The identification of possible future provincial boundaries for South Africa based on an intramax analysis of journey-to-work data

National census data contain information on place of residence and place of work. It is possible to combine this information and create journey-to-work flows. The process of establishing these flows are presented in this paper. The intramax method is explained and used to identify functional regions based upon these flows. Interesting applications, such as the demarcation of regions in South Africa are considered and solutions to disputed areas are put forward. The process of the creation of the current provincial boundaries are discussed. New boundaries, based on the intramax analysis of the journey-to-work data are proposed for four or five new provinces. Results compare favourably with those from a principal component and cluster analysis, which has previously been used to demarcate the South African space economy into a hierarchy of development regions.


Introduction
On 28 May 1993, the Negotiating Council of the Multiparty Negotiating Process established a fifteen-person commission to make proposals for new internal boundaries in South Africa [7].The resulting Commission on the Demarcation/Delimitation of Regions (the CDDR) held its first meeting on 8 June 1993 and reached a decision by 31 July 1993.After six weeks, the commission more than doubled the number of provinces, from the initial four to the current nine provinces [7].No meaningful time was allotted for public consultation, and the commissioners took as the initial draft the nine planning regions established by the Development Bank of Southern Africa between 1982 and 1988 [7].Only one month of the CDDR's itinerary was devoted to gathering of testimony, and in reaction to broad Boundaries should be drawn so as to minimise the splitting of communities.South Africa's current spatial organisation and delineation are characterised by internal conflicts.Figure 1 shows, on a national level, the disputed areas after the 1993 delineation of provincial boundaries.Ramutsindela and Simon [19] described the process of negotiating between the provinces in the time period after 1993 as "horse-trading."Northern Province (currently Limpopo Province), for example, demanded that the towns of Groblersdal and Marble Hall, which are part of Mpumalanga, be transferred to the Northern Province to compensate for relinquishing Bushbuckridge.On the other hand, the people of Bushbuckridge have been campaigning for years to be incorporated into Mpumalanga and not Limpopo Province.While belonging to Limpopo Province, research has shown that many (95-98%) of the residents prefer incorporation into Mpumalanga, with their reasons advanced being geographical proximity and economic ties.Residents argue that this is where they work and undertake their shopping [19].
According to Smith [21], the former chairperson of the ANC, Mosiuoa Lekota, became the most senior member of the party to date to suggest that a reduction in number from the current nine provinces should be considered seriously.According to Ngalwa [18] a discussion document, which moots a four or five province option, was drafted and circulated in government during 2007.Some ministers in the previous cabinet, including Finance Minister Trevor Manuel, Defence Minister Mosiuoa Lekota and Minister Sydney Mufamadi have publicly suggested that the number of provinces should be reduced.They also requested that proper research should be conducted to review the performance of the provincial system before deciding on their future.
It is clear that the process of demarcation cannot be examined without taking political motives into consideration, whilst the needs of people living and working in the provinces should also be considered.Functional regions based on activities of households and businesses are the people's way of deciding to which areas they belong.

Functional regions
The concept of a functional region or functional area may be described in many ways.Feldman et al. [4] described it as an area defined by business and economic activities rather than by administrative or historic boundaries.A functional region was also defined by Brown and Holmes [1] as an area or locational entity which enjoys more interaction or connection within its boundaries than with outside areas.
Functional regions may also be seen as areas in which the businesses concerned recruit most of their labour force.The quality of functional region demarcation has a strong influence on both productivity and prosperity.The functional region is a phenomenon arising exclusively from human activity, and is best described as a community of interests.In respect of human activity, specific reference is paid to transport, work and residential choice and therefore functional regions are a spatial manifestation of social organisation.Functional regions represent the day-to-day regions in people's lives, i.e. they are created by the various choices and decisions of individual people and enterprises.
Feldman et al. [4] noted that the best-established technique for a functional approach to area grouping is to identify boundaries across which relatively few people commute.Mitchell et al. [17] reasoned that journey-to-work data provide information about the interaction between spatial units and are a useful basis for defining functional regions.A commuting area is conceived as a geographical area within which there is a high degree of interactivity and may be seen as an appropriate spatial region to capture the interplay between labour supply and demand.Mitchell et al. [17] concluded that aggregations of journey-to-work data reflect economic behaviour rather than administrative structures.
The objective of this paper is to analyse journey-to-work flow data and to use intramax analysis to establish functional regions in South Africa in general, but specifically at provincial level.The purpose is to demonstrate how functional regions differ from administrative regions (which are more than likely demarcated in terms of political or ideological philosophy).A further objective is to test whether the functional regions or provinces identified by the intramax analysis are economically viable regions.

Literature review on analysis of flow data
Journey-to-work data may be captured in a network flow problem, which consists of a collection of transhipment nodes connected by directed arcs in both directions.Figure 2 contains an example of journey-to-work data between four regions.A schematic representation of a so-called interaction matrix is provided in Table 1, where rows are designated as origins and columns are destinations.Marginal totals may be interpreted as follows: O i = j a ij and D j = i a ij represent the total outflow from region i and total inflow into region j respectively.
Ward [26] developed a hierarchical aggregation procedure which is a routine for searching through groups of data to find which pair of basic data units shows the greatest mutual similarity with respect to specified characteristics.Given k subsets, this method permits their reduction to k − 1 mutually exclusive subsets by considering the union of all possible are in one group.
Ward [26] defines a functional relation that provides a "value reflecting" number as an objective function.It is common practice to use the mean value to represent all scores.
The loss in information that results from treating scores as one group may be indicated by a "value-reflecting" number such as the Error Sum of Squares (ESS).The ESS is given by where x i is the score of the i-th individual and where m denotes the number of individuals.
If scores are classified in groups, the grouping can be evaluated as the sum of the ESS values, that is ESS Groups = ESS (Group 1) + ESS (Group 2) + . . .
The same procedure can be used for aggregation of flow data if the objective function is respecified in terms of the two-directional flow between two regions.It will be necessary to consider two entries for this purpose, namely a ij and a ji , for all i = j.
Masser and Brown [14] formulated as objective the maximisation, at each stage of the grouping process, of the difference between the observed values, a ij , and "expected values" a * ij , which are derived similarly to the expected frequency of the cell in row i and column j in a contingency table for the Chi-square test, namely The objective is therefore to The entries a ij are standardised so that i j where a ij = a ij /n.It can be shown that the standardised objective is to Contiguity constraints may be introduced to restrict the search for potential pairings.These constraints may take the form c ij = 1, if movement of a basic data unit from i to j is allowed, and c ij = 0 otherwise.
The intramax analysis is a stepwise analysis.During each step two areas are grouped together and the interaction between the two areas becomes internal (or intrazonal) interaction for the new resulting area.This new area now takes the place of the two parent areas at the next step of the analysis.So with N areas, all areas are grouped together into one area after N − 1 steps and all interaction is intrazonal.The outcome of an intramax analysis may be presented in dendrogram form.
According to Tyree [24], the alternative concept of mobility ratios was developed by three sociologists, Natalie Rogoff, David Glass and Gösta Carlsson [24], working independently on the problem of intergenerational occupational mobility.A matrix of frequencies of occupations of respondents by occupations of fathers may be converted into matrices of inflow and outflow percentages.The mobility ratio M ij is simply the ratio of the frequency observed in a cell to the frequency expected under the assumption of statistical independence.Hollingworth [13] studied migration between Scottish executive areas and also defined the mobility index as (1).The value of the objective function in this case is then M ij + M ji , which was used as a symmetric measure of the mutual association of areas i and j.
Hirst [12] noted that both the objective functions defined by Masser and Brown [14] and Hollingworth [13] is inappropriate, because of the influence of unequal marginal distributions which define the expected frequencies.For example, the ratio or difference between the observed and expected values will tend to increase for cells in those rows and columns with large sums.Since the objective function is recalculated after each step in the grouping procedure, this bias will be cumulative.
Tyree [24] suggested that the interaction matrix should first be adjusted to achieve an arbitrary origin-destination distribution.This may be accomplished iteratively by standard matrix operations: rows are scaled initially to sum to a given total, and then columns are scaled to sum to the same total.This procedure is repeated until sufficient convergence occurs to a matrix in which all row and column sums are simultaneously equal.
Hirst [12] claimed that it can be proved that this matrix exists, is unique, and that the iterative procedure is convergent.He suggested that a possible solution would be to divide a ij − a * ij by a * ij , with a * ij corrected for blank entries in the interaction matrix as proposed by Goodman [7], but noted that results will still tend to favour small zones, because of the differences between the values obtained for small as opposed to large zones.Hirst also remarked that an increasing number of heuristic techniques has become available, and that a need for comparative evaluation of their respective merits and areas of application has arisen.
Masser and Scheurwater [15] evaluated three methods for functional regionalisation, namely the functional distance method [1] (not discussed in this paper), the iterative proportional fitting based procedure (IPFP) [20] (not discussed in this paper) and the intramax procedure [14].Their conclusion was that the intramax procedure is the only one of the three procedures which explicitly identifies regions that have more (direct) interaction with each other than with other areas at each stage of the grouping process.It has a practical advantage over the other two methods, because it only involves a series of direct comparisons between the observed and expected values that are calculated by the multiplication of the respective row and column totals.This avoids the complex set of matrix manipulations that are required for the other two methods.The intramax procedure may be more readily applied to large data sets and may be adapted more easily to deal with large, sparse matrices.Masser and Scheurwater [15] also noted that stronger connections would appear between pairs of smaller zones containing a relatively low proportion of intrazonal interaction than between pairs of larger zones containing a relatively high proportion of intrazonal interaction and that the former would tend to fuse together before the latter.They reason that this bias noted by Hirst [12], far from being a disadvantage, is in fact advantageous and that it is a reflection of the inherent characteristics of the structure of spatial interaction in the matrix.
Fischer et al. [5] compared the intramax procedure with the IPFP-based graph approach (not discussed in this paper) and came to the conclusion that the intramax approach is superior to the IPFP-based graph-theoretical one, because the results are easily interpretable in terms of functional regions.The intramax approach also leads to spatial groupings which show more interaction with each other than with other regions.
Brown and Pitfield [2] noted that the objective function was reformulated in literature appearing after the comment of Hirst [12] to maximise i =j They remarked that this revised form of the objective function was employed in all subsequent applications of the procedure, and may be re-expressed a little more simply as The reason for this is that the part that is subtracted in each term is constant and may thus be ignored.This objective function is also discussed by Brown and Pitfield [2].The resulting formula is strikingly similar to the mobility ratios employed by Hollingworth [13], where and specialises in displaying interaction data (such as commuting and migration flows), interaction analysis (such as accessibility analysis), network analysis, and interaction modelling.The program uses several kinds of data, which may be grouped into three classes: maps, flow data and distance tables.
Flowmap uses intramax analysis to identify functional regions from an interaction matrix."The objective of the intramax procedure is to maximise the proportion within the group interaction at each stage of the grouping process, while taking account of the variations in the row and column totals of the matrix " [22].This implies that in this particular case two areas are grouped together for which the objective function is maximised where T ij is the interaction between origin location i and destination location j, and where This is similar to (2) and the method of Hollingsworth [11], but the constant n is omitted.
The objective function in (3) can only be calculated for all D j > 0 and for all O i > 0.
In Flowmap actual flow values are used, hence T ij instead of a ij , but that should not have any effect on the results as no comparisons are made; the maximum relationship is merely sought at each aggregation step.The use of the above objective function is also substantiated in a thesis by Floor and de Jong [6].

Methodology and data
The methodology employed and the data used in this paper are described in this section.

Journey-to-work data and intramax analysis
The data used in this paper all derive from the 2001 South African Census [22].The question was asked "In the seven days before 10 October did (the person) do any work for pay (in cash or in kind), profit or family gain, for one hour or more?If "Yes," does (the person) work in the same sub-place in which s/he usually lives?"If "No," the main place of work was recorded.The definition of work includes formal, informal and seasonal work.The database of all persons between the age of 15 to 65 represented 28 427 129 individuals.A subdatabase was prepared at the request of the authors containing amongst others, the following fields: main place code and main place of work code.For reasons of confidentiality, records were totalled and frequencies in each category, defined by the field names, were calculated.The resulting subdatabase contained a total of 1 890 827 records.Part of the confidentialising process was to change frequencies of 1 and 2 according to an algorithm, as follows: • Change a frequency of 1 to 0 in two thirds of the cases; • Change a frequency of 1 to 3 in one third of the cases; • Change a frequency of 2 to 0 in one third of the cases and • Change a frequency of 2 to 3 in two thirds of the cases.
Certain records were not considered for the intramax analysis1 .The records not considered included 198 758 records (18 792 972 individuals) for which the main place of work were marked as not applicable, due to the fact that these records represent persons unemployed or not economically active.A further 65 556 records (156 899 individuals) were deleted, because the main place of work was "unspecified."Of the remaining records, a further 107 818 records (182 237 individuals) were removed due to the fact that they replied "No" to the question "Is this your usual place of stay?"A further 3 997 records (9 679 individuals) were deleted because their economic activity was marked "Not economically active."Some further 32 290 records (59 933 individuals) were deleted because the main place names could not be matched (the province code was given instead of the code of a specific main place).
The following data cleanup was also performed and the interaction data were adjusted accordingly: • 7 islands were removed, • 638 fully embedded regions were dissolved, • 24 main places without interaction were dissolved, • 46 main places with only intrazonal interaction were dissolved.
Intramax analysis was therefore applied to a total of 861 939 records involving 2 393 extended main places.

Principal component and cluster analysis
It is important to validate the results, e.g. to use different methods with different variables to establish whether boundaries and regions defined by the intramax analysis may be viewed as socio-economic functional regions.Harmse [10], using mainly 1996 Census data, demarcated the South African space economy into a hierarchy of five development regions, i.e. a highly developed metropolitan core region, an upward transitional region, a downward transitional region, a resource frontier region and special problem regions.Harmse et al. [11] reapplied this technique on 2001 Census data, using the following socioeconomic variables: A data matrix consisting of variables and municipalities as spatial units was compiled as input for the multivariate analysis.Using principal component analysis, the large number of correlated variables was reduced to fewer variables that captured most of the variation in the original variables.Cluster analysis was then used to identify groups of similar main places in order to reduce the number of spatial units to a more manageable number, using the scores of the different principal components.By applying Ward's cluster analysis, the semi-partial R 2 values generated was used to identify a significant grouping.The mean score on principal component I for these different groups was calculated in order to determine how the groups may be assigned to the different regional types [10].The results are reported in the following section.
The Community Profile database [23] of Census 2001 was accessed in SuperCross format at main place level.The weighted mean, median and inter quartile range of some socio-economic variables were calculated for a proposed five-province scenario and were compared using Bonferroni multiple comparisons.

Intramax analysis
A total of 2 392 iterations were required in the intramax process.At each stage of the clustering process, two regions with the strongest possible commuting ties were aggregated.These two regions were then seen as one region, and commuting between these two regions become intrazonal.The total number of regions was thus reduced by one region and the interaction matrix was reduced by one row and one column.This process was repeated until only one region remained (theoretically), in which all commuting is intrazonal.
During this process, there were 18 minor areas exhibiting unusually large flows, which were not clustered -they remained original main places.For example, the Kgalagadi Park (main place 39 302) in the Northern Cape has only outside commuter links and comprises a total of 7 persons all residing/working in the Saldanha area over 800 km away.The flows to/from the 18 problem main places were removed.Other surviving unlinked main places were also removed or dissolved, yet ensuring that this process did not impact on the boundaries of the remaining clusters.
The clustering process continued until 80% of the interzonal interaction internalised with 70 functional areas (blocks) remained.The results are shown in the dendrograms in Figures 3-7 and the map in Figure 8.
In Figure 3, the Nama Khoi region includes the town of Springbok and the Richtersveld National Park.This fuses with the Matzikama region, which includes Van Rhynsdorp,  In Figure 3, the Nama Khoi region includes the town of Springbok and the Richtersveld National Park.This will fuse with the Matzikama region, which includes Van Rhynsdorp, Vredendal, Calvinia, Sutherland, Carnarvon and others.About 31% of all the journey-to-work flows in and out these regions are intrazonal for this new aggregation.In the next step, this region will fuse with the Witzenberg region, which includes places such as Ceres, Tulbach and Clanwilliam (47% intrazonal).The Cape Town region (including Stellenbosch, Strand, Paarl, etc) will fuse with the Swartland region (30% intrazonal), which includes Moorreesburg, Malmesbury, Saldanha and others.The Breede River/Winelands area (Montagu, Swellendam, etc) will fuse with the Breede Valley area (Worcester, Robertson, etc) (31% intrazonal), which will then fuse with the Cape Town/ Swartland cluster (44% intrazonal).This cluster will then fuse with the Theewaterskloof cluster (63% intrazonal), which includes the Overberg region.The George cluster (which includes most of the Garden Route) will merge with the bigger Cape Town cluster (64% intrazonal), and finally this will merge with the Witzenberg/ Nama Khoi/ Matzikana cluster (68% intrazonal).('First Province' of 9 last clusters shown in Figure 8) In Figure 4, the Paradise Beach area (Jeffreys Bay, Tsitsikamma National park, Stormsriver area) merges with the Port Elizabeth area (30% intrazonal), and fuses in the next step with the Ubuntu area (which includes places such as Victoria West, Richmond, and others) and the Inxuba Yethemba region (Cradock, Middelburg and others) (31% intrazonal).This region then fuses with the Graaff Reinet area (47% intrazonal).The Grahamstown and East London (31% intrazonal), fuse with the Lusikisiki (including Flagstaff), Queenstown, Kokstad and Marburg (Port Shepstone and others) regions (63% intrazonal).This region then fuses with the greater Port Elizabeth cluster (67% intrazonal).('Second Province' of the nine last clusters shown in Figure 8) In the second part of Figure 4, the Durban and Pietermaritzburg regions (34% intrazonal) merge with the Umvoti (Greytown, Kranskop and more) and Stanger regions (43% intrazonal).The Myeni/Ntsindi area (Jozini and more) fuses with the Richardsbay area (31% intrazonal), and this region fuses next into the greater Durban region, followed by the Mkhambathini region, which looks like a region on its own (Camperdown and more) (69% intrazonal).('Third Province' of the 9 last clusters shown in Figure 8).Vredendal, Calvinia, Sutherland, Carnarvon and others.Approximately 31% of all the journey-to-work flows in and out of these regions are intrazonal for this new aggregation.In the next step, this region fuses with the Witzenberg region, which includes places such as Ceres, Tulbach and Clanwilliam (47% intrazonal).The Cape Town region (including Stellenbosch, Strand, Paarl, etc.) fuses with the Swartland region, which includes Moorreesburg, Malmesbury, Saldanha and others (30% intrazonal).The Breede River/Winelands area (Montagu, Swellendam, etc.) fuses with the Breede Valley area (Worcester, Robertson, etc.) (31% intrazonal), which then fuses with the Cape Town / Swartland cluster (44% intrazonal).This cluster then fuses with the Theewaterskloof cluster (63% intrazonal), which includes the Overberg region.The George cluster (which includes most of the Garden Route) fuses with the larger Cape Town cluster (64% intrazonal), and finally this fuses with the Witzenberg / Nama Khoi / Matzikama cluster (68% intrazonal).('First Province' of the nine last clusters shown in Figure 8.) In Figure 4, the Paradise Beach and Kouga areas (Jeffreys Bay, Tsitsikamma National park, Stormsriver area) merges with the Port Elizabeth area (30% intrazonal), and fuses in the next step with the Ubuntu area (including Victoria West, Richmond, etc.) and the Inxuba Yethemba region (Cradock, Middelburg, etc.) (31% intrazonal).This region then fuses with the Graaff Reinet area (47% intrazonal).The Grahamstown and East London regions (31% intrazonal) fuse with the Lusikisiki (including Flagstaff), Queenstown, Kokstad and Marburg (Port Shepstone and others) regions (63% intrazonal).This region then fuses with the greater Port Elizabeth cluster (67% intrazonal).('Second Province' of the nine last clusters shown in Figure 8.) In the second part of Figure 4, the Durban and Pietermaritzburg regions (34% intrazonal) merge with the Umvoti (Greytown, Kranskop, etc.) and Stanger regions (43% intrazonal).The Myeni/Ntsinde area (Jozini, etc.) fuses with the Richards Bay area (31% intrazonal), and this region fuses next into the greater Durban region, followed by the Mkhambathini region, which looks like a region on its own (Camperdown, etc.) (69% intrazonal).('Third Province' of the nine last clusters shown in Figure 8.) The third part of Figure 4 consists of the Ladysmith region (including Escourt, etc.) and the Newcastle region (including the Volksrust and Standerton areas in the current Mpumalanga Province) (45% intrazonal).('Fourth Province' of the nine last clusters shown in Figure 8.)In Figure 5 the Dukathole (including Jamestown in the current Eastern Cape and Aliwal North) and Kopanong (Bethulie, Philippolis, etc.) areas fuse (31% intrazonal).This region then fuses with the Naledi (Van Stadensrus, Wepener, etc.) and Bloemfontein regions and the resulting region results in 47% intrazonal flows.The Setsoto (Clocolan, Ficksburg, Senekal), Nketoana (Lindley, Reitz, Petrus Steyn), Phuthaditjhaba and Phumelela (Memel, Vrede and Warden) regions merge (47% intrazonal flow) and this region fuses with the greater Bloemfontein region (68% intrazonal).The Tswelopele (Bultfontein and Hoopstad), Maquassi Hills (Leeudoringstad and Makwassie regions in the current North West Province), Thabong (Odendaalsrus and Welkom), Nala (Bothaville regions), Moqhaka (Kroonstad and Steynsrus) and Klerksdorp region in the current North West Province fuse (48% intrazonal) which then fuse with the previous region, including Bloemfontein, (69% intrazonal) to form the 'Fifth Province' of the nine last clusters shown in Figure 8.
In Figure 6, the regions of Kai !Garib (Augrabies, Kakamas and other regions in the Northern Cape) and !Kheis (Groblershoop, Grootdrink, etc. in the Northern Cape) merge with the Kimberley, Letsemeng (Petrusburg, Jacobsdal, etc. in the Free State) and Vryburg (also Schweizer-Reneke and other regions in the North West Province) regions (47% intrazonal).This region merges with the Rustenburg and Mafikeng fusion (69% intrazonal), resulting in the 'Sixth Province' of the nine last clusters shown in Figure 8.
The Modderfontein region merges with the Boksburg, Johannesburg fusion (28% intrazonal), and the Evaton (Vaal Triangle, including Sasolburg in the Free State) and Lesedi (Heidelberg, Nigel Springs) regions then fuse into the Johannesburg region (46% intrazonal), then follow the Pretoria region, the Randfontein region and lastly the Merafong (Carltonville, Khutsong and others) region (66% intrazonal).This results in the 'Seventh  In Figure 6, the regions of Kai !Garib (Augrabies, Kakamas and other regions in the Northern Cape) and !Kheis (Groblershoop, Grootdrink and more in the Northern Cape) merged with the Kimberley, Letsemeng (Petrusburg, Jacobsdal and more in the Free State) and Vryburg (also Schweizer-Reneke and other regions in the North West Province)regions (47% intrazonal).This region merges with the Rustenburg ( ) Mafikeng ( ) fusion (69% intrazonal), resulting in the 'Sixth Province' in figure 8. Province' of the nine last clusters shown in Figure 8.
The Modderfontein region merges with the Boksburg, Johannesburg fusion (28% intrazonal), and the Evaton (Vaal Triangle, including Sasolburg in the Free State) and Lesedi (Heidelberg, Nigel Springs) regions then fuse into the Johannesburg region (46% intrazonal), then follow the Pretoria region, the Randfontein region and lastly the Merafong (Carltonville, Khutsong and others) region.(66% intrazonal).This results in the 'Seventh Province' in Figure 8.     Finally the Greater Tzaneen (including Haenertsburg, Letsitele, etc.) and Phalaborwa (including Gravelotte, Die Eiland, etc.) regions merge (28% intrazonal).The Pietersburg region (Polokwane) fuses with the Tzaneen region (45% intrazonal), followed by a fusion with the Tshivhase region (Thohoyandou, Gijana, etc.) and lastly the Bela-Bela region (Warmbaths, Nylstroom, etc.), with a total of 67% intrazonal flow, resulting in the 'Ninth Province' in Figure 8. Finally the Greater Tzaneen (including Haenertsburg and Letsitele for example) and Phalaborwa (also Gravelotte and Die Eiland for example) regions merge (28% intrazonal).The Pietersburg region (Polokwane, fuses with the Tzaneen region (45% intrazonal), followed by the fusion with the Tshivhase region (Thohoyandou, Gijana regions) and lastly the Bela-Bela region (Warmbaths, Nylstroom regions), with a total of 67% intrazonal flow, resulting in the 'Ninth Province' in Figure    The dots in Figure 8 are proportional to the volume or level of intrazonal interaction per new functional block and the nine-province division shown in the figure has been constructed by means of the intramax method from the interaction between the remaining 70 blocks.
Tables 2 and 3 show the commuter flows crossing provincial boundaries in the current context and the proposed new situation with nine provinces.The number of boundarycrossing commuters is reduced in the intramax solution by over 45% from 287 000 to approximately 157 000.The total workforce is approximately 9.4 million, but only some 2.7 million workers commute daily between different main places.The difference between the total workforce and the part of the workforce that actually commutes explains the difference between the numbers in Tables 2 and 3 and the numbers given in §5.1.

Reducing the number of provinces to four or five
The dendrograms in Figures  the Eastern Cape and KwaZulu-Natal region would fuse with the Western Cape region (three provinces) and the Free State region would amalgamate with the Gauteng region (two provinces).The country thus becomes consolidated into a final North-South division.
The boundaries between the Western Cape region and the Eastern Cape / KwaZulu-Natal region are mountainous regions, but it seems that rivers, such as the Orange River and the Vaal River, which were historical boundaries, do not impact as much on the boundaries any longer, because of the accessibility via roads to the nearest major centres.
Figure 9 shows the reduction from nine provinces to four provinces.

Disputed areas
The disputed area of Bushbuckridge is used as an example to demonstrate how the intramax analysis may be used to resolve similar contentious situations.The current provincial boundary crosses straight through the Buschbuckridge functional area and generates five times more cross-boundary commuting than the alternative suggested by intramax.The map in Figure 10 The disputed regions of Groblersdal and Marble Hall (Shown in Figure 1) were allocated to Mpumalanga, but transferred to Limpopo province in December 2005 [8].The intramax analysis indicates that these regions will actually fuse with the Gauteng region.Sasolburg will also fuse into the Gauteng region, and not with the Free State, where it is currently situated.
Kuruman, Postmasburg and Hartswater (currently in the Northern Cape) will be allocated to the North West region, but the boundaries of the North West region will move further south, and include more regions of the Northern Cape, even regions such as Upington, Prieska and De Aar.This is because of the accessibility to Kimberley, which will also be located in the North West region.
The Namaqualand (currently in the Northern Cape), Clanwilliam and Van Rhynsdorp (currently in the Western Cape) regions will be allocated to the Western Cape, and again, here, the N7 route ensures accessibility to the Cape Metropole.
The Pondoland, East Griqualand (currently in the Eastern Cape) and Umzimkulu (currently in the KwaZulu Natal) regions will fuse initially with the Eastern Cape region, but in a four and five province scenario, the Eastern Cape region will fuse with the KwaZulu-Natal region, leaving these disputed areas in the middle of the new province.The calculated PC I scores for each of the 249 spatial units comprised a new data set.
Cluster analysis was performed on this data set and the most effective grouping of the 249 spatial units resulted in 18 groups, which were then assigned to four regional types.Discriminant analysis was conducted to determine the effectiveness of the groupings.
Figure 11 shows the results of the demarcation of socio-economic development regions in the South African space economy.The 2001 development regions in South Africa ranged from the highly developed core region, through the upward-transitional and downwardtransitional regions, to the special problem regions.According to Harmse et al. [11], the core region has the highest level of development and, in 2001, 69.2% of the country's total income was earned by people living in the core region.The core region housed 38% of the country's population on only 5.45% of the land area.The non-contiguous core region consists of the following regions (in descending order per province): City of Johannesburg In dept analysis of Bush Buck Ridge as example.
Map1 shows "major "commuter flows; many crossing current provincial border (in blue) Intramax results in a two way split of the area where the whole Bushbuckridge functional area is allocated to the southern province.The new provincial boundary is indicated in black.
Of the 60420 commuters in the area 3771 (6.24%) cross the provincial border in the current situation.This number would be reduced to 679 (1.12%) in the proposed provincial split.

22 Figure 2 :
Figure 2: Example of a network of flows between 4 regions.

Figure 3 :
Figure 3 : Dendogram of last 8 regions in Western Cape

Figure 3 :
Figure 3: Dendrogram of the last nine regions in the Western Cape.

Figure 4 :
Figure 4: Dendrogram of the last twenty-one regions in the Eastern Coastal region.Kouga and Paradise Beach merge at the very start of the procedure resulting in less than 0.5% intrazonal interaction.

Figure 5 :
Figure 5: Dendogram of the last 14 regions in the Central region

Figure 6 :
Figure 6 : Dendogram of the last 15 regions in the Gauteng region

Figure 6 :
Figure 6: Dendrogram of the last fifteen regions in the Gauteng region.

Figure 7
Figure 7 is a fusion of the remaining regions of the Limpopo Province and the Mpumalanga Province.Msukaligwa (Ermelo region) and Mkhondo (Piet Retief region) merge with Embalenhle (Kinross, Leslie, Evander regions) and Witbank region (48% intrazonal).The Highlands (Dullstroom, Machadodorp regions) and Mbombela (Nelspruit region) regions merge with the greater Witbank region (67% intrazonal) which completes the 'Eighth Province' of the nine last clusters shown in Figure 8.

Figure 7 :
Figure 7 : Dendogram of the last 11 regions in the Northern region

Figure 7 :
Figure 7: Dendrogram of the last eleven regions in the Northern region.

Figure 8 :
Figure 8: The remaining nine clusters with dots indicating the relative size of the intrazonal interaction per functional region.
(a)  shows "major" commuter flows, many crossing the current provincial boundary.The map in Figure10(b) shows the intramax analysis results after a cleanup into eleven functional areas just before the Buskbuckridge area fuses with the South Kruger Park.The map in Figure 10(c) shows several larger commuter flows into / out of Bushbuckridge across the current provincial boundary.The intramax results shown in Figure10(d) allocate the whole of the Bushbuckridge functional area to the southern province and the proposed boundary follows the boundary of the building block instead of cutting through it.Of the 60 420 commuters in the area 3 771 (6.24%) currently cross the provincial boundary.This number would be reduced to 679 (1.12%) in the proposed provincial split.

Figure 9 :
Figure 9: Reduction from nine to four provinces using intramax analysis.
(a) Flows of 10 or more commuters between MPs.Intramax analysis results after cleanup in 11 functional areas just before Buskbuckridge area clusters with the South Kruger Park (b) Just before Bushbuckridge merges with a neigbouring area the 74 MPs in the region have clustered to 11 functional areas.Several larger commuter flows into/out of Bushbuckridge cross current provincial boundary (in blue) (c) Flows of 100 or more commuters between the functional areas Bushbuckridge has stronger ties to the east (Kruger Park South).
An intramax two way split of the area results in a new provincial boundary.

Figure 10 :
Figure 10: In-depth analysis of Bushbuckridge as example of a disputed area.