The shape structure of agricultural fields signi­ficantly affects the farm profitability. Proper setup and arrangement of farm parcels is the primary criterion enabling an efficient production strategy ( Gawroński & Jasnowska, 2007 ). Growing spatial irregularities of the field structure, negatively affected the process of their agricultural operations with an increased required time and forcing to undertake relevant works arrangement ( Schultz, 1964 ). The availability of technology now allows the use of advanced solutions to improve efficiency of field operations ( Kwinta & Gniadek, 2017 ). For example, most agricultural field operations involve a number of highly interconnected tasks executed by co-operating heterogeneous agri­cultural machines. Such a multiple machinery sys­tem is normally involved in both, output material flow operations (such as harvesting), as well as in input material flow ones ( e.g. , spraying and fertilizing). These operations require considerable efforts in terms of management and planning tasks ( Bochtis & Sørensen, 2009 ). Some research works, mainly related to the implementation of agricultural automated navigation systems, tackle the problem of the optimization of the working time through linear and non-linear algorithms ( Backman et al. , 2012 ; Kraus et al. , 2013 ; Kayacan et al. , 2014 ). The implementation of technological advanced solutions purposed by actual research studies ( i.e. , modern data processing algorithms contained in the GIS software) can help building in decision-making tools to improve the agricultural work arrangement.

The efficiency of field operations has traditionally been analyzed by manually measuring the time spent for each agricultural field operation such as the number of turnings, distance travelled in headlands, vehicle characteristics (for example the larger the turning radius is the larger the headlands and the larger the overhead would be) etc. The estimation of operating costs of agricultural and forestry machineries is a key factor in both planning agricultural policies and farm management, and, for this estimation, the time spent in field operations is strictly necessary ( Guerrieri et al. , 2016 ).

Generally, agricultural field is not uniform in sha­pe. As reported by Oksanen (2013) , for rectangular field, the analysis of agricultural operation effi­ciency and path planning for machines is pretty straightforward and can be copied from field to field. However, in regions where the agricultural lands are not rectangular, no two identical shapes exist, and the coverage path planning has to be specific for each agricultural unit. However, often complex shapes can be assimilated to a multiplicity of simple shapes.

In this study, only several items of the Commi­ssion Internationale de l'Organisasion Scientifique du Travail en Agriculture (CIOSTA) ( Manfredi, 1971 ; Biondi, 1999 ) method were taken into consideration for the estimation of agricultural operation efficiency starting from their shapes. These operations regard the effective work time (ET) and the tractor turning-around time (TAT) (which together represent the net time, NT). These variables (ET, TAT and NT) were determined by means of theoretical methods for agricultural fields with different surface areas (from 1 ha to 50 ha) and shapes: square, circle (quarter of a circle and semicircle), rectangle and triangle (isosceles and scalene).

The aim of this study was to find a model able to extract the NT from different agricultural fields of basic shapes and other variables to be easily collected by the farmer. Being this a non-linear problem, an artificial neural networks (ANNs) approach is proposed. This method allows a higher performance in predicting NT when compared with a multivariate linear approach and could serve to implement other similar multivariate approaches used to estimate, costs, consumes and emissions. The model was trained using an artificial dataset calculated for different shapes and then was tested on a real field dataset.

Table 3 shows the results regarding the perfor­man­ces of the GRNN model (training and test) to estimate the NT constructed from the artificial dataset. Both the r (correlation coefficient) of training (80% of the sample size) and internal test (20% of the sample size) appears to be excellent being equal to 0.98 ( R ² = 0.98). Also, the RMSE (training and test) resulted to be very low being equal to 1.008 and 1.020 respectively. The comparison with the r values obtained by the MLR model (0.14 training dataset and 0.13 internal test) being very low, confirm that the GRNN model was best performing, that the problem was non-linear, and that the x-block was not strictly related with the y-block ( i.e. , NT).

Fig. 3 shows the scatter plot of the observed vs predicted NT obtained from the GRNN model for both training (left side of the Fig.) and test. It is possible to observe that, for both training and internal test regression between observed and GRNN predicted NT values, the records are very proximal to the bisectrix ( i.e. , perfect attribution), confirming the high r values (Table 3).

Regarding the variable impact on the GRNN model (Fig. 4) it must be underlined as the "number of turnings" return the higher impact (44.34%) for the NT estimation. This variable was followed by "side length orthogonal to working direction" and the "side length parallel to working direction" (18.49% and 11.04% respectively) which also have a high impact. Field gross area also returned lower but with an important contribution (3.70%).

Fig. 5 shows the scatter plot of the observed vs predicted NT obtained from the GRNN model applied on the external test. As also reported in Fig. 5, the regression coefficient is quite high ( R ² = 0.63). It is possible to observe as the points that move away from the bisectrix belong to a shredding and two kinds of plowing (simple and two furrow plow - one-way working). This is probably due to the work being carried out on a small area with a particular shape (quarter circle).

This work gives an important contribute in predic­ting the effective working time needed considering different basic field shapes and the main agricultural operations. The working time calculation is directly related with the farm productivity. The ability to predict in advance the operational agricultural efficiency ( i.e. , fuel consumption, emissions, pollution, labor, costs, consumes, production, etc.) is an important task which contribute to optimize the resources use. This is a major aim in advanced precision agriculture ( Lundström & Lindblom, 2016 ). The estimation of operating costs of agricultural machinery and the definition of economic competitiveness gap (conditionality standards on agri­cultural farms and short- and medium-term business planning of the agricultural farm activities) are a key factor in planning policies of Rural Development Programs (RDPs) both at national and European level ( Guerrieri et al. , 2016 ).

Furthermore, the work reports the advantage of using a non-linear approach when compared with a liner one. As confirmed by the comparison of the ANN approach with the MLR one, the relationship between independent variables and the dependent one ( i.e. , NT) is non-linear. This is an unexpected result considering the basic geometrical problem. A liner multivariate approach (MLR) applied to the artificial dataset returned poor performances in predicting the NT ( r values lower than 0.15). Non-linear (ANN) approach, to the other hand, returned excellent performances ( r values equal to 0.98 and 0.80 for the internal and external tests respectively) interesting also from an applicative point of view. These applications could involve the precision agriculture framework. The introduction of automatic guidance of agricultural production machines for the improvement of the accuracy of field operations ( Kayacan et al. , 2014 ), requires the implementation of specific algorithms for the optimization of path planning and, more in general, of the working time. Some authors hence identified non-linear approaches to better fulfil these aims ( Backman et al. , 2012 ; Kraus et al. , 2013 ; Kayacan et al. , 2014 ).

Generally, shape analysis of agricultural fields is of interest in many agricultural and farm management areas and most of the few studies in the literature concern the analysis of the land fragmentation for rural control and administration purposes. Janus & Taszakowski (2015) reported as functioning conditions of agriculture are closely related to the spatial structure of rural areas, one of the most important factors influencing profitability of agricultural production. For example, in the study of Demetriou et al. (2013) , a new parcel shape index which integrates GIS with a decision-making method was presented. The study of Gąsiorowski & Bielecka (2014) provided a contribution to an area-wide quantitative statistical description and classification of morphometric parameters such as shape, area and slope of existing agricultural parcels. In this light, the presented study contributes furnishing a model for extracting the NT, that is directly proportional to the agricultural efficiency. In details, the analysis starts from basic field shapes (square, circle, rectangle and triangle) and from other easily collected parameters ( i.e. , field gross area, working speed, number of turnings, side length parallel and orthogonal to the working direction and working direction type). These variables are easily to collect by operators and, for this reason, the approach may result extremely useful for both farmers (in terms of economic advantages) and policy makers at institutional level.

The shapes taken into consideration, are very simple (Fig. 1), but they could be associated with each other to analyze complex ones. A complex shape could be figured out as a combination of many simple shapes and the NT calculation could result as the addition of many NTs calculated on simple shapes. As reported in Fig. 4, the proposed applicative model is mainly influenced (variables impact in the GRNN) by parameters such as "number of turnings" (44.34%), "side length orthogonal to working direction" (18.49%) and the "side length parallel to working direction" (11.04%). This means that the number of turnings is more important than other variables because it depends on the working width. For this reason, more complex is the shape, more working steps and turnings (TAT) are required. Also, the field shape variable could affect the accuracy of estimation of NT in the model even if in a reduced but important percentage (7.56%; Fig. 4) with respect to the one-way working type and to the field gross area.

As conclusions, in this scenario, the presented study finds a model for extracting the NT (directly proportional to the previously mentioned agricultural efficiency) starting from basic field shapes (square, circle, rectangle and triangle) and from other easily collected parameters ( i.e. , field gross area, working speed, number of turnings, side length parallel and orthogonal to the working direction and working direction type). The results of the study could be used to implement models predicting fuel consumption and costs. Indeed, NT is estimated with a high precision through a model built with easily collectable data. This kind of modelling approach could be implemented on a web platform and made available to all the stakeholders and policy makers interested. Such instruments may lead to economic and environmental benefits.