Ann Oper Res (2006) 146:119–134

DOI 10.1007/s10479-006-0051-6

Using error bounds to compare aggregated generalizedtransportation models

Igor S. Litvinchev · Socorro Rangel

Published online: 24 June 2006C© Springer Science + Business Media, LLC 2006

Abstract A comparative study of aggregation error bounds for the generalized transportation

problem is presented. A priori and a posteriori error bounds were derived and a computational

study was performed to (a) test the correlation between the a priori, the a posteriori, and the

actual error and (b) quantify the difference of the error bounds from the actual error. Based on

the results we conclude that calculating the a priori error bound can be considered as a useful

strategy to select the appropriate aggregation level. The a posteriori error bound provides a

good quantitative measure of the actual error.

Keywords Clustering . Network models . Approximation algorithms

Introduction

An important issue in optimization modelling is the trade-off between the level of detail to

employ and the ease of using and solving the model. Aggregation/disaggregation techniques

have been developed to facilitate the analysis of large models and provide the decision maker a

set of tools to compare models that have a high degree of detail with smaller aggregated models

that have less detail. Aggregated models are simplified, yet approximate, representations of

the original complex model and may be used in various stages of the decision making process.

Frequently, a number of aggregated models can be considered for the same original model,

and it is then desirable to select an aggregated model providing the tightest (in some sense)

approximation.

In optimization-based modelling, the difference between the actual optimal objective

function value and the aggregated optimal objective function value can be used as a measure

I. S. LitvinchevComputing Center Russian Academy of Sciences, Vavilov 40, Moscow 119991, Russia

S. Rangel ( )Department of Computer Science and Statistics, UNESP, Rua Cristovao Colombo, 2265,S.J. Rio Preto, SP 15054-000, Brazile-mail: socorro@ibilce.unesp.br

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120 Ann Oper Res (2006) 146:119–134

of the error introduced by aggregation. Many upper bounds on this error have been developed

for the aggregated optimization models and are referred to as error bounds (Leisten, 1997;

Litvinchev and Rangel, 1999; Litvinchev and Tsurkov, 2003, and references therein). Two

types of error bounds are considered: a priori error bounds and a posteriori error bounds.

A priori error bounds may be calculated after the model has been aggregated but before

the smaller aggregated model has been optimized. A posteriori error bounds are calculated

after the aggregated model has been optimized. Error bounds allow the modeler to compare

various types of aggregation and, if the error bounds are not adequate, modify the aggregation

strategy in an attempt to tighten the error bounds (Rogers et al., 1991).

There are two main issues associated with the error bounds. The first one is addressed in

the difference of an error bound from the actual error. The closer the error bound to the actual

error, the better. The second issue is how to compare different aggregation strategies based

on the error bounds. It is not obvious that the aggregation yielding the tightest error bound

has the smallest actual error (Rogers et al., 1991). Moreover different bound calculation

techniques may result in a different relationship (correlation) between the error bound and

the actual error.

The first issue above is relatively well explored, at least for the case of a posteriori error

bounds for linear programming (see Leisten, 1997; Litvinchev and Tsurkov, 2003; Rogers

et al., 1991, and the references therein). The second area is much less investigated. To our

knowledge the first comprehensive study in this direction has been made recently in Norman,

Rogers, and Levy (1999) for the classical transportation problem (TP) where they explored,

among others, the following question:� If a model is aggregated using different methods to create several different smaller models

and an error bound (a priori and/or a posteriori) is calculated for each aggregated model,

does the model with the tightest error bound(s) give the closest approximation of the

objective function value?

Many applications of aggregation/disaggregation theory in optimization involve network

problems such as location models (Francis, Lowe, and Tamir, 2002), the generalized assign-

ment model (Hallefjord, Jornsten, and Varbrand , 1993) and, in particular, the TP. For the

latter both a priori and a posteriori error bounds were developed and investigated (Rogers

et al., 1991; Zipkin, 1980a). In network flow problems the nodes traditionally correspond

to constraints, and arcs to variables. For most aggregation schemes, aggregation of nodes

(constraints) implies or necessitates a corresponding aggregation of arcs (variables). Usually,

only variables and constraints of the same type (e.g., sources or destinations) are aggregated.

The TP is for minimizing the cost of transporting supplies from the set of sources to the

set of destinations, and it is assumed that the transportation flow is conserved on every arc

between the source and the destination. In the generalized transportation problem (GTP) the

amount of flow that leaves the source can differ from the amount of flow that arrives to the

destination. A certain multiplier is associated with each arc to represent a “gain” or a “loss”

of a unit flow between the source and the destination. There are two common interpreta-

tions of the arc multipliers (Ahuja, Magnanti, and Orlin, 1993). For the first interpretation

the multipliers are employed to modify the amount of flow of some particular item. There-

fore it is possible to model situations involving physical or administrative transformations,

for example, evaporation, seepage, or monetary growth due to the interest rates. For the

second interpretation, the multipliers are used to transform one type of item into another

(manufacturing, machine loading and scheduling, currency exchanges, etc.). For a more de-

tailed discussion on the GTP and its applications see Ahuja, Magnanti, and Orlin (1993) and

Glover, Klingman, and Phillips (1990). In contrast to the TP, much less work has been done

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Ann Oper Res (2006) 146:119–134 121

in aggregation for the GTP. Only a posteriori error bounds for aggregation in the GTP have

been developed in Evans (1979).

In this paper we present a study of the correspondence between the error bounds and

the actual error for the aggregated GTP. The paper is organized as follows. In Section 1,

we state the aggregated GTP and consider its properties. In the next section, a family of a

priori error bounds for a class of the GTP is derived, and new a posteriori error bounds are

proposed. A numerical study carried out to compare the quality of the error bounds and to

determine if there are relationships between the error bounds and the actual error is presented

in Section 3. Based on the results of this study we can conclude that there is a significant

difference among the various methods of calculating error bounds. Moreover, the improved a

priori and a posteriori error bounds have a statistically significant correlation with the actual

error.

1. The original and the aggregated models

The original GTP before aggregation (OP) is considered in the following form Evans (1979):

(OP) z∗ = min∑i∈S

∑j∈T

ci j xi j

s. t.∑j∈T

di j xi j ≤ ai , (i ∈ S),

∑i∈S

xi j = b j , ( j ∈ T ),

xi j ≥ 0, (i ∈ S, j ∈ T ).

Here S and T are the set of all sources and all destinations (customers), respectively; xi j is

the amount of flow from a source node i to a destination node j ; ci j is the transportation cost;

di j is the flow gain; ai is the node supply, and b j is the node demand. It is assumed that all

the coefficients in (OP) are positive.

The role of sources and destinations in (OP) can be reversed (Evans, 1979) by a simple

transformation of variables: yi j = di j xi j . This transformation removes the arc multipliers

from the first group of constraints in (OP) and places them in the second group. Taking into

account this correspondence between the sources and the destinations we consider here only

aggregation of the destinations (customers) in (OP). The aggregation of the sources can be

treated similarly.

The aggregation is defined by a partition of the set T of the destinations into a set T A

of clusters Tk, k = 1, . . . , K such that Tk ∩ Tp = ∅, k = p and⋃ K

k=1 Tk = T . Note that

the columns in the (OP) are also partitioned/clustered respectively. The fixed-weight variable

aggregated linear programming problem (Zipkin, 1980b) is derived by replacing the columns

in each cluster by their weighted average. The objective coefficients are combined similarly.

For each k, consider the non-negative weights gi j , i ∈ S, j ∈ Tk fulfilling the following

normalizing conditions:

∑j∈Tk

gi j = 1. (1)

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122 Ann Oper Res (2006) 146:119–134

The aggregated problem can be obtained by fixing the normalized weights gi j and substituting

the linear transformation (disaggregation):

xi j = gi j Xik, ( j ∈ Tk) (2)

into (OP) (Litvinchev and Rangel, 1999). Thus the aggregated problem is:

z = min∑i∈S

∑k∈T A

Xik

∑j∈Tk

ci j gi j (3)

s.t.∑k∈T A

Xik

∑j∈Tk

di j gi j ≤ ai , (i ∈ S), (4)

∑i∈S

gi j Xik = b j , ( j ∈ Tk, k ∈ T A), (5)

Xik ≥ 0, (i ∈ S, k ∈ T A). (6)

To write (3)–(6) we have used the observation that∑

j∈T = ∑k∈T A

∑j∈Tk

.

From (1) and (2) it follows that for any normalized weights we have Xik = ∑j∈Tk

xi j and

hence Xik is the supply from the source i to the aggregated customer Tk . However, (3)–(6)

still has |T | restrictions (5) corresponding to destinations, and hence cannot be considered

as some GTP associated with S sources and T A destinations. The number of constraints (5)

can be reduced as follows.

Let the weights be chosen as gi j = b j/∑

j∈Tkb j , j ∈ Tk . Then for each k ∈ T A in (5)

we obtain |Tk | conditions of the same form∑

i∈S Xik = ∑j∈Tk

b j . The redundant constraints

can be removed without changing the primal solution to form the problem which is usually

refereed to as the aggregated GTP (Evans, 1979):

(AP) z = min∑i∈S

∑k∈T A

cik Xik

s.t.∑k∈T A

dik Xik ≤ ai , (i ∈ S),

∑i∈S

Xik = bk, (k ∈ T A),

Xik ≥ 0 (i ∈ S, k ∈ T A),

where cik = ∑j∈Tk

ci j gi j , dik = ∑j∈Tk

di j gi j , bk = ∑j∈Tk

b j .

We will denote by (AP) the aggregated problem obtained for the weights gi j before

removing the redundant constraints. The primal solutions Xik of (AP) and (AP) are the same

and the fixed-weight disaggregated solution xi j = gi j X ik, j ∈ Tk is feasible to the original

problem (OP). The dual problem to (AP) always has multiple optimal solutions, even if the

optimal dual solution to (AP) is unique, as supported in Proposition 1:

Proposition 1. Let {ui , vk} be an optimal dual solution to (AP). Denote

D(v) ={

v j , j ∈ T

∣∣∣∣ ∑j∈Tk

v j b j = vkbk, k = 1, . . . , K

}. (7)

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Ann Oper Res (2006) 146:119–134 123

Then any pair {u, v} with v ∈ D(v) is an optimal dual solution to (AP) and

z = −∑i∈S

ai ui +∑k∈T A

∑j∈Tk

v j b j . (8)

Proof: Consider the dual to (AP) or, which is the same, the dual to (3)–(6) for gi j = gi j :

(DAP) −∑i∈S

ai ui +∑k∈T A

∑j∈Tk

v j b j → max

s.t. cik + dikui −∑j∈Tk

v j gi j ≥ 0, (i ∈ S, k ∈ T A),

ui ≥ 0, (i ∈ S).

The dual to (AP) is:

(DAP) −∑i∈S

ai ui +∑k∈T A

vk

∑j∈Tk

b j → max

s.t. cik + dikui − vk ≥ 0, (i ∈ S, k ∈ T A),

ui ≥ 0, (i ∈ S). �

These two dual problems should have equal optimal value since the primal problems (AP)

and (AP) have the same optimal solutions. Let {ui , vk} be an optimal solution to (DAP).

Construct {ui , v j } as follows: ui = ui and v j is such that∑

j∈Tkv j gi j = vk . Comparing the

restrictions of (DAP) and (DAP) we see that this {ui , v j } is feasible to (DAP). Moreover,

since gi j = b j/∑

j∈Tkb j it is not hard to verify that the objective value of (DAP) for {ui , v j }

is equal to the optimal objective of (DAP). Hence this {ui , v j } is optimal to (DAP).

Note, that the nonuniqueness of the optimal duals in (AP) follows from the way the

aggregation was performed, no matter whether (AP) has a unique optimal dual solution or

not. The nonuniquiness of the duals will be used further to derive a priori error bounds for

the GTP.

2. The error bounds

If the fixed weights gi j are used to form the aggregated problem (AP), the optimal value

z of the objective function for (AP) is also an upper bound for the optimal objective z∗ to

the original problem, such that z − z∗ ≥ 0. This follows from the fact that the disaggregated

solution xi j = gi j X ik, j ∈ Tk is feasible to the original problem.

To get an upper bound on the value z − z∗ we will use the following result established

in Litvinchev and Rangel (1999) for the fixed-weight aggregation in convex programming:

if W is a closed convex and bounded set containing an optimal solution to the original

problem (such a W is called a localization), then z − z∗ ≤ minx∈W L(x, τ ). Here L(·, ·) is

the standard Lagrange function associated with the original problem and τ is a vector of

Lagrange multipliers. Assume that the condition x ≥ 0 is always included in the definition

of W .

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124 Ann Oper Res (2006) 146:119–134

For the GTP, τ = {u, v}, the Lagrangian has the form:

L(x, u, v) = −∑i∈S

ui ai +∑j∈T

v j b j +∑i∈S

∑j∈T

(ci j + ui di j − v j )xi j , (u ≥ 0)

and hence for any fixed dual multipliers u ≥ 0 and v we have:

z − z∗ ≤(

z +∑i∈S

ui ai −∑j∈T

v j b j

)+ max

x∈W

∑i∈S

∑j∈T

(v j − ci j − ui di j )xi j ≡ ε(u, v, W ).

(9)

In particular, if ui , v j is an optimal solution of the dual to (AP), then:

−∑i∈S

ui ai +∑j∈T

v j b j = z and

ε (u, v, W ) = maxx∈W

∑i∈S

∑j∈T

(v j − ci j − ui di j )xi j . (10)

2.1. A priori error bounds

To get an a priori error bound for z − z∗ by (10) we need to estimate the objective coefficients

of the maximization problem in (10) without solving the aggregated problem. For the TP this

can be done by choosing ui = ui , v j = vk, j ∈ Tk, that is v ∈ D(v). To see this, note that

for di j = 1, the term in the brackets in (10) is vk − ci j − ui and by the restrictions of (DAP)

this is less than or equal to cik − ci j , j ∈ Tk, which gives:

z − z∗ ≤ maxx∈W

∑i∈S

∑j∈T

(cik( j) − ci j

),

where k( j) is the cluster to which j belongs, k( j) ∈ T A. If the localization W is defined beforean aggregated problem has been solved (we will refer to such W as a priori localization),

then the inequality above provides the a priori error bound for the TP. However, if di j = 1,

we can’t estimate the coefficients in (10) under the same choice of the duals u, v. For this

reason it was conjectured in Evans (1979) and Rogers et al. (1991) that “. . . a priori boundscannot be obtained for the GTP as is possible for simple transportation problems”. To cope

with this problem, we will use the nonuniqueness of the optimal dual multipliers for the (AP),

as stated in Proposition 1.

Proposition 2. Let di j = pi t j , where pi > 0, t j > 0. Then there exists v ∈ D(v) such thatfor the pair u, v we have:

v j − ci j − ui di j ≤ (cikdi j − ci j dik)/dik, (i ∈ S, j ∈ Tk).

Proof: From the restrictions of (DAP) we have ui ≥ (vk − cik)/dik and hence:

v j − ci j − ui di j ≤ [(cikdi j − ci j dik) − (vkdi j − v j dik)]/dik .

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Ann Oper Res (2006) 146:119–134 125

By the definition of dik and the correspondence between vk and v j , j ∈ Tk we have:∑j∈Tk

b j (vkdi j − v j dik) = 0 (∀i).

Since b j > 0, then the case (vkdi j − v j dik) > 0 (<0) for all j ∈ Tk is impossible.

Now choose v j to fulfill conditions vkdi j − v j dik = 0 for all j ∈ Tk . Resolving the latter

system of equations with respect to v j for di j = pi t j and dik = (∑

j∈Tkb j di j )/(

∑j∈Tk

b j )

we obtain:

v j = vk t j

∑j∈Tk

b j∑j∈Tk

b j t j, ( j ∈ Tk).

It is simple to check that for this v j we have∑

j∈Tkv j b j = vk

∑j∈Tk

b j . Hence v ∈ D(v)

and the claim follows. �

Proposition 2 together with (10) yields immediately the following expression for the a

priori error bound:

Corollary . For di j = pi t j we have:

z − z∗ ≤ maxx∈W

∑i∈S

∑j∈T

δai j xi j ≡ εa(W ),

where

δai j ≡

(cik( j)

dik( j)

− ci j

di j

)di j

and k( j) is the cluster to which j belongs, k( j) ∈ T A.

Remark 1. Under the assumption di j = pi t j we have∑

j∈T t j xi j ≤ ai/pi , i ∈ S in the

“source” restrictions of the original problem. This means that the “gain” (“loss”) along the

arc (i, j) is the same for all sources i connected with a destination j . We can treat this case as

an intermediate one between the TP, where the gains are the same for all arcs, and a general

form of the GTP, where the gains vary from arc to arc. Interpreting the GTP as the machine

loading problem (see, e.g., Ahuja, Magnanti, and Orlin (1993), and Glover, Klingman, and

Phillips (1990)), producing one unit of product j on machine i consumes di j hours of the

machine’s time. If pi is interpreted as a production “velocity” of the machine i, and t j is a

“length” of the product j measured in suitable units, then the assumption di j = pi t j holds.

A priori localizations can be defined, for example, by manipulating the constraints of the

original problem. Let

Wb = {xi j | 0 ≤ xi j ≤ ri j , i ∈ S, j ∈ T }, ri j = min{b j , ai/di j },

Ws ={

xi j

∣∣∣∣ ∑j∈T

di j xi j ≤ ai , i ∈ S, xi j ≥ 0 ∀i, j

},

Wd ={

xi j

∣∣∣∣ ∑i∈S

xi j = b j , j ∈ T, xi j ≥ 0 ∀i, j

},

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126 Ann Oper Res (2006) 146:119–134

where the lower indices b, s, and d show that the restrictions defining the respective localiza-

tion are associated with bounds, sources and destinations. Obviously Ws, Wd are localizations

as well as Wbs = Wb ∩ Ws and Wbd = Wb ∩ Wd . For these four localizations the optimization

problem to calculate εa(W ) can be solved analytically. Due to the decomposable structure

of the localizations, the computation of εa(Ws), εa(Wd ) results in a number of independent

single-constrained continuous knapsack problems, which yields

εa(Ws) =∑i∈S

ai maxj∈T

[δa

i j

]+/di j ,

εa(Wd ) =∑j∈T

b j maxi∈S

δai j ,

where [α]+ = max{0, α}. Similarly, the calculation of εa(Wbs), εa(Wbd ) is reduced to the

solution of independent single-constrained knapsack problems with upper bounds for the

variables. The respective solutions can also be obtained analytically (see, e.g., Litvinchev

and Rangel, 1999).

By construction we have min{εa(Wb), εa(Ws)} ≥ εa(Wbs) and similarly for Wd . Then

εaBest is defined as

εaBest = min{εa(Wbs), εa(Wbd )}.

It is interesting to compare the obtained a priori error bounds with the results known for the

TP (Zipkin, 1980a). Let di j = 1, i ∈ S, j ∈ T and all “≤ ” in the linear restrictions of (OP)

be substituted for “=”. Then δai j = cik( j) − ci j and to get the estimation stated in proposition

2 we should put v j = vk, j ∈ Tk . The expressions for our four a priori error bounds will

continue to be valid for the classical transportation problem if we change [δai j ]

+ for δai j . Then

εa(Ws) and εa(Wd ) coincide with the a priori error bounds obtained in Zipkin (1980a) for

the customer aggregation in the TP. Our error bound εaBest is at least as good as min{εa(Ws),

εa(Wd )}.

2.2. Tightening the a priori error bounds

From the definition of εa(W ) it follows that the tighter the localization W , the better (smaller)

the error bound εa(W ). Defining W by the original constraints, we should retain as many

constraints as possible. Alternatively, localization W should be simple enough to guarantee

that the computational burden associated with the calculation of the error bound is not too

expensive. A compromise decision is to utilize “easy” restrictions directly, while dualizing

“complicated” restrictions by the Lagrangian.

Suppose that the localization Wall is defined by all constraints of (OP) and the destination

constraints are treated as “easy”, while the source constraints are considered as “complicated”.

Then by the Lagrangian duality we have :

εa(Wall ) ≡ maxx∈Wall

∑i∈S

∑j∈T

δai j xi j = min

u≥0

{ ∑i∈S

ui ai + maxx∈Wd

∑i∈S

∑j∈T

(δa

i j − ui di j)xi j

}≡ min

u≥0εa(u, Wd ) ≤ εa(0, Wd ) ≡ εa(Wd ),

where εa(u, Wd ) denotes the expression in the figure brackets.

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Ann Oper Res (2006) 146:119–134 127

After the error bound εa(0, Wd ) = εa(Wd ) has been calculated, it is natural to look for

another value of u ≥ 0 to diminish the current error bound. If s is the search direction, we

get u = ρs, where the stepsize ρ ≥ 0 and the components of the search direction should be

nonnegative.

Let

εa(W ) = maxx∈W

∑i∈S

∑j∈T

δai j xi j ,

and denote by x(W ) its optimal solution. Consider that εa(Wd ) has been calculated and let

x(Wd ) be unique. Then by the marginal value theorem, εa(u, Wd ) is differentiable at u = 0,

and ∇ui εa(0, Wd ) = ai − ∑

j∈T di j xi j . Now take one step of the projected gradient technique

for the problem minu≥0 εa(u, Wd ) starting from u = 0. This gives

ui (ρ) = ρ max

{0,

∑j∈T

di j xi j − ai

}(i ∈ S, ρ ≥ 0).

The stepsize ρ associated with the steepest descent can be defined by a one-dimensional

search: under the uniquiness of x the choice of ui (ρ) guarantees a strict decrease of the a

priori error bound, that is either εa (u, Wd ) < εa(0, Wd ) = εa(Wd ) or εa(Wd ) = εa(Wall ). The

other a priori error bounds can be strengthened similarly. We will refer to the a priori error

bound εa(W ), tightened by searching the duals in the direction of the projected anti-gradient

as εa∇ (W ).

2.3. A posteriori error bounds

Suppose now that (AP) has been solved and the values u and v are known. The expression

for ε (u, v, W ) in (10) still provides a valid error bound and we only need to construct u, v

optimal to (AP). By Proposition 1 we may put u = u and v j = vk, j ∈ Tk .

We can calculate the a posteriori error bounds ε p (u, v, W ) analytically for the same

localizations used in Section 2.1 since:

z − z∗ ≤ maxx∈W

∑i∈S

∑j∈T

δpi j xi j ≡ ε p (u, v, W ),

where δpi j ≡ v j − ci j − ui di j . Hence we only need to substitute δ

pi j for δa

i j in the expressions

for the a priori error bounds obtained earlier. Note that the a posteriori error bound is valid

for the GTP in its general form, without any assumptions about the specifics of the arc

multipliers.

For the choice of u, v described above, the error bounds ε p (u, v, Ws) and ε p (u, v, Wd )

coincide with the a posteriori error bounds obtained in Evans (1979). The new error bound

εpBest = min{ε p (u, v, Wbs), ε p (u, v, Wbd )} obtained for the same choice of u, v is at least as

good as ε p (u, v, Ws), ε p (u, v, Wd ), and a numerical study below demonstrates that in many

cases it is significantly better.

After the a posteriori error bound has been calculated for some fixed u ≥ 0, v, it is natural

to look for another dual solution to decrease the error bound. Let the duals {u, v} be changed

in the direction {s, q} with the stepsize ρ, such that u + ρs ≥ 0. Then the best (i.e., error

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128 Ann Oper Res (2006) 146:119–134

bound minimizing) duals in this direction are defined from the problem:

minρ:u+ρs≥0

ε p (u + ρs, v + ρq, W ).

The dual-searching approaches to tightening the a posteriori error bound are well known for

linear programming aggregation (see Leisten (1997) and the references therein). Mendelssohn

(1980) proposed to look for the new dual solution in the direction of the dual aggregated

solution, i.e., for our case s = u, q = v. Shetty and Taylor (1987) used the right-hand side

vector of the original problem as the search direction. In Litvinchev and Rangel (1999)

one step of the projected gradient technique was used, similar to what was done in Section

2.2 for the a priori error bounds. In all cases the stepsize ρ can be obtained either by a

one-dimensional search or by analyzing the non-differentiability points of the expression for

ε p (u + ρs, v + ρq, W ) as the function of the stepsize. The detailed description and examples

of usage of error bounds for Mendelssohn, Taylor and Shetty, and Litvinchev and Rangel can

be found in Litvinchev and Tsurkov (2003). We will denote these error bounds by εpM (W ),

εpST (W ), and ε

p∇ (W ), respectively, while ε p(W ) is used to denote the bound ε p (u, v, W ).

Thus for each of the four localizations Wd , Ws, Wbs, and Wbd we can calculate two a priori

error bounds εa(W ), εa∇ (W ) and four a posteriori error bounds ε p(W ), ε

pM (W ), ε

pST (W ), and

εp∇ (W ).

3. Numerical study

We tested the error bounds derived in this paper for the GTP with di j = pi t j . The objective of

the study is to compare the performance of the proposed error bounds regarding the difference

from the actual error and the correlation with it. Our numerical study is based on the study

design presented in Norman, Rogers, and Levy (1999). To analyze the results we used the

Pearson Correlation Coefficient, descriptive statistics and the hypothesis test for the mean of

two independent samples (t-test) (Montgomery, 2001).

The problem instances were generated using the open source code GNETGEN, which is

a modification by Glover of the NETGEN generator (Klingman, Napier, and Stutz, 1974).

Five problem sizes grouped in two sets were considered: SET1 of relatively small problems

and SET2 of relatively large problems. The problems have the following dimensions: SET1

(30×50, 50×100, 100×150) and SET2 (50×1000, 50×2000), such that the number of desti-

nations is greater (or significantly greater) than the number of sources. For each problem size

ten instances were randomly generated using random number seeds. The available supplies

ai were randomly generated with∑

ai = 104 for all instances in SET1 and∑

ai = 1.8 · 106

for all instances in SET2. The cost coefficients ci j and the arc multipliers di j were randomly

generated in the intervals [2.0, 20.0] and [0.5, 3.0], respectively.

The destinations are clustered in 5 levels: 80, 60, 40, 20 and 10% of its total number.

This clustering is referred to as level 1 - level 5 in the tables below (the higher the level,

the smaller the number of destinations). Each cluster has �|T |/|T A|� destinations sorted by

their index, except the last cluster that contains all the others (�α� denotes the integer part

of α). There are a total of (5 problem sizes)×(10 problem instances)×(5 cluster levels) =250 aggregated instances (150 in Set1 and 100 in SET2). Further we refer to the aggregated

instances as simply instances.

Four localizations were used to calculate the aggregation error bounds: Wd , Ws, Wbs, and

Wbd . For each localization we calculated two a priori error bounds εa(W ), εa∇ (W ) and four

a posteriori error bounds ε p(W ), εp∇ (W ), ε

pST (W ) and ε

pM (W ). We will pay special attention

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Ann Oper Res (2006) 146:119–134 129

to the localizations Wbd and Wd . We conjecture that since only demands were aggregated,

the bounds calculated by allocating the reduced costs based on demands may be tighter and

may have a higher correlation with the actual error.

The dimension of the original problem allows the exact solution for all problems sizes.

Correlations between the error bounds and the actual error, z − z∗, were calculated using the

Pearson coefficient and are presented in Table 1 for all localizations. In all correlation tables to

be presented the p-values are less than 0.0001 if not indicated. In Table 1 the exceptions are:

in SET1, εa(Ws) (p = 0.0003); and in SET2, εa∇ (Ws) (p = 0.0005), εa(Wbs) (p = 0.1357),

ε p(Ws) (p = 0.2876), and εpST (Ws) (p = 0.3112).

Table 1 Correlation between the actual error and the error bounds

SET1 SET2

Wd Ws Wbd Wbs Wd Ws Wbd Wbs

εa(W) 0.6609 0.2929 0.7196 0.4441 0.5615 −0.4840 0.5617 0.1502

εa∇ (W) 0.7131 0.4043 0.7376 0.6471 0.5626 −0.3418 0.5624 0.5611

εp(W) 0.9826 0.5016 0.9864 0.5738 0.9985 −0.1074 0.9985 0.4239

εp∇ (W) 0.9897 0.9897 0.9910 0.7288 0.9998 0.9998 0.9998 0.6958

εpST(W) 0.9827 0.4979 0.9864 0.5753 0.9986 −0.1023 0.9986 0.4243

εpM(W) 0.9989 0.9765 0.9997 0.9660 1.0000 0.9419 1.0000 0.9086

The a priori error bounds calculated for the localization Wd and Wbd are correlated with

the actual error at a statistically significant level (p < 0.0001) for both SET1 and SET2,

and the highest correlation is achieved for the error bound εa∇ . The correlation coefficient is

smaller for the instances in SET2 than for the ones in SET1.

The a posteriori error bounds for SET1 are strongly and statistically significantly correlated

with the actual error for all localizations used. For the instances in SET2 the bounds ε p(Ws)

and εpST (Ws) are not correlated with the actual error. For both SET1 and SET2 the highest

correlation is achieved for the bounds εp∇ (W ) and ε

pM (W ) calculated for the localizations Wd

and Wbd . The error bounds calculated using localizations Ws and Wbs have lower correlation

values and for some error bounds the correlation value is not statistically significant.

The correlations between the a priori error bounds, εa∇ and εa , and the a posteriori error

bounds are presented in Table 2, with the values for εa∇ in bold. For the localizations Wd

and Wbd both a priori error bounds are correlated with the a posteriori error bounds, the

Table 2 Correlation between the a priori error bounds, εa∇(W) and εa(W), and the a posteriori

error bounds

SET1 SET2

Wd Ws Wbd Wbs Wd Ws Wbd Wbs

ε p(W) 0.7831 0.5937 0.7905 0.8207 0.5953 0.5289 0.5951 0.3776

0.7259 0.6513 0.7677 0.7695 0.5944 0.4141 0.5945 0.6970

εp∇ (W) 0.7382 0.5905 0.7582 0.8263 0.5743 −0.1365 0.5741 0.1677

0.6788 0.6319 0.7364 0.5913 0.5733 −0.1815 0.5734 0.0748

εpST(W) 0.7830 0.6080 0.7900 0.8218 0.5933 0.5267 0.5930 0.3781

0.7258 0.6383 0.7671 0.7680 0.5924 0.4071 0.5924 0.6972

εpM(W) 0.7176 0.4687 0.7402 0.7259 0.5617 −0.2326 0.5614 0.4849

0.6693 0.3719 0.7230 0.5418 0.5607 −0.3261 0.5606 0.2847

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130 Ann Oper Res (2006) 146:119–134

correlations for the instances in SET2 are weaker than for the instances in SET1. For these

localizations εa∇ is more correlated with a the posteriori error bounds than εa . For the instances

in SET2, the correlation with εp∇ and ε

pM when using localizations Ws and Wbs are very low

and in some cases not significant (e.g., p-values for the correlation between εa∇ (Ws) and

εp∇ (Ws) and for εa(Wbs) and ε

p∇ (Wbs) are 0.1757 and 0.4596 respectively).

In Table 3 we present, sorted by aggregation level, the correlations between the actual

error and the error bounds calculated for localization Wbd , the tightest localization based on

demands. The a priori error bounds are highly and statistically significantly correlated with

the actual error for all levels of aggregation. The lowest correlation was obtained for level

5. The correlations for the a priori error bound εa∇ are higher (except for level 4 in SET1)

than for εa . Comparing with a priori error bounds, the a posteriori error bounds are more

correlated with the actual error for all levels of aggregation. The error bound εpM has the

strongest correlation which is almost insensitive to the aggregation level.

Table 3 Correlation between the actual error and the error bounds by level of aggregation (localization Wbd )

SET1 SET2

level 1 level 2 level 3 level 4 level 5 level 1 level 2 level 3 level 4 level 5

εa 0.6871 0.7660 0.6625 0.6297 0.5636 0.9849 0.9883 0.9080 0.9281 0.6920

εa∇ 0.7100 0.9647 0.8000 0.6281 0.5803 0.9849 0.9887 0.9101 0.9288 0.7261

εp 0.9858 0.9954 0.9837 0.9912 0.9557 0.9984 0.9992 0.9956 0.9786 0.9710

εp∇ 0.9863 0.9965 0.9867 0.9948 0.9663 0.9998 0.9997 0.9992 0.9974 0.9948

εpST 0.9858 0.9954 0.9835 0.9908 0.9556 0.9984 0.9992 0.9957 0.9783 0.9706

εpM 0.9997 0.9997 0.9995 0.9993 0.9992 0.9997 0.9995 0.9996 0.9993 0.9997

Similar to Leisten (1997) and Litvinchev and Rangel (1999), the numerical quality of the

error bounds was characterized by the upper and the lower bounds on z∗: z ≥ z∗ ≥ L Bε,

where L Bε = z − ε(W ) and ε(W ) is an error bound. We used the indicators (z − z∗)/z∗

and (z∗ − L Bε)/z∗ to represent the proximity ratio. In particular, the smaller the proximity

ratio (z∗ − L Bε)/z∗ is, the tighter is the error bound. The average values of these indicators,

together with standard deviations, are presented in Tables 4 and 7, sorted by localizations

and levels of clustering. In Table 5 we present the relative number of instances having W as

the best (bound minimizing) localization for a fixed type of error bound (in % of the total

number of instances). The localization Ws is not represented in Table 5 since it was never

the best. In Table 6 we give the relative number of instances having a particular combination

bound-localization as the best among all the error bounds calculated.

Note that by construction, for any fixed localization W we have εa(W ) ≥ εa∇ (W ) and

ε p(W ) ≥ max {ε pM (W ), ε

pST (W ), ε

p∇ (W )} since the improved error bounds εa

∇ (W ), εpM (W ),

εpST (W ) and ε

p∇ (W ) are at least as good as the original error bounds εa(W ) and ε p(W ), respec-

tively. To compare different localizations, observe that in Mendelssohn (1980)’s approach

we look for the new duals in the direction of the dual aggregated solution, while Shetty and

Taylor (1987) scheme uses the right-hand side vector of the original problem as the search

direction. In both cases the search direction is independent on the localization used. Hence,

εpM (Wbd ) ≤ ε

pM (Wd ) (ε

pST (Wbd ) ≤ ε

pST (Wd )) since Wbd ⊆ Wd and the objective functions to

calculate εpM (Wbd ) and ε

pM (Wd ) are the same. This observation is also valid for the local-

izations Wbs and Ws . But this is not the case for the gradient-based improvement scheme

used to obtain the bound εp∇ (W ). Here the dual search direction is defined by the optimal

solution x(W ) of the problem used to calculate the bound ε p(W ) and hence depends on the

localization used. Thus the objective functions used to calculate εp∇ (Wbd ) and ε

p∇ (Wd ) are not

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Ann Oper Res (2006) 146:119–134 131

Table 4 Proximity ratio

(average

standard deviation

)SET1 SET2

(z − z∗)/z∗ 1.6137 1.0293

0.6463 0.4953

(z∗ − LBε)/z∗ Wd Ws Wbd Wbs Wd Ws Wbd Wbs

εa(W) 1.4870 7.1782 1.2440 5.4950 1.1410 10.9971 1.1377 3.4889

0.8922 5.9698 0.7905 4.5655 0.9503 13.1592 0.9481 1.7821

εa∇ (W) 1.2335 4.4308 1.1452 2.9979 1.1226 4.5688 1.1219 2.3655

0.8105 3.4014 0.7532 1.5581 0.9316 1.7838 0.9318 0.6100

ε p(W) 0.2707 9.2814 0.2076 5.3673 0.0495 6.0126 0.0493 1.9566

0.1610 7.7396 0.1349 4.4300 0.0352 6.8493 0.0351 1.8763

εp∇ (W) 0.1684 3.5182 0.1228 2.1924 0.0140 1.1395 0.0138 0.9779

0.1095 4.4342 0.0991 1.6776 0.0115 2.7697 0.0114 0.7742

εpST (W) 0.2706 8.9535 0.2068 5.3447 0.0481 5.7335 0.0478 1.9549

0.1609 7.4195 0.1348 4.4138 0.0346 6.5401 0.0344 1.8762

εpM (W) 0.1043 0.6292 0.0595 0.5476 0.0125 0.6359 0.0109 0.4226

0.0323 0.1434 0.0169 0.1756 0.0047 0.1673 0.0039 0.2197

Table 5 Relative number ofinstances (in %) having W as thebest localization

SET1 SET2

Wbd Wd Wbs Wbd Wd Wbs

εa(W) 98 0 2 100 0 0

εa∇ (W) 89.33 5.33 5.33 77 23 0

εp(W) 100 0 0 100 0 0

εp∇ (W) 98.67 1.33 0 94 6 0

εpST(W) 100 0 0 100 0 0

εpM(W) 100 0 0 100 0 0

the same and we cannot guarantee that εp∇ (Wbd ) ≤ ε

p∇ (Wd ). Similarly for the localizations

Wbs and Ws as well as for the a priori error bound εa∇ (W ).

It can be seen from Table 4 that the use of localizations Wd and Wbd give the lowest

average proximity ratios for the a priori error bounds and there is no statistically significant

difference between εa and εa∇ calculated using these localizations, according to the t-test run

to check the differences. However, for localizations Ws and Wbs , εa∇ gives a smaller proximity

ratio than εa . As was expected, the a posteriori error bounds are much tighter than a priori

error bouns, giving the best results for localizations Wd and Wbd . Comparing the different

types of the error bounds, εpM and ε

p∇ are tighter than ε p and ε

pST for any fixed localization,

with the best proximity ratio obtained for εpM (Wbd ). The second best is ε

p∇ (Wbd ).

From Table 5 we can see that there are a few instances in SET1 to which the localization

Wbs results in tighter a priori error bounds than Wbd and Wd . For the a priori error bounds

calculated in SET2 and all types of a posteriori error bounds the tightest results were obtained

for localizations Wbd and Wd . For the a posteriori error bounds ε p(W ), εpST (W ), and ε

pM (W )

the localization Wbd was the best for all instances. This is not the case for the gradient-based

error bounds εa∇ (W ) and ε

p∇ (W ) where for some instances the localization Wd provides better

results, especially in SET2. Comparing the best error bounds in Table 6 we see that for the

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132 Ann Oper Res (2006) 146:119–134

Table 6 Relative number ofinstances (in %) having a givencombination (error bound,localization) as the best

SET1 SET2

εa∇ (Wbd ) 89.33 77.00

εa∇ (Wd ) 5.33 23.00

εa∇ (Wbs ) 5.33 0

εpM(Wbd ) 90.67 48.00

εp∇ (Wbd ) 9.33 49.00

εp∇ (Wd ) 0 3.00

Table 7 Proximity ratio by level of aggregation (localization Wbd )

(average

standard deviation

)SET1 SET2

level 1 level 2 level 3 level 4 level 5 level 1 level 2 level 3 level 4 level 5

(z − z∗)/z∗ 1.0332 1.7219 1.3672 1.6523 2.2939 0.7235 1.4655 0.3400 0.9615 1.6557

0.5283 0.5319 0.5210 0.4520 0.4527 0.1196 0.1189 0.0869 0.0966 0.1257

(z∗ − LBε)/z∗

εa 0.5469 0.4652 1.4389 1.9824 1.7865 0.0695 0.0588 1.2281 2.2021 2.1301

0.5043 0.4212 0.4980 0.4607 0.5297 0.0263 0.0207 0.0668 0.0667 0.0940

εa∇ 0.5092 0.3213 1.2852 1.9223 1.6881 0.0695 0.0586 1.2237 2.1723 2.0854

0.4757 0.1522 0.3354 0.4587 0.5213 0.0263 0.0202 0.0666 0.0683 0.0895

εp 0.0975 0.1350 0.2040 0.2873 0.3141 0.0244 0.0339 0.0213 0.0669 0.1000

0.1061 0.0678 0.1067 0.0603 0.1621 0.0074 0.0053 0.0082 0.0228 0.0312

εp∇ 0.0869 0.0927 0.1255 0.1468 0.1619 0.0061 0.0071 0.0074 0.0199 0.0286

0.1030 0.0579 0.0983 0.0464 0.1422 0.0026 0.0030 0.0035 0.0077 0.0130

εpST 0.0975 0.1348 0.2028 0.2851 0.3138 0.0244 0.0339 0.0199 0.0633 0.0977

0.1061 0.0679 0.1073 0.0614 0.1623 0.0074 0.0053 0.0082 0.0236 0.0314

εpM 0.0469 0.0583 0.0610 0.0653 0.0663 0.0110 0.0132 0.0072 0.0105 0.0127

0.0134 0.0123 0.0163 0.0170 0.0182 0.0029 0.0037 0.0025 0.0037 0.0040

relatively small problems in SET1 the error bound εpM (Wbd ) outperforms the bound ε

p∇ (Wbd )

for most, but not all instances. In SET2 of relatively large problems the gradient-based bound

εp∇ provided tightest results slightly more frequently than Mendelssohn’s bound ε

pM .

The proximity ratios as a function of clustering are shown in Table 7 for localization Wbd .

Concerning the a priori error bounds, εa∇ is slightly tighter than εa . The error bounds do

not increase monotonously with respect to the clustering level. The a priori error bounds for

levels 1 and 2 are significantly tighter than for the highly aggregated instances in levels 4 and

5. This holds for both SET1 and SET2. Comparing the a posteriori error bounds, εpM and ε

p∇

have the lowest proximity ratio with εp∇ outperforming ε

pM only for levels 1 and 2 in SET2.

The bound εpM is very stable, while ε

p∇ increases with the clustering level.

To summarize:

� The a posteriori error bounds εpM , ε

p∇ are highly and statistically significantly correlated

with the actual error and are the tightest error bounds for all instances and localizations.� εpM is less sensitive than ε

p∇ to the level of clustering and problem dimension, and it is

tighter than εp∇ for most instances (90%) in SET1, while only for 48% of instances in SET2

of larger problems.

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Ann Oper Res (2006) 146:119–134 133� Localization Wbd gives the tightest results for all, but a few, a posteriori error bounds. The

exceptions are for the error bound εp∇ where Wd performs better for (3%) of the instances

in SET2.� For localizations Wbd and Wd the a priori error bounds εa∇ and εa are statistically signifi-

cantly correlated with the actual error and with all the a posteriori error bounds.� For all instances εa∇ provides the best results for localizations Wbd and Wd , except for

5.33% of the instances in SET1 to which εa∇ (Wbs) performs better.

4. Conclusions

In this paper we derived new a priori and a posteriori aggregation error bounds for the

GTP and studied the relationships between the error bounds and the actual error. Only

customers (destinations) were aggregated and the clustering level was the only element of

the aggregation strategy that was varied in the numerical study.

Our numerical study demonstrates that the localization Wbd almost always performs bet-

ter, resulting in higher correlations between the error bounds and the actual error, and giving

tighter error bounds. This was expected, since in our case the error is due to incomplete

(aggregated) information in the aggregated problem about particular demands. Thus, us-

ing all demand constraints explicitly, as in Wbd and Wd , may provide tighter estimation.

Moreover, Wbd ⊆ Wd and strengthening the localization improves the bound. It can be seen

from the numerical results that strengthening the localization also increases the correlation

between the error bound and the actual error. The new a priori bound εa∇ (Wbd ) is signifi-

cantly correlated with the actual error (α = 0.05). That is, if the customers in the GTP are

aggregated using the strategy outlined in this study, and the prescribed localization is cho-

sen, the a priori error bound can be considered as an appropriate guide to use in selecting

the clustering level. Constructing various (including nonlinear, say, ellipsoidal (Litvinchev

and Rangel, 1999) localizations for structured problems is an interesting topic for future

research.

Comparing different techniques to calculate the a posteriori error bounds, surprisingly

good results were provided by Mendelssohn (1980)’s approach. We would like to stress that

this is not the case for unstructured problems. In particular, for the continuous knapsack prob-

lems, the approach of Shetty and Taylor (1987) gives a tighter error bound than Mendelssohn

(1980)’s one as reported in Leisten (1997) and Litvinchev and Rangel (1999). The good

performance of Mendelssohn (1980)’s bound for the GTP needs to be further investigated

and explained.

In summary, for our set of randomly generated problems, the aggregated model having

the tightest a priori error bound εa∇ (Wbd ) tends to give the best approximation of the optimal

objective for the non-aggregated model. After the aggregated model (the level of clustering)

was selected by the a priori error bound and solved, Mendelssohn (1980)’s a posteriori error

bound, εpM , as well as the gradient-based error bound, ε

p∇ , provide a good quantitative measure

of the actual error.

Acknowledgments The first author was partially supported by Russian foundation RFBR (02-01-00638)and Mexican foundation PAICyT(CA857-04), while the second author was partially funded by Brazilianfoundations FAPESP (2000/09971-9) and FUNDUNESP (00509/03-DFP). The authors thank the anonymousreferees for a careful reading of the article and helpful suggestions. Thanks are also due to Adriana B. Santosfor useful comments related to the Numerical Study section.

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134 Ann Oper Res (2006) 146:119–134

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