HIGHER TERRITORIAL UNITS AND TRANSPORTATION ACCESSIBILITY OF THEIR CENTERS

Článok sa zaoberá územnosprávnym členením Slovenska z hľadiska dopravnej dostupnosti sídelných miest vyšších územných celkov (krajov). Dopravnú dostupnosť definujeme ako priemernú vzdialenosť, ktorú musí obyvateľ prekonať pri cestovaní z miesta svojho bydliska do najbližšieho sídelného mesta vyššieho územného celku. Porovnávame optimálny návrh členenia s návrhmi, kedy je sídlo niektorého kraja pevne stanovené.


Introduction
Every change in an administration partition of the state territory and location of higher territorial unit centers has a lot of political, social and economic consequences. The corresponding process of decision and ratification is subjected to opposing priorities of various groups of citizens on local as well as countrywide level. Every decision, which is profitable for one group of citizens, is injurious for another one. That is why the mentioned process is a sequence of negotiations and compromises among public administration, self-administration and other groups of citizens. Citizen's priorities are often omitted in this argumentation in spite of that the impact of the form of higher territorial units and center locations can be easily evaluated. Our goal in this paper is not to design an ideal partition satisfying all concerned people and we are aware of that not all factors influencing partition quality can be evaluated. We are not going to analyze the number of higher territorial units either, because a serious analysis would suppose that we are able to calculate costs caused by location a higher territorial unit center at a given place. We come out from the initial governmental design of twelve higher territorial units in our considerations, one of which corresponds to Bratislava with its urban parts. We just want here to highlight that some factors playing role in partition quality can be evaluated and mathematics and computers can help in finding answers of question, what decisions should be made to accomplish the best results. In this paper, we take into account that the execution of the particular design of higher territorial units and center locations impacts the distance, which an inhabitant has to travel to access services that are usually located at a regional center [6]. This transportation aspect of administration partition is called transportation meste vyššieho územného celku [6]. Tento dopravný aspekt územnosprávneho členenia budeme nazývať dopravnou dostupnosťou. Dopravnú dostupnosť môžeme považovať za mieru kvality územnosprávneho členenia. Definujeme ju ako priemernú vzdialenosť, ktorú musí obyvateľ prekonať pri cestovaní z miesta svojho bydliska do najbližšieho sídelného mesta vyššieho územného celku. Takto chápané kritérium kvality možno vyhodnotiť pre ľubovoľný návrh územnosprávneho členenia charakterizovaného umiestnením sídelných miest vyšších územných celkov za rozumného predpokladu, že vyššie územné celky sa vytvárajú priradením obcí najbližšiemu sídelnému mestu. Samotné kritérium dostupnosti však nepokrýva úplne ekonomický aspekt tvorby vyšších územných celkov. Rešpektovanie ekonomického hľadiska neumožňuje výber ľubovoľného počtu sídelných miest, a tým i počtu úradov a rovnako nedovoľuje umiestniť sídlo v ľubovoľnej obci. Preto v nasledujúcej štúdii a analýze budeme považovať počet vyšších územných celkov za daný. Budeme študovať prípad územnosprávneho členenia s optimálnou dostupnosťou a porovnávať ho s rôznymi politicky či inak motivovanými modifikáciami.
accessibility. Transportation accessibility can be regarded as a measure of the quality of administration partition. It is defined as an average distance, which has to be traveled by an inhabitant from his place of residence to the nearest higher territorial unit center. This quality criterion is possible to evaluate for any partition design given by higher territorial unit center locations under consideration that higher territorial units are formed by assigning villages to the nearest higher territorial unit center. The criterion of accessibility itself does not include the economic aspect of higher territorial unit creation. To respect the economic point of view, any number of higher territorial unit centers and thereby any number of offices cannot be selected and the center cannot be located at any place, as well. That is why in the next study and analysis we will consider the number of higher territorial units to be given. We will study a partition instance with optimal accessibility and compare it with modifications motivated politically or in a different way.

Transportation accessibility of higher territorial unit centers
Transportation accessibility calculation is based on the existing road network in the Slovak Republic described in [1] and [2]. This network includes n dwelling places with known number of inhabitants b j at each place j ϭ 1, …, n. The distance between each pair of places i and j is given by cell d ij of matrix of distances. Let us suppose that p higher territorial units was created and that each higher territorial unit k ϭ 1, …, p is formed by set S k of dwelling places and by higher territorial unit center i k . Then we can formulate accessibility for the whole Slovak Republic using the abovementioned data: Term (1) expresses the average distance, which has to be traveled by an inhabitant of Slovak Republic to the center of his higher territorial unit.
The allocation of place j to higher territorial unit center i contributes to the value of term (1) by increment c ij , where When we express the accessibility given by term (1) using coefficients c ij , we get The less is the value of terms (1) or (3), the better is accessibility of services for inhabitants of Slovakia.

Model of the problem of transport-optimal higher territorial units
We suppose the set of cities, which can be taken into account as higher territorial unit centers, is finite and known in advance. In the Slovak Republic, the higher territorial unit center location is hardly supposed to be at a dwelling place, which is not at least a district center in the present public administration system. In our next considerations, we will come out from the fact that m candidates for higher territorial unit centers are given and the number p of centers, which should be selected from candidates, is stated. Bivalent variable y i ʦ {0, 1} models the decision if candidate i from the set {1, …, m} is or is not selected, where value 1 implies the positive decision and value 0 the negative one. To model the decision if place j belongs to the unit, which is defined by candidate i, we use bivalent variable x ij ʦ {0, 1} having value 1 in the case of assignment j to i and value 0 in the opposite case. Now a mathematical model of transport-optimal higher territorial units forming can be completed in the following way: x ij Յ y i for i ϭ 1, …, m, j ϭ 1, …, n (6) Α m iϭ1 y i Յ p (7) x ij , y i ʦ {0, 1} for i ϭ 1, …, m, j ϭ 1, …, n (8) This is the well-known p-median problem in a network with positive edge weights.
In the model, constraints (5) ensure that each dwelling place must be assigned to exactly one candidate for higher territorial unit center. Constraints (6) cause a higher territorial unit center to be placed at candidate i whenever at least one dwelling place is assigned to the candidate. Finally, constraint (7) inhibits selecting more candidates than the given number of higher territorial units.
This program can be further transformed to a simpler form.
Solving problem (4)-(8) or (9), (5), (6), (8) respectively to optimality, we get design (x*, y*) of transport-optimal partition of the Slovak Republic to higher territorial units including location of their centers. This design will be determined by resulting values

Design of transport-optimal higher territorial units in the Slovak Republic
To solve problem (9), (5), (6), (8), we have implemented and tested procedure BBdual [3,4] based on a branch-and-bound approach using a dual ascending method and a dual variables adjusting method to compute a lower bound. To find an appropriate parameter f, we used an approach from [3] implemented as a procedure. Distances were drawn from the road network of the Slovak Republic described in [1] and [2], properties of which were proved in [5].
Bratislava and its urban parts as a capital with special status were excluded from the set of dwelling places. After this operation, n ϭ 2889 dwelling places were taken into consideration. As a result of excluding Bratislava the number of planned higher territorial units decreases from 12 to p ϭ 11. The set of all 70 present district centers without Bratislava and urban parts of Košice was regarded as set {1, ..., m} of candidates for higher territorial unit centers in this study.

Analysis of some politically motivated designs
The transport-optimal design of higher territorial units from the previous section does not include any traditional county centers or cities preferred by some political groups. The problem model and solution method allow to perform an analysis in terms of accessibility what the cost of, for example, a higher territorial unit center location in Komárno would be (creation of Komárno higher territorial unit), or what the cost of keeping a county center in Banska Bystrica or Trenčín as a territorial unit center would be.
This analysis can be performed in such a way that one of the desired higher territorial unit center location is fixed and after then a transport-optimal location of the remaining ten centers including assignment of dwelling places to all eleven higher territorial units is computed. The difference between the computed accessibility and the optimal accessibility represents the lower bound of the accessibility increase caused by performing a politically motivated decision. If this decision were performed without a change in location of the other higher territorial unit centers, the accessibility would be even worse than the computed lower bound.
Vykonaná analýza sa týka výhradne otázok dopravnej dostupnosti možných sídiel dvanástich vyšších územných celkov a vyčísľuje len túto dopravnú dostupnosť pre politickým rozhodnutím určený počet územných celkov. and Lučenec. The results are summarized in table 1, where each row corresponds to one instance. The first column contains the analysis number, the second one the fixed city name. The column labeled d refers to the accessibility value for the particular instances. The column labeled dd indicates the difference between the accessibility computed for the analyzed instance and the optimal accessibility. The column labeled dd% reports the percentage increase of accessibility against the optimal accessibility from section 4. In the column t, there are reported algorithmic running times in seconds obtained on a Pentium 100MHz PC. Table 2 lists the higher territorial unit centers for the particular instances. The first column lists all cities occurred in, at least, one design. The subsequent columns refer to designs 1 to 4. In these columns, sign * indicates if the corresponding city belongs to the given design.
Assignment of dwelling places to the higher territorial units for designs 1 and 2 is shown in figures 2 and 3.

Conclusion
As we have mentioned in the introduction, to design a good administration partition of the state territory is a complex problem and it cannot be reduced to the only problem of transport accessibility. From point of view of costs required to establish the higher territorial unit residence, there are substantial differences among candidate cities in their current infrastructure, as well as in human resources needed for higher territorial unit center function.

Oznámenie
Príspevok bol podporovaný grantom VEGA 1/7211/20. It is possible that somebody can consider the differences among analyzed designs (see table 2) to be too small, specially when the greatest difference is 3.3 % against the optimal partition. This view of analysis can be greatly misleading because it is necessary to realize that increasing of accessibility just by 1 % will impact each of 4 925 751 inhabitants of eleven higher territorial units under consideration, and not for one or two years but for time of one generation or even longer. Therefore, we believe that the developed way for analysis of various designs and their comparison to the transport-optimal administration partition is a useful decision support tool allowing to evaluate at least those parameters, which can be figured out.