Applied Spatial Analysis and Policy

, Volume 11, Issue 4, pp 657–667 | Cite as

The Varying Zone Size Effect and Dual Variables for Entropy Maximising Models of Spatial Interaction

  • M. L. SeniorEmail author
  • H. C. W. L. Williams


Prior weighted entropy functions compensating for the arbitrary partitioning by the zoning system of the origin and destination quantities in entropy maximising models of spatial interaction are presented and justified. This type of prior weighting is essential if the dual variables derived from the entropy maximising derivation of these models are to be used and interpreted as spatial prices in empirical studies. Failure to use prior weighted entropy functions taking account of varying zone sizes results in the biasing of dual variable values. Empirical illustrations of zone size biasing of house price and consumer welfare dual variable values are presented using a Herbert-Stevens model of a spatial housing market.


Entropy Prior weights Zone size Spatial interaction Dual variables 


The year 2017 marks the 50th anniversary of Sir Alan Wilson’s seminal paper (Wilson 1967) reinterpreting spatial interaction models using the principles of entropy maximising (EM) and information theory. As described in Boyce and Williams (2015 section 3.5) this paper did far more than simply provide an alternative to the Newtonian gravity model. Rather it was a work of synthesis and development which identified major issues of that time; it and later key publications (for example Wilson 1970, 1971) set the scene for a large amount of innovative research.

Over this period there have been many developments of the approach, as for example that of Batty and March (1976), Erlander (1977) and Snickars and Weibull (1977), which have enriched the theoretical basis and practical usefulness of a wide range of models, notably for spatial housing and labour markets and the associated journey to work, and for the demand for services. In the basic entropy maximizing approach spatial interaction models are the outcome of a nonlinear programming problem (Balinski and Baumol 1968). This comprises a primal problem, (the maximisation of an entropy function expressed in terms of the primal spatial interaction variables) and a dual minimisation problem which extends the interpretation and usefulness of the models by producing valuations (the dual variables or Lagrangian multipliers) of the origin and destination quantities used as constraints in the primal problem. These dual variables are transformations of the more commonly recognised balancing factors which are often interpreted as accessibility measures.

In this paper we wish to comment on a specific technical issue in the development of entropy based models which has significant implications both for the formulation of the basic spatial interaction model and for the determination and interpretation of the variables in the dual problem. The issue relates to the varying size of zones in a study area. Although this topic attracted comments many years ago in relation to the development of the primal problem (e.g. Batty 1974; Wilson et al. 1981) we are unaware of any literature relating to the dual problem, and specifically to the problem of deriving meaningful interpretation and unbiased results for the dual variables, which form the basis of spatial prices and economic evaluation of changes arising from policy.

In the next section we consider the process of adopting prior weights according to the zoning system adopted in developing the family of spatial interaction models proposed by Wilson (1967). Then we turn to the dual representation and examine the implication of prior weights for the dual variables. In both primal and dual representations we shall refer to the transportation model of linear programming as the ‘optimal’ limiting form of the entropy models in order to motivate an analysis of its extension to a ‘sub-optimal’ situation based on entropy maximisation. In the penultimate section we illustrate the implications of prior weighting schemes for the dual representation of the Herbert-Stevens model of a competitive housing market and note the biases arising from unweighted schemes. Finally, we draw conclusions and describe policy applications.

Compensating for Varying Zone Sizes in Entropy Maximising Models

The issue of zone size effects in EM models is not new. For example, Batty (1974) reinterpreted the EM model with zone size measured directly as each zone’s areal extent, and both he and Openshaw (1977) have discussed idealized or optimal zoning systems. Here, however, we consider the zoning systems typically used in spatial interaction modelling exercises which arbitrarily partition the origin and destination quantities, say Oi and Dj, in doubly constrained models, and attractiveness sizes, say Wj, in singly constrained models. In 1977 Williams flagged up the consequences of the zone size effect in an unpublished note and showed how the use of prior “weights” in the expression for entropy can offer a solution to any bias that this may create (see also Wilson et al. 1981 pages 28–29). This zone size issue was also illustrated in a pedagogic guide to EM by Senior (1979), who then used a prior weighted entropy term to derive the doubly constrained model. However, this weighting of the entropy expression in EM has not, apparently, become routine practice.

To illustrate the contemporary relevance of the problem consider, for example, the doubly constrained model as used by Stillwell et al. (2016) for migration. The Oi and Dj terms are known totals, respectively, of out-migrants from zones i and in-migrants to zones j. Now assume for the moment that migration distances, dij, had no influence whatsoever on migration then, in the absence of any further information, expected migration flows \( {\overset{\sim }{T}}_{ij} \) would be shaped only by the zoning system which is imposed. So defining prior probabilities in terms of the proportions qi of an out-migrant being in zone i and qj of an in-migrant being in zone j as:
$$ {q}_i=\frac{O_i}{\mathrm{N}};{q}_j=\frac{D_j}{\mathrm{N}} $$
where N is the total number of migrants in the study area, then expected migrant flows \( {\overset{\sim }{\mathrm{T}}}_{\mathrm{ij}} \) would be:
$$ {\overset{\sim }{T}}_{ij}=\mathrm{N}{q}_i{q}_j=\mathrm{N}.\left(\frac{O_i{D}_j}{{\mathrm{N}}^2}\right)=\left(\frac{O_i{D}_j}{\mathrm{N}}\right) $$
This expression is akin to a ‘main effects’ log-linear model of migration flows of the various ‘push’ and ‘pull’ factors. The \( \left(\frac{O_i{D}_j}{\mathrm{N}}\right) \) terms can be used as prior weights in a revised entropy measure, namely:
$$ Maximise\left\{{T}_{ij}\right\}:-\sum \limits_{ij}{T}_{ij} lo{g}_e\frac{T_{ij}}{{\tilde{T}}_{ij}}=-\sum \limits_{ij}{T}_{ij} lo{g}_e\frac{T_{ij}}{\left(\frac{O_i{D}_j}{N}\right)} $$
Equation (3) is a version of the Kullback-Leibler divergence or relative entropy measure (Kullback 1959). Batty and March (1976) used a similar approach, but their prior weights comprised a gravity model incorporating not just the Oi and Dj terms but also distance decay effects and with total interactions constrained to a known total. To compensate for arbitrary zone size effects, all spatial interaction models could be derived using Eq. (3) as a “prior weighted” entropy measure (suitably modified according to which member of the family of models is being used) rather than the more familiar “unweighted” entropy:
$$ Maximise\left\{{T}_{ij}\right\}:-\sum \limits_{ij}{T}_{ij} lo{g}_e{T}_{ij} $$
The differences in the doubly constrained interaction models derivable by maximising separately entropy expressions (4) or (3) subject to the usual origin, destination and total cost constraints:
$$ {\sum}_j{T}_{ij}={O}_i $$
$$ {\sum}_i{T}_{ij}={D}_j\kern0.5em $$
$$ {\sum}_{ij}{T}_{ij}{c}_{ij}=\mathrm{C} $$
can be gauged from the following. Introducing Lagrange multipliers associated with the various constraints, the conventional “unweighted” model is:
$$ {T}_{ij}={e}^{-{\lambda}_i+{\mu}_j-\beta {c}_{ij}}={A}_i{B}_j{O}_i{D}_j{e}^{-\beta {c}_{ij}} $$
$$ {A}_i{O}_i={e}^{-{\lambda}_i}\kern0.5em and\kern0.5em {B}_j{D}_j={e}^{\mu_j} $$
where Ai and Bj are determined in the usual way from:
$$ {A}_i=\frac{1}{\sum_j{B}_j{D}_j{e}^{-\beta {c}_{ij}}}\ and\ {B}_j=\frac{1}{\sum_j{A}_i{O}_i{e}^{-\beta {c}_{ij}}}\kern0.5em $$
While the “prior weighted” model may be expressed as:
$$ {T}_{ij}=\left(\frac{O_i{D}_j}{\mathrm{N}}\right){e}^{-{\overset{\sim }{\lambda}}_i+{\overset{\sim }{\mu}}_j-\beta {c}_{ij}}={\overset{\sim }{A}}_i{\overset{\sim }{B}}_j{O}_i{D}_j{e}^{-\beta {c}_{ij}} $$
$$ {\overset{\sim }{A}}_i=\frac{e^{-{\overset{\sim }{\lambda}}_i}}{\sqrt{\mathrm{N}}}\kern0.75em and\ {\overset{\sim }{B}}_j=\frac{e^{{\overset{\sim }{\mu}}_j}}{\sqrt{\mathrm{N}}} $$
The right-hand sides of expressions (8) and (11) imply that the two models are effectively identical as calibration and use of both models with the same data set will result in:
$$ {\overset{\sim }{A}}_i={A}_i=\frac{1}{\sum_j{B}_j{D}_j{e}^{-\beta {c}_{ij}}}\kern0.75em and\ {\overset{\sim }{\ B}}_j={B}_j=\frac{1}{\sum_j{A}_i{O}_i{e}^{-\beta {c}_{ij}}} $$

This equivalence of course raises the question: why bother with the prior weighted EM derivation? The answer lies in the extraction, interpretation and use of dual variables in practical applications of spatial market models.

Dual Variables from Doubly Constrained Spatial Interaction Models

It is well known, as shown by Evans (1973), that, in the limit as the β value in the doubly constrained spatial interaction model tends to infinity, (one of) the solutions of the transportation problem of linear programming is obtained. The latter has the primal form:
$$ Minimise\ \left\{{T}_{ij}\right\}:\kern0.5em C={\sum}_{ij}{T}_{ij}{c}_{ij} $$
subject to origin and destination constraints (5) and (6) and the non-negativity conditions:
$$ {T}_{ij}\ge 0 $$
This primal transportation problem has a dual problem with variables αi and νj, namely:
$$ Maximise\left\{{\alpha}_{i,}{v}_j\right\}:\sum \limits_j{v}_j{D}_j-\sum \limits_i{\alpha}_i{O}_i $$
subject to:
$$ {\nu}_j-{\alpha}_i\le {c}_{ij} $$

Interpretations of these dual variables as valuations of location advantages or location rents is given, for example, in Stevens (1961) and White and Senior (1983 pages 177-184).

The entropy maximising formulation can also be viewed as a primal nonlinear programming problem with nonlinear objective function (3) or (4) and linear constraints (5), (6), (7) and (15). Wilson and Senior (1974) used the Lagrangian approach, and what are now known as the Karush-Kuhn-Tucker conditions (Boyce and Williams 2015 page 299), to show that this primal entropy maximising problem also has a dual programming problem with dual variables that are the Lagrangian multipliers λ, γ and β and dual objective function:
$$ Minimise\left\{{\lambda}_i,{\gamma}_j,\beta \right\}:\sum \limits_i{\lambda}_i{O}_i-\sum \limits_j{\gamma}_i{D}_j+\beta C+\sum \limits_{ij}{T}_{ij} $$
subject to dual constraints which are, given use of the “prior weighted” entropy term (3) in the primal:
$$ {\mathit{\log}}_e{T}_{ij}={\mathit{\log}}_e\left(\frac{O_i{D}_j}{\mathrm{N}}\right)-{\overset{\sim }{\lambda}}_i+{\overset{\sim }{\mu}}_j-\beta {c}_{ij} $$
or, equivalently, to aid comparison with the transportation problem dual constraints (17):
$$ \frac{{\overset{\sim }{\mu}}_j}{\beta }-\frac{{\overset{\sim }{\lambda}}_i}{\beta}\kern0.5em ={c}_{ij}+\frac{{\mathit{\log}}_e\left(\frac{\ \mathrm{N}\ {T}_{ij}}{O_i{D}_j}\right)}{\beta } $$

It should be noted that equality constraints always apply in (19) and (20) because all Tij are greater than zero for finite β parameters. Expression (19) or (20) is just another way of writing the doubly constrained spatial interaction model (11), and the expression for Tij may be substituted into (18) to give an unconstrained dual objective function. So both primal (Tij) and dual variable information can be extracted from this model, the latter by examining expressions (12) for the balancing factors \( {\overset{\sim }{A}}_i \) and\( {\overset{\sim }{\ B}}_j \).

Now, what justifies model derivation using the “prior weighted” entropy formulation can be explained by comparing expressions (12) for the “prior weighted” model with the equivalent ones (9) for the “unweighted” one (Table 1). It is immediately apparent that the “unweighted” dual variables are biased by the zone size terms logeOi and logeDj, while the “prior weighted” ones are not as loge √N is just a constant. Indeed this constant can be ignored as the dual variable values in both the transportation problem and entropy formulation are not unique, because multiplying one balancing factor by a constant and dividing the other one by the same constant does not change the solution in the doubly constrained model.
Table 1

Formulas for the dual variables in a doubly constrained model


“unweighted” model

“prior weighted” model


Equations (9)

Equations (12)

Origin zone dual

λi =  − logeAiOi = −logeAi − logeOi

\( {\overset{\sim }{\lambda}}_i=-{\mathit{\log}}_e{\overset{\sim }{A}}_i-{\mathit{\log}}_e\sqrt{\mathrm{N}} \)

Destination zone dual

μj = logeBjDj = logeBj + logeDj

\( {\overset{\sim }{\mu}}_j={\mathit{\log}}_e{\overset{\sim }{B}}_j+{\mathit{\log}}_e\sqrt{\mathrm{N}} \)

We stress that use of “prior weighted” doubly constrained spatial interaction models is necessary if dual variables are to be extracted and suitably interpreted, as we illustrate below.

An Illustration of the Zone Size Effect on Dual Variables

This section considers the entropy maximising version of the Herbert-Stevens residential location model (Senior and Wilson 1974). This is a doubly constrained spatial interaction model with Hik representing houses in residential zones i disaggregated by house type k, with Ejw denoting workers in employment zones j by social class w, and with N now denoting total workers in the study area. The model adapts Alonso’s (1964) theory of the urban land market, with workers making bids (bijkw-cij) for available housing which take account of commuting costs cij.

To derive the model using a “prior weighted” entropy term, let q(i) be the probability of a house being in zone i, q(k|i) be the conditional probability of a house in zone i being of type k, q(j) be the probability of a worker being in zone j, and q(w|j) be the conditional probability of a worker in zone j being in category w. Then (with * denoting summation over the appropriate index):
$$ {q}_i^k=q(i)q\left(k|i\right)=\frac{H_i^{\ast }}{\mathrm{N}}\frac{H_i^k}{H_i^{\ast }}=\frac{H_i^k}{\mathrm{N}}\kern0.5em and\kern0.5em {q}_j^w=q(j)q\left(w|j\right)=\frac{E_j^{\ast }}{\mathrm{N}}\frac{E_j^w}{E_j^{\ast }}=\frac{E_j^w}{\mathrm{N}} $$
$$ {\overset{\sim }{\boldsymbol{T}}}_{\mathrm{ij}}^{\mathrm{kw}}=\mathrm{N}{q}_i^k{q}_j^w=\mathrm{N}\frac{H_i^k{E}_j^w}{{\mathrm{N}}^2}=\frac{H_i^k{E}_j^w}{\mathrm{N}} $$
The \( {\overset{\sim }{\boldsymbol{T}}}_{\mathrm{ij}}^{\mathrm{kw}} \) are the expected spatial interactions arising solely from the varying zone size and housing and employment category sizes by zone and they can be used as prior weights in the entropy formulation as follows.
$$ Maximise\left({T}_{ij}^{kw}\right):-\sum \limits_{ij}{T}_{ij}^{kw}{\log}_e\frac{T_{ij}^{kw}}{{\overset{\sim }{T}}_{ij}}=\mathit{\operatorname{Max}}:-\sum \limits_{ij}{T}_{ij}^{kw}{\log}_e\frac{T_{ij}^{kw}}{\left(\frac{H_i^k{E}_j^w}{N}\right)} $$
subject to:
$$ {\sum}_{jw}{T}_{ij}^{kw}={H}_i^k $$
$$ {\sum}_{ik}{T}_{ij}^{kw}={E}_j^w $$
$$ {\sum}_{ij kw}{T}_{ij}^{kw}\left({\mathrm{b}}_{\mathrm{ij}}^{\mathrm{kw}}-{c}_{ij}\right)=\mathrm{Z} $$
where Z is the sum of all bid prices/rents for houses by all workers in the system.
Lagrangian maximisation gives:
$$ {T}_{ij}^{kw}=\frac{H_i^k{E}_j^w}{\mathrm{N}}{e}^{-{\overset{\sim }{\lambda}}_i^k}{e}^{{\overset{\sim }{\lambda}}_i^w}\exp \left\{\upmu \left({\mathrm{b}}_{\mathrm{ij}}^{\mathrm{kw}}-{c}_{ij}\right)\right\}={\overset{\sim }{A}}_i^k{\overset{\sim }{B}}_j^w{H}_i^k{E}_j^w\exp \left\{\upmu \left({\mathrm{b}}_{\mathrm{ij}}^{\mathrm{kw}}-{c}_{ij}\right)\right\} $$
$$ {\overset{\sim }{A}}_i^k=\frac{e^{-{\overset{\sim }{\lambda}}_i^k}}{\sqrt{\mathrm{N}}}\kern0.5em and\ {\overset{\sim }{B}}_j^w=\frac{e^{\overset{\sim }{\gamma_j^w}}}{\sqrt{\mathrm{N}}} $$
Model (27) can be re-expressed in dual variable form as:
$$ \frac{{\overset{\sim }{\lambda}}_i^k}{\upmu}-\frac{{\overset{\sim }{\gamma}}_{\mathrm{j}}^{\mathrm{w}}}{\upmu}\kern0.75em =\left({\mathrm{b}}_{\mathrm{ij}}^{\mathrm{kw}}-{c}_{ij}\right)-\frac{{\mathit{\log}}_e\left(\frac{\mathrm{N}\ {T}_{ij}^{kw}}{H_i^k{E}_j^w}\right)}{\mu}\kern0.5em $$
The EM Herbert-Stevens model derived using the “unweighted” entropy term gives a model with the same expression as that on the far right-hand side of Eq. (27), but with:
$$ {A}_i^k{H}_i^k={e}^{-{\lambda}_i^k}\ and\kern1.25em {B}_j^w{E}_j^w={e}^{\gamma_j^w}\kern0.5em $$
and dual variable form:
$$ \frac{\lambda_i^k}{\upmu}-\frac{\gamma_{\mathrm{j}}^{\mathrm{w}}}{\upmu}\kern0.75em =\left({\mathrm{b}}_{\mathrm{ij}}^{\mathrm{kw}}-{c}_{ij}\right)-\frac{{\mathit{\log}}_e\left({T}_{ij}^{kw}\right)}{\mu}\kern0.5em $$

The dual variables can be interpreted as market prices/rents for housing, λik/μ, and consumer welfare terms for workers, γjw/μ (a subsidy indicating a welfare loss or a surplus indicating a welfare gain depending on the sign of the dual variable). The terms deducted from bids on the right-hand sides of (29) and (31) may be interpreted as sub-optimal welfare terms.

Again it is seen (Table 2) that the dual variables in the “unweighted” version of the model are biased by the zone size dependent Hik and Ejw terms. We illustrate this empirically by comparing the dual housing market prices/rents with observed housing prices/rents in 1966 for the Leeds study area used by Senior and Wilson (1974). Here, the observed price of housing of type k in any zone is taken as the simple average of weekly prices or rents of a sample of such houses in that zone. It is readily seen (Figs. 1 and 2) that the dual house prices from the “prior weighted” model (27) or (29) provide a much closer fit to the observed prices than the dual house prices from the “unweighted” model (31). The dual house prices from both models are adjusted by a constant to have an identical minimum price to the observed minimum price, but this does not necessarily imply that these lowest prices refer to the same houses by type or zone.
Table 2

Dual variables and sub-optimal welfare terms in the Herbert-Stevens model


“unweighted” model

“prior weighted” model


Equations (30) and (31)

Equations (28) and (29)

Market prices for housing

\( \frac{\lambda_i^k}{\upmu}=\frac{-{\log}_e{A}_i^k-{\log}_e{H}_i^k}{\upmu} \)

\( \frac{{\overset{\sim }{\lambda}}_i^k}{\upmu}=\frac{-{\log}_e{A}_i^k-{\log}_e\sqrt{\mathrm{N}}}{\upmu} \)

Consumer surplus/subsidy

\( \frac{\gamma_j^w}{\upmu}=\frac{-{\log}_e{B}_j^w-{\log}_e{E}_j^w}{\upmu} \)

\( \frac{\gamma_j^w}{\upmu}=\frac{-{\log}_e{B}_j^w-{\log}_e\sqrt{\mathrm{N}}}{\upmu} \)

Sub-optimal welfare term

\( \frac{{\mathit{\log}}_e{T}_{ij}^{kw}}{\mu } \)

\( \frac{{\mathit{\log}}_e\left(\frac{\mathrm{N}\ {T}_{ij}^{kw}}{H_i^k{E}_j^w}\right)}{\mu } \)

Fig. 1

Observed and dual house prices from the “unweighted” model (re-drawn from Senior and Wilson 1974)

Fig. 2

Observed and dual house prices from the “prior weighted” model

Senior and Wilson (1974 pages 221-2 and 235-6) interpret the logeTijkw/μ terms as suboptimal surpluses (indicating benefit gains) to workers as their values are positive for Tijkw > 1 (but zero for Tijkw = 1) and they are always subtracted from bids in (31). However, once the zone size effects are compensated for in the “prior weighted” model (29), these suboptimal welfare terms incorporate division of the Tijkw by the prior weights (HikEjw)/N. The effect is often to reduce surpluses and, for smaller values of Tijkw and/or larger values of HikEjw, this suboptimal term may be negative, indicating that those making the most suboptimal choices suffer a disbenefit as they have to increase their bids to meet the market price of housing. This is illustrated in Table 3 which examines intermediate social class workers employed in the university zone Westfield. They are predicted to prefer better owner-occupied housing in, for example, Far Headingley in suburban Leeds, where they benefit from a much smaller surplus of 53 pence compared with the biased value of £2.42. Effectively only one worker chooses poorer private rented housing in another suburban zone, Roundhay, and suffers a loss of 50 pence.
Table 3

Sub-optimal welfare values (in £) for intermediate workers in Westfield ward (near Leeds city Centre) predicted to choose housing in four neighbourhoods in 1966


“unweighted” model


“prior weighted” model


Sub-optimal welfare


Sub-optimal welfare

Owner-occupied housing in

\( \frac{{\mathit{\log}}_e{T}_{ij}^{kw}}{\mu } \)


\( \frac{{\mathit{\log}}_e\left(\frac{\mathrm{N}\ {T}_{ij}^{kw}}{H_i^k{E}_j^w}\right)}{\mu } \)

Hyde Park








Far Headingley




Private rented housing in






Conclusions: Recommendations and Policy Applications

The conclusions to be drawn from this paper are twofold. First, EM derivation of all doubly constrained models in the spatial interaction family can use the type of prior weights presented here to compensate for the arbitrary effects of the imposed zoning system. Although not illustrated in this paper, other models in the spatial interaction family can also use appropriately modified forms of such prior weights. This recommendation applies also to the group surplus approach to deriving the same type of spatial interaction models (Williams and Senior 1978).

Secondly, prior weighted entropy formulations of doubly constrained models allow dual variables to be analysed and interpreted in terms of spatial prices and benefit indices, unbiased by zone size, which can be used in land use and/or transport policy evaluation exercises (Williams and Senior 1978). For example, Williams and Senior (1977) use combined spatial interaction and mode choice models to compare each of four transport policies for West Yorkshire (highway investment, free-fares public transport, marginal increase in parking charge in Leeds city centre and car exclusion in that centre) against a “do nothing” strategy and derive changes in total consumer surplus from the dual variables for each policy. While this latter study considers only policies affecting travel costs and presents only study area evaluation measures, the zonal dual variables will place values on changes in accessibility arising from the transport policies. As measures of spatial variations in benefits, the dual variables will be particularly useful in land use policy contexts. Thus the Herbert-Stevens model can be used to obtain house price and welfare changes arising from planned zonal changes in housing supply and/or in employment and/or from changes in travel costs associated with transport policy proposals. Similarly, in the spatial interaction shopping model (Wilson 1976) the impacts of changes in retail provision and/or resident populations are recorded in dual variables interpreted as retailer profits and shoppers’ welfare. Moreover, the zonal dual variables indicate where marginal changes in land use are beneficial, thus directing future growth or relocation. Such considerations lead to optimal plan design issues (Wilson et al. 1981 chapter 8) where spatial interaction sub-models are embedded in problems seeking the optimal size and location of various land uses.


Compliance with Ethical Standards

Conflict of Interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.RosemountLlangwm, UskUK
  2. 2.School of Geography and Planning, Glamorgan BuildingCardiff UniversityCardiffUK

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