This paper presents an annual multiobjective crop-mix planning as a problem of concurrent maximization of net profit and maximization of crop production to determine an optimal cropping pattern. The optimal crop production in a particular planting season is a crucial decision making task from the perspectives of economic management and sustainable agriculture. A multiobjective optimal crop-mix problem is formulated and solved using the generalized differential evolution 3 (GDE3) metaheuristic to generate a globally optimal solution. The performance of the GDE3 metaheuristic is investigated by comparing its results with the results obtained using epsilon constrained and nondominated sorting genetic algorithms—being two representatives of state-of-the-art in evolutionary optimization. The performance metrics of additive epsilon, generational distance, inverted generational distance, and spacing are considered to establish the comparability. In addition, a graphical comparison with respect to the true Pareto front for the multiobjective optimal crop-mix planning problem is presented. Empirical results generally show GDE3 to be a viable alternative tool for solving a multiobjective optimal crop-mix planning problem.
The overarching objective of this work is to investigate the performance of the generalized differential evolution metaheuristic for multiobjective optimal crop-mix planning decision in the agricultural domain. The purpose of agricultural crop planning decision is generally to guarantee sufficient food resources for the human population, which is increasingly growing at a fast rate. In addition, the global demand for food items is growing at an accelerated rate. However, most of the available techniques for expanding agricultural systems have a serious long-term implication for the human environment [
The intensification of agricultural practices such as clearing land for massive crop production, achieving higher yields through increased agricultural inputs, and promoting innovations through the application of information communication technology could improve crop production and agricultural value chain [
The current work explores an approach based on evolutionary metaheuristics to solve a multiobjective optimal crop-mix optimization problem. This could suggest an effective tool to support farmers in optimal crop planning decision making. There are numerous reasons for using evolutionary metaheuristics to solve optimization problems. One of such reasons is that evolutionary metaheuristics need little problem specific knowledge and can be applied to a broad range of problem types [
Previous studies on crop planning have used single and multiobjective optimization models, including linear programming [
This section presents the mathematical formulation of the optimal crop-mix planning problem investigated in this work. The optimal crop-mix planning model is designed to maximize total net profit that can be produced by maximizing total crop production. The objective is to make an optimum use of the available limited resources in order to determine the land allocation for several competing crops required to be planted in a year. The soil characteristics, cropping patterns, crop produced, planting region, and cropping method are factors that contribute to the production cost, yield rate, and earning realized by a decision farmer. The crop-mix planning model is considered for a large scale planning incorporated with dataset collected from the South African abstract of agricultural statistics [
Maximize
The two objectives functions are to be concurrently solved, subject to the following constraints: food delivery, land allocation, labor cost, capital cost, and nonnegativity of decision variables.
The multiobjective evolutionary metaheuristics are population based techniques for solving complex multiobjective optimization problems. A metaheuristic is an iterative master process that guides and modifies the operation of a subordinate heuristic to efficiently produce high quality solutions by exploring and exploiting a solution search space [
In this study, we investigated a set of metaheuristics to test the performance of generalized differential evolution for multiobjective optimal crop-mix planning problem. These metaheuristics are
The generalized differential evolution 3 (GDE3) [
In the case of comparing feasible, incomparable, and nondominating solutions, both offspring and parent vectors are saved for the population of the next generation. This mechanism reduces the computational costs of the metaheuristic and improves its efficiency. The population size may increase at the end of a generation based on a similar selection method as used in NSGA-II; the population is reduced back to the original size. The sorting of the population members is based on the goal for a
The
The nondominated sorting genetic algorithm (NSGA) [
The multiobjective optimal crop-mix planning problem was solved using GDE3, NSGA-II, and
The GDE3 was compared to NSGA-II and
In order to compare the performance of GDE3 with performances of NSGA-II and
Table
Additive epsilon indicator metric (
GDE3 | NSGA-II |
|
|
---|---|---|---|
Best | 7.59 | 8.29 | 9.50 |
Average | 8.24 | 9.16 | 10.90 |
Worst | 9.65 | 10.5 | 13.10 |
Std. dev. | 0.431 | 0.726 | 9.83 |
Table
Generational distance metric (
GDE3 | NSGA-II |
|
|
---|---|---|---|
Best | 2.15 | 2.73 | 2.94 |
Average | 3.45 | 4.06 | 4.12 |
Worst | 6.87 | 7.10 | 7.92 |
Std. dev. | 1.48 | 1.74 | 1.91 |
Table
Inverted generational distance metric (
GDE3 | NSGA-II |
|
|
---|---|---|---|
Best | 2.40 | 2.62 | 2.73 |
Average | 2.81 | 3.52 | 3.88 |
Worst | 3.15 | 4.39 | 5.01 |
Std. dev. | 0.171 | 0.398 | 0.512 |
Table
Spacing metric (
GDE3 | NSGA-II |
|
|
---|---|---|---|
Best | 1.07 | 1.23 | 1.62 |
Average | 1.47 | 1.53 | 1.92 |
Worst | 1.69 | 2.03 | 2.84 |
Std. dev. | 0.415 | 0.653 | 0.976 |
In general, because performance metrics can sometimes be misleading in multiobjective optimization, it is always desirable to consider a graphical comparison whenever possible [
The first evaluation criterion requires us to determine the extent to which the points on the Pareto front are linearly correlated. Consequently, the metaheuristic that gives a Pareto front with points more linearly correlated is judged to be the best performing in solving the multiobjective optimal crop-mix planning problem. This approach is effective because it could indicate a natural association between crop production and net profit. The strength of the linearly of an association between two variables such as crop production and net profit can be determined by calculating the Pearson correlation coefficient. The correlation coefficient is a number between −1 and 1 that indicates the strength of the linear association between two variables, for instance, crop production and net profit. The higher positive value indicates a strong linear association in the same direction; that is, increase/decrease in one variable leads to increase/decrease in the other variable. If there is evidence of strong linearity, we would likely expect higher values of crop production to yield higher values of net profit. The second evaluation criterion suggests that solutions on the Pareto front should be uniformly distributed.
Figure
Pareto optimal set for GDE3, NSGA-II, and
The crop planning system based on the generalized differential evolution 3 (GDE3) was implemented using the C-Sharp programming language in VISUAL-STUDIO. The purpose of the implementation was to provide a software tool to assist farmers in optimal crop planning decision making. The testing of the crop planning software was done with 10000 fitness function evaluations. The combination of parameters chosen for the testing is appropriate to have a reasonably good performance. This can be corroborated by checking the original sources of the GDE3. The varied operations such as capturing crop information and managing the information on crop combination as provided by the crop planning system could be used for crop planning related activities such as land allocation and crop selection. The recorded data are stored in a database for easy accessibility and could be used in various planning and decision making processes. The prototype system is relatively easy to use and simple to accommodate basic users with very little literacy levels.
The system was tested with a scenario where a household farmer has a working capital of R10,000 (unit in South Africa rand) with the land mass of 1 hectare. The farmer chooses to plant crops that could be planted along with cotton and maize such that the crop combination should be of order 3; that is the farmer decided to plant on a tricropped land. The farmer supplied all the necessary inputs and clicked on the button (view combination group) to view the crop combination group, consisting of crops that could be planted with the selected crops (cotton, maize). In order to view the number of possible crop combinations that could be obtained, the farmer selected any of the crop combination group of his/her choice and clicked the button (possible crop combination). Figure
The screenshot of the process page of the decision support system.
The system allocated a land portion to each crop combination; working with the scenario where the farmer decided to choose both combination groups, the system produced the result in Table
The land allocation result.
Optimization | ||
---|---|---|
Serial number of crop combination | Crop combination | Allocated land portion |
1 | Cotton, dry beans, maize | 0.0538847660170879 |
2 | Cotton, dry beans, soya beans | 0.0552823360239594 |
3 | Cotton, dry beans, sugar | 0.0554001961353126 |
4 | Cotton, dry beans, tomatoes | 0.0537548193766635 |
5 | Cotton, maize, soya beans | 0.054378675341399 |
6 | Cotton, maize, tomatoes | 0.0537169683262593 |
7 | Cotton, cabbages, dry beans | 0.0560629255563752 |
8 | Cotton, cabbages, maize | 0.0529411764705882 |
9 | Cotton, cabbages, tomatoes | 0.0529411764705882 |
10 | Cotton, tomatoes, potatoes | 0.0530232065275474 |
11 | Maize, soya beans, dry beans | 0.053340093038395 |
12 | Maize, soya beans, potatoes | 0.054408535208579 |
13 | Maize, cabbage, dry beans | 0.0577712841409234 |
14 | Maize, cabbage, tomatoes | 0.0529411764705882 |
15 | Maize, dry beans, sugar | 0.0535396709191825 |
16 | Maize, dry beans, tomatoes | 0.0529411764705882 |
17 | Maize, tomatoes, potatoes | 0.0532893345994195 |
Output of the optimization process.
Net profit (ZAR) | Total crop production (tons) | Total land utilization (ha) |
---|---|---|
|
31.5857454386647 | 0.919617517093457 |
This work suggests that generalized differential evolution 3 (GDE3) is a useful multiobjective optimization tool for optimal crop-mix planning decision support. The metaheuristic is able to produce improved results when compared to those generated by other two metaheuristics that are representatives of the state-of-the-art in evolutionary multiobjective optimization. The GDE3 uses a simple mechanism to deal with constraints and the results computed by the metaheuristic generally indicate that such mechanism, despite its simplicity, is effective in practice.
The following conclusions can be made about the performance of GDE3: (i) GDE3 is able to produce most of the true Pareto fronts of the optimal crop-mix planning problem considered and it has the best performance; (ii) the GDE3 is able to produce a good distribution of solutions of the multiobjective optimal crop-mix planning problem; and (iii) GDE3 is ranked first with respect to the selected
Future work will extend GDE3 for crop planning decision under uncertainty. This will produce a novel approach to deal with practical situations for which profit coefficients of agriculture are uncertain. The optimization approach can help farmers to efficiently utilize the available meager resources, including planting area, time, and money. The approach combines indigenous farming with information communication technology to optimize crop production, support efficient planning, and help farmers determine the possible combination of crops to plant on the same planting land year by year. As part of the future work, other optimization techniques can be compared to GDE3 to establish its superiority over many other techniques for crop planning decision making.
The authors declare that there is no conflict of interests regarding the publication of this paper.