**1. Introduction**

The excessive fragmentation of plots owned by a farm is one of the major factors adversely affecting the profitability of agricultural production [1]. The spatial arrangemen<sup>t</sup> of land owned by individual farmers in the rural areas of southern and southeastern Poland, developed by historical processes, is characterized by farms covering a small area of land and made up of fragmented and scattered plots. Fragmentation of plots, or in other words land fragmentation, is discussed both in the domestic literature [2–10] and international references [11–15]. Normally, excessive fragmentation of land has its roots in history and social and economic reality [10]. The present-day plot boundaries are a result of long-term transformations. In the reference literature, four types of land fragmentation are distinguished depending on the type of use, form of ownership, and geometric structure of plots (area, number of plots per farm, plot shape, plot elongation, lack of access to the plot, and distance from the farmer's dwelling) [12]. All the above-mentioned defects have a

**Citation:** Str˛ek, Z.; Le ´ ˙ n, P.; Wójcik-Le ´n, J.; Postek, P.; Mika, M.; Dawid, L. A Proposed Land Exchange Algorithm for Eliminating the External Plot Patchwork. *Land* **2021**, *10*, 64. https://doi.org/ 10.3390/land10010064

Received: 24 November 2020 Accepted: 7 January 2021 Published: 13 January 2021

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very negative effect on agricultural production and income derived from such production. This is mainly due to the cost of transport, workload, and losses of crops connected with a small area and unfavourable plot shape [16–19]. However, it cannot be claimed that the fragmentation of land is an adverse phenomenon in all countries. The authors give some examples of where exogenous fragmentation is seen as an advantage. This is the case, for instance, in Ethiopia [20], China [21], and India [22].

In turn, long-term studies in Poland show that the plot patchwork is closely linked to the fragmentation of land (plot patchwork). It is one of the factors negatively affecting the organization, costs, and level of agricultural production [23]. With regard to the administrative division, an internal patchwork (within the limits of the village) and an external one can be distinguished. The external patchwork can occur both between respective villages and between communes, counties, voivodeships, and even between countries [24]. The analysis of the origin of plot patchworks in Poland and in other countries shows that this phenomenon is a result of a long-term historical process. Their emergence and development were a result of various causes of a legal, economic, and socio-economic nature [17,23–25]. The external patchwork is a negative phenomenon manifested in decreased labour efficiency due to time lost for accessing the scattered plots. This increases the cost of transport and, as a consequence, all agricultural production. A term associated with plot patchwork is "non-resident owners", coined by Rabczuk (1968) and later specified by Noga [23]. The division was introduced into local non-resident owners, who are owners whose land is not situated in the analyzed village in which they reside, and out-of-village non-resident owners whose land is situated in the analyzed village but who live elsewhere.

A land surveying tool used for improving the arrangemen<sup>t</sup> of land is the operation of consolidation and exchange of land which occurs both in Poland and many countries of Europe and the world: Netherlands [26], Cyprus [12], Slovakia [16], Czech Republic [27], China [28], Finland [29], and Northern Ghana [30], Ethiopia [31], Turkey [32]. Land consolidation is a rural managemen<sup>t</sup> procedure aimed at creating more favourable managemen<sup>t</sup> conditions in agriculture and forestry by improving the territorial structure of farms, forests and forestland, reasonable configuration of land, aligning the limits of real properties with the system of water irrigation structures, roads and terrain.

The problem in the procedure of land consolidation and exchange is the fact that consolidation mainly occurs within the administrative boundaries of villages. Then, the land of out-of-village non-resident owners is usually situated at the outer boundary of the village, which does not completely improve the existing plot patchwork of farms. It would be advisable to carry out land consolidation and exchange in a manner ensuring the possibility of land exchange between local non-resident owners. It should be added that in the past (in the years after World War II) exchange of land was carried out almost entirely in order to increase the surface area of land owned by the state or a cooperative. Few works were carried out to eliminate the external plot patchwork.

Such an understanding of the issue gives rise to the objective of this paper being the development and presentation of a universal land exchange algorithm for eliminating the external plot patchwork. The work contains a detailed analysis of the external plot patchwork in the study area. It pays attention to the spatial dimensions of the plot patchwork in developing the methodology of its elimination. The elimination of land, especially that owned by local non-resident owners, in the process of land exchange makes it possible to bring the land situated outside the village closer to the dwelling of the owner of such land. Previous solutions regarding land consolidation works involved only the study of out-of-village non-resident owners that, in principle, provide information about the existing defects but does not eliminate this phenomenon [23].
