2.3. AHP Methodology
AHP methodology aids decision-making and is widely used in the business world. It allows for establishing a priority list for a set of possible alternatives through pairwise comparisons between elements with a scale designed specifically for the method. The numerical scale was established by Saaty for the use of the method [
29] (
Table 2).
By comparing the alternatives pairwise for each criterion and using the scale from the previous table, square decision matrices are obtained that must satisfy the properties of reciprocity (if aij = x, then aji = 1/x), homogeneity (if i and j are equally important, aij = aji = 1, and also aii = 1 for all i), and consistency (the matrix must not contain contradictions in the valuation). The dimensions of the decision matrix depend on the number of criteria and alternatives selected, always being square as both criteria and alternatives are compared pairwise. The eigenvector vai of the proposed matrix indicates the importance or weighting of each alternative for each criterion.
The methodology allows evaluation of the inconsistency of expert decisions through the consistency ratio (
CR). The ability to measure the consistency of results is one of the strengths of the method, as it ensures that the information entered into the model is correct. The calculation is carried out as follows:
where
Generally, the following values can be taken depending on the range of the matrix (
Table 3).
In case these values are not met, the obtained judgments must be reviewed, or the matrix must be discarded.
Regarding the criteria to be used, a high number of them can, in some cases, be quite complex. Miller sets this limit at seven, which is called the ‘magic number’, as higher sets are difficult to handle [
31]. In this case, four criteria have been selected, an appropriate number that encompasses the entirety of the issues to be addressed.
This method is empirically validated in various applications, practically in almost all knowledge sectors such as economics and transportation [
32,
33,
34], resource localization [
35,
36,
37], healthcare [
38,
39], marketing [
40], agriculture [
41,
42], project management [
43], and hydrology [
44].
In this case, the method will be used to determine the best layout for each segment of the irrigation network. Various studies exist where the method is utilized to find solutions for the least costly routes for transportation and movement [
45] and where combined methods with GIS are applied for pipeline layout [
46].
Alternatives
At this point, the possible layout alternatives that can be found in the design of pressurized irrigation networks are established. Identifying these alternatives for each specific case study is important because the uniqueness of each area will lead to different results for each alternative. In this case, those considered primary and prioritized are established. Not all of them are present in the case studies, so for their evaluation, only the ones detected in the cases are used. Shown in
Table 4.
Criteria
Once those geographical elements that can be used as pathways for the layout of pressurized irrigation pipelines are defined, it is time to determine the importance or weight of each of them. Given their characteristics, not all alternatives can be used under the same conditions, because each of them has its particularities, making the layout more difficult or easier depending on the criteria considered.
The relevance of each layout option will be defined by the pairwise comparison of all participating criteria in the study, as well as the comparison of the different alternatives among themselves. For this case study, the following criteria have been defined.
Criterion A: Type of roadway
This criterion aims to determine the negative impact that the temporary occupation or cutting of the roadway will have on the installation of pipelines. The complexity of carrying out work on different types of roads, such as highways, main roads, paths, etc., varies significantly depending on various factors. These factors are related to the difficulty of access to the roads and the possible impacts that may arise during the execution of the works. Roads with high traffic intensity, such as highways or main roads, can generate more problems of impact and access than roads with lower traffic, such as a rural path.
Criterion B: Type of pavement
The type of pavement and composition of the road or pathway to be laid out affects the costs of demolition and replacement of the road, as well as the total execution time of the work [
18]. In terms of optimizing the layout, it is advisable to avoid those that involve higher costs, both for trenching and replacement, and longer execution times.
Simplifying all types of pavements [
47] found in the case studies, they can be included in the following three main groups shown in
Table 5.
In this case, experts must assess both the demolition task of the existing pavement, if any, and the replacement based on the type of roadway along which the pipeline is traced.
Criterion C: Permission obtaining
Obtaining permits and licenses is a relevant issue when carrying out any type of construction. There are studies stating that it is one of the main causes of construction delays [
48,
49]. This criterion aims to consider the problems and delays in execution that obtaining a construction permit can trigger, depending on the relevant authority. General aspects of the permit application are considered. Experts must consider the following aspects:
Number of permissions to be requested;
Functioning of the issuing authority;
Ease of obtaining the permission;
Number of documents per permission;
Resolution time.
Criterion D: Accessibility to the roadway
Although there are many definitions for the concept of spatial accessibility, here we refer to it as the ‘ease with which a service can be reached from a location’ [
50]. We will measure this ease as the speed at which the network manager or technician can access the tracing element in question using mobile vehicles. This implies that for each layout alternative, objective criteria must be established for later pairwise comparison. In case the element cannot be traversed, the speed of the roadway will be determined as 0 km/h. To allow experts to provide their assessments, the following maximum speed values are established in
Table 6 for each of the roadways detected in the case studies.
Expert panel
Finally, once the criteria and alternatives were obtained, categorized, and analyzed, the selection of experts took place. This phase is a critical step in the application of the methodology. The selection of the right experts ensures the quality and reliability of the results [
52]. The criteria considered for their selection were as follows:
Solid knowledge directly related to the modernization of irrigation;
Practical experience in the design phase of such projects;
Impartiality and without personal interest in decision-making;
Familiarity with the methodology to be applied.
Five interviews were conducted with technical specialists with extensive and recognized experience in the design, execution, and maintenance of irrigation and irrigation modernization projects. The profiles of the five experts were men and women between 30 and 60 years old with degrees in Agricultural Technical Engineering, Agricultural Engineering, and Ph.D., professionals with project management capabilities.
Hierarchical Structure
The hierarchical structure applied to the problem in question is shown in
Figure 2. The highest level locates the decision problem, i.e., the objective. Elements affecting the decision are represented at lower levels so that criteria occupy the intermediate level. Finally, at the lowest level, decision options or alternatives are represented. Consensus among all parties involved is required for construction.
Example of Matrices.
The methodology established for value determination by the experts is as follows. Each expert was provided with a criteria and alternatives matrix for assessment using the Saaty proposed scale. An example matrix (
Table 7) is attached.
After each expert’s assessment, the following sequence of checks must be carried out:
Therefore, only the assessments of experts 1, 2, 3, and 5 will be considered since expert 4 obtained values of unacceptable inconsistency. After these checks, it is possible to obtain eigenvectors and resistance values that will be part of the irrigation pipes tracing process.
2.4. Use of GIS. Methodology Applied to Graphs
In this study, a combination of finding paths with minimal resistance and the use of Geographic Information Systems will be employed. This methodology is based on graph theory concepts, allowing the search for an optimal path to be solved as a graph search problem [
54]. The algorithm obtains a path of minimum resistance from one vertex to each of the vertices composing the graph.
The strategy of the Dijkstra algorithm [
55] consists of growing a tree, starting from a previously established origin node, by adding, in each iteration, a border edge whose vertex not belonging to the tree is as close as possible to the initial one.
The Dijkstra algorithm uses a strategy called “minimum cost search” to gradually determine the shortest path from the origin node to all other nodes. It starts by assigning an initial infinite cost to all nodes except the origin node, to which an initial cost of zero is assigned. Then, the node with the lowest cost is chosen, and all its adjacent edges are examined. If the total cost of reaching a neighbor node through the current node is lower than the currently assigned cost to that node, the cost is updated. This process is repeated until all nodes in the graph have been examined [
56]. The following
Figure 3 shows a series of algorithm iterations to find the path with the least resistance.
The resistance values obtained act as tracing resistances in the arcs that form each of the nodes. In this case, the nodes correspond to points already defined in the network, such as the collection point, multi-user hydrants (elements that provide service to a group of plots with flow and pressure measurement and/or regulation), and the final water outlet in each plot. Given the magnitude of data represented by the irrigation surfaces of the study cases or any network of this kind to be analyzed, it is essential to have computer tools that automate these processes. The goal is to achieve a tracing base that allows movement in any direction without isolated areas that lead to longer traces.
At a second methodological level, and for the graphical representation of resistances and to create the map, it was decided to use raster objects instead of vector objects. A raster is nothing more than an image where each of its pixels takes a different value based on its characteristics. In this case, the value of each pixel is derived from the resistance to the trace defined for that type of route. In this case, their use improves the traces and their logic, as it creates a network of points without gaps where the program can advance in any direction. In contrast, with vector objects, vertices are required, which are not always connected, leading to fewer logical traces. In the following image (
Figure 4), it can be noticed how tracing with raster objects improves trace length and has a greater logic than with vector objects.
As observed, vector tracing is unable to cross the path since the vertices are disconnected, thus resulting in greater length. On the other hand, the raster object, without gaps, can follow the path of least resistance and offers better and more coherent results, so it was decided to use raster objects with the QGIS 3.X software.
Another value to be decided is the pixel size to use. In this work, three pixel sizes were tested, namely 2.0 m, 1.0 m, and 0.5 m. The pixel size influences both the execution time of the traces and their zig-zag effect. It was found that with a smaller pixel size (0.5 m side length), the trace generates more vertices, creating a zig-zag effect that is not representative of reality (
Figure 5). On the other hand, larger pixel sizes, such as 2.0 m side length, result in almost no vertices, sometimes generating traces outside the paths or boundaries as their values need to be extrapolated.
In the above figure, it is observed that tracing with pixel sizes of 1.0 m and 0.5 m is similar, following the course of the path and providing a representation of reality. The blue trace generates fewer vertices, thus reducing execution times for very similar results. A pixel size of 2.0 m is too large for this experiment, as it generates unrealistic traces with long straight sections not representing the actual trace of the pipes.
In the following graph (
Figure 6), the average number of vertices generated per 100 m of tracing and its evolution based on the pixel size can be seen.
On the other hand, the pixel size also influences the software execution time and the automatic tracing of pipes. With a smaller pixel size, execution times increase because the number of tracing possibilities between each pixel increases exponentially, as summarized in the following
Figure 7.
Therefore, given the problem of representing reality that comes with a high pixel size, such as 2.0 m × 2.0 m, and the long execution times associated with a small pixel size, it is considered appropriate to use a medium pixel size, such as 1.0 m × 1.0 m (
Figure 8).
Larger values create too many gaps, with the subsequent zig-zag pattern in the layout, while smaller values result in very heavy files that slow down the process without leading to better solutions.
Now, with the rasterized object, each pixel centroid acts as a node, which implies a large number of arcs through which the layout of pipes can pass. These nodes provide greater precision in the layout and prevent long layouts from nodes without adjacent elements.
With defined resistance values implemented on a raster object, it is now time to obtain the layout of each candidate intake between the defined water outlet and the candidate hydrants. This process is automated using the Model Builder tool of QGIS 3.X.
This tool allows, through a graphical interface, the assignment of different chained calculations where input data are introduced, and a final result is obtained with measurements and execution costs.