant L such that |g(U)− g(V )| ≤ L‖U −V ‖ for any U,V ∈ S. (‖U −V ‖ is the Euclideannorm.) In this paper we apply the above only in the case when S = Rn+.
The following is a basic result directly derived from the Mean Value Theorem.
Theorem 1 Suppose that S is compact and g is differentiable on S. If the gradient of g isuniformly bounded on S, then g is Lipschitzian.
Consider the location model f (X : A) = g(D(X,A)), where g satisfies the conditions ofTheorem 1 over the compact set S = {(u1, . . . , un) : 0 ≤ ui ≤ D′}, where D′ is the diameter
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of the network, or, in the planar case, the diameter of the convex hull of the DPs. We nowobtain the following Lipschitzian condition for the function f (X : A):
The two inequalities above are due respectively to Theorem 1 and to the fact that the clos-est distance p-server min-max distance triangle inequalities give |D(X,aj ) − D(Y,aj )| ≤�(X,Y ) for j ∈ N and any two p-servers X and Y , (Note that if the DPs belong to somemetric space T ∗, then f (X : A) is a mapping from the Cartesian space T ∗ × T ∗ × · · · × T ∗to the real line.)
The above Lipschitzian result is quite general and does not depend on any subadditivityor homogeneity properties of the costing function g. However, when we consider generalSAND costing functions, differentiability is no longer guaranteed. For example, the centercosting objective function g(U) = max{u1, . . . un} is not differentiable. Nevertheless, thisconvex function is Lipschitzian with L = 1 on Rn+.
More generally, Theorems 10.4 and 24.7 in Rockafellar (1970) ensure that if g is a finiteand convex function and S is a compact set in the relative interior of the domain of g, then g
is Lipschitzian. In particular, the convex ordered median and the �p-norm costing functionsare Lipschitzian. Note that these functions are positively homogeneous, and by Theorem1.4.7 of Rosenbaum (1950) and Theorem 4.7 of Rockafellar (1970), every homogeneoussubadditive function is convex. (In fact, a necessary and sufficient condition that a finiteconvex function is subadditive is that it is quasi-homogeneous, Theorem 1.4.6 in Rosenbaum1950).
Focusing on Lipschitz conditions of SAND functions, the remaining question is whetherall SAND functions are Lipschitzian on compact domains. The answer is no. Let g be de-fined on the real line by g(u) = 1 if u < 1, and g(u) = 2 if u ≥ 1. Then g is SAND but itis not Lipschitzian since it is not continuous at u = 1. (If we omit the requirement that g isND and assume that it is only SA, such a function may even be everywhere discontinuous,as illustrated by the following Example 1.1.8 from Rosenbaum 1950: let g be defined on thereal line, by g(u) = 0 if u is rational, and g(u) = 1 otherwise.)
Finally, we note the following set of conditions ensure a Lipschitz property on the loca-tion model f (X : A) = g(D(X,A)), without assuming explicitly that g is convex or differ-entiable.
Suppose that g is SAND. As in Sect. 4.2 (Lipschitz-Like- Bound), we obtain |f (X :A) − f (Y : A)| ≤ g(�(X,Y )e), where e = (1, . . . ,1) ∈ Rn+. If we further assume that theSAND function g is subhomogenous of degree 1 along the direction e = (1, . . . ,1) (i.e.,for any nonnegative r , such that re ∈ S,g(re) ≤ rg(e)) we get the Lipschitzian inequality|f (X : A) − f (Y : A)| ≤ �(X,Y )g(e).
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