Field path planning using capacitated arc routing problem

https://doi.org/10.1016/j.compag.2020.105401Get rights and content

Highlights

  • Agricultural operations are modeled as a capacitated arc routing problem.

  • A solution algorithm based on adaptive large neighborhood search (ALNS) is employed.

  • The proposed algorithm achieves the optimal harvesting pattern efficiently.

  • The ALNS algorithm surpasses the conventional harvesting pattern.

Abstract

Duration of Agricultural operations is a crucial issue in fleet management as it directly affects the operation costs. Agricultural fleet usually traverses paths, covering the whole field to complete the operation. The agricultural operations are not necessarily performed efficiently due to the various variables including field shape, presence of obstacles in the field, number of vehicles, and their specifications. In this study, agricultural operations are modeled as a capacitated arc routing problem (CARP). CARP is a combinatorial optimization problem, which determines the harvest paths in a field and finds the best order of traversing. By dividing the paths into two categories of required and non-required, CARP demonstrates a field as a graph. Using CARP, no extra or invalid edge on the graph would be needed. As CARP is an NP-Hard problem, a solution algorithm based on adaptive large neighborhood search (ALNS) is employed to solve large instances of the problem under study. The proposed ALNS algorithm is applied to well-known problem instances as well as a realistic case study to achieve the optimal harvesting pattern efficiently. The computational results demonstrate that the ALNS algorithm surpasses the conventional harvesting pattern. The amount of improvement obtained by the proposed method could vary due to the several factors, including the depot location, vehicle capacity, and field shape.

Introduction

Most of the agricultural operations, such as harvesting, spraying, and seeding, are performed using agricultural machinery. For large scale operations, it is often necessary to hire a fleet of machines to complete the operations on time. The required time to fulfill the agricultural operations depends on how machines are managed. The completion time is important for the farmers as it directly affects the operation costs and therefore, the final price of the products. Thus, fleet management has a significant role in preventing extra expenses during operations.

Machine working time consists of operation time and non-productive time. Operation time is defined as the time required by a machine to perform its special duty on the field like harvesting, mowing, trilling, and spraying. The non-productive time is the time that a machine spends moving across the headland, maneuvering, and loading or unloading the hopper. To improve the working time efficiency of the machines, it is recommended to minimize the non-productive time (Srivastava et al., 1993). To achieve this goal, different operations research methods have been proposed by researchers such as coverage path planning problems (Oksanen and Visala, 2009), routing problems (Bochtis and Sørensen, 2009), as well as simulation (Zhou et al., 2015).

Farmers may not use operations research methods due to several reasons. First, in many countries, traditional farmers manage most fields using old-fashioned methods based on the farmers’ judgment and experience (Jian et al., 2016). Second, agricultural operations such as harvesting and spraying have simple, intuitive solutions that could perform well enough on small-sized fields that have a small economy of scale. Such solutions do not necessarily perform well on large-sized fields. Nevertheless, the farmers may not use operations research methods to find the best possible solutions because they underestimate the improvements that could be achieved. They might believe that the cost saving obtained via optimization could not be sufficient to cover analysis and consultancy costs. However, operations research could take into account the intricacies of the problem and yield better results (i.e., less vehicle costs as well as fuel consumption). In addition, as autonomous vehicles become more common in agriculture, operations research techniques would be more applicable in planning the farm operations. Finally, agricultural economists could use operations research as a useful tool for analyzing various agricultural decisions and policies (Arfini et al., 2016, p. 14).

In this study, the arc routing problem (ARP) approach is used to optimize the paths that agricultural machines traverse in the field. Routing problems belong to a class of combinatorial optimization problems that have various applications in logistics. Routing problems seek to satisfy all the customers’ demands at the minimum cost. In routing problems, there are some customers with positive demands that are served by a fleet of vehicles. Responding to customers’ demands requires traversing multiple paths, which in turn imposes traveling costs to the system. Each vehicle is stationed at a predetermined depot (or depots) and ready to meet the customers’ demands through a road network, which could be depicted as a graph. Each vehicle makes a tour by beginning from the depot, serving a subset of customers while observing its capacity, and then returning to the depot. The main objective of routing problems is to determine the best tours with the minimum traveling cost. The routing problems include two major classes of problems, which are different in how to deal with the demands on the graph.

In the first class, entitled vehicle routing problem (VRP), customers’ demands are considered on vertices of a graph. In VRP, customers and depots are located on the vertices of the graph. The edges show the path between the vertices and their associated costs. This problem has real-world applications such as goods and drink distribution, waste collection, as well as school bus routing. Some other extensions of the problem are vehicle routing problems with time windows, split-delivery VRP, and VRP with backhauls.

The second class of routing problems is called the arc routing problem (ARP). In this problem, the customers and their demands are considered on the edges of the graph. Each edge has a non-negative amount of demand which must be served by vehicles. The edges that have positive demands are called required edges or tasks. The remaining edges are called non-required or no-service edges (Corberán and Laporte, 2013). In addition, the cost of each edge is predetermined. The goal of arc routing problem is to determine a set of tours for the vehicles so that traversing all the required edges be done at the minimum cost. Meter reading, snow plowing, salt spreading, and road maintenance are some of the applications of this problem (Corberán and Laporte, 2013). Other extensions of CARP have been considered in the literature including but not limited to multi-depot CARP, multi-compartment CARP, and open CARP.

Agricultural operations, such as harvesting, planting, and spraying, are intrinsically logistics problems. In these operations, it is necessary to transport a bulk of material from the warehouse to the farm or vice versa. Thus, routing problems can be applicable in modeling these operations. These models could increase the efficiency of the agricultural fleets. While both classes of routing problems could be applied to modeling agricultural operations, these operations are inherently similar to arc routing problems. The advantage of ARP is that there is no need for major changes in data preparation and model building.

Contrary to the current literature on the application of VRP models to agricultural operations, this research focuses on modeling the agricultural operations carried out by machines using an ARP approach to reduce the total cost/time of such operations. Due to transferring agricultural products in such operations, it is necessary to take into account the capacity of the machines. Therefore, the capacitated arc routing problem (CARP) is an appropriate approach to model agricultural operations. CARP can be used in many agricultural operations like harvesting, seeding, and spraying. CARP was introduced by Golden and Wong (1981) and proved to be an NP-hard problem (Corberán and Laporte, 2013). Hence, obtaining optimum results in polynomial time is impossible. Metaheuristic algorithms could be employed to obtain high-quality solutions in a reasonable amount of time. Inspired by the notable performance of the adaptive large neighborhood search in routing problems, this method is employed to solve the agricultural path planning problem.

The remainder of the article is organized as follows. Section 2 presents the literature review and previous methods. The problem definition is given in Section 3. In Section 4, the CARP mathematical model is presented. In Section 5, the adaptive large neighborhood search is described. The computational results on CARP instance problems and a realistic example filed are reported in Section 6. Finally, sensitivity analysis and conclusion are presented in Sections 7 and 8 respectively.

Section snippets

Literature review

In order to optimize the infield and inter-field agricultural operations, researches have used different approaches. These approaches could be categorized into three categories, namely coverage path planning, linear programming, and routing problems. It is worth mentioning that combinations of these approaches have also been used in some studies.

The path planning problem is a practical approach for covering the whole field surface. It has many applications in robotics as well. This method was

Problem definition

The agricultural operations, such as harvesting, spraying, and seeding, are accomplished using one or several tractors, harvesters, or robots. Completing an operation usually involves transferring material (i.e., products, chemicals, or seeds) from warehouses or silos to the field and vice versa. In this study, several assumptions are considered to perform an operation as follows:

  • The crop rows are predetermined considering the field characteristics (i.e., field slope, geographical position,

Mathematical formulation

The general form of CARP is described on an undirected graph. Consider graph G=V,E, where V is the set of vertices and E is the set of edges. Er is a subset of E that contains the required edges with positive demand of qij. Corresponding to each edge a cost cij is defined to show traversal cost from node i to node j. In asymmetric applications of CARP (i.e., cij ≠ cji), a directed graph is used, thus two arcs with opposite directions are defined instead of each edge. One of the vertices,

Solution method

As previously mentioned, CARP is an NP-hard problem. Hence, obtaining its optimal solution for large instances using exact algorithms (such as branch & bound) could take a long time and even sometimes is impossible. In this study, the adaptive large neighborhood search (ALNS) algorithm is applied to the problem, to obtain high quality solutions. Proposed by Ropke and Pisinger (2006), ALNS is an improvement algorithm that starts with an initial solution and improves it in each iteration to find

Computational results

The ALNS algorithm described above is coded in Matlab 2017a and run on Intel Corei5 CPU and 4 GB of RAMs to solve the in-filed or inter-field agricultural operations. Before solving the agriculture problem, the solution method is validated on a set of arc routing problem instances, including the well-known KSHS (Kiuchi et al, 1995) and GDB (Golden et al., 1983). Adaptive large neighborhood search can solve all the six instances of KSHS optimally (see Table 1). The GDB contains 23 instances. The

Sensitivity analysis

The previous section presented the applicability of the CARP method for optimizing a realistic operation. Further, assuming the depot as a temporary facility, the notable role of the depot location in accomplishing an operation is confirmed. In the following, the effects of other factors such as vehicle capacity and field shape are investigated.

Conclusion

Due to the various numbers of variables in agricultural operations, using conventional harvesting patterns is not necessarily the most efficient way. Completing these operations involves both monetary costs and time. If the operations are conducted more efficiently, fewer costs will be imposed. Therefore, optimizing agricultural operations matters.

In this study, following the logistical nature of the agricultural operations, the capacitated arc routing problem is introduced to plan the infield

CRediT authorship contribution statement

Amin Khajepour: Conceptualization, Writing - original draft. Majid Sheikhmohammady: Supervision, Writing - review & editing. Ehsan Nikbakhsh: Methodology, Formal analysis.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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