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Optimising profitability and environmental trade-offs in managing cropping systems using differential evolution

Peter DeVoil1, W.A.H. Rossing2 and Graeme Hammer1,3

1 Agricultural Production Systems Research Unit , Department of Primary Industries, Qld, Australia. Email peter.devoil@dpi.qld.gov.au
2
Biological Farming Systems Group, Wageningen University, The Netherlands. Email Walter.Rossing@wur.nl
3
School of Land and Food Sciences, The University of Queensland, Brisbane, Australia. Email graeme.hammer@dpi.qld.gov.au

Abstract

Whilst traditional optimisation techniques based on mathematical programming techniques are in common use, they suffer from their inability to explore the complexity of decision problems addressed using agricultural system models. In these models, the full decision space is usually very large while the solution space is characterized by many local optima. Methods to search such large decision spaces rely on effective sampling of the problem domain. Nevertheless, problem reduction based on insight into agronomic relations and farming practice is necessary to safeguard computational feasibility.

Here, we present a global search approach based on an Evolutionary Algorithm (EA). We introduce a multi-objective evaluation technique within this EA framework, linking the optimisation procedure to the APSIM cropping systems model. The approach addresses the issue of system management when faced with a trade-off between economic and ecological consequences.

Media summary

Cropping system design is formulated as a multi-objective optimisation problem. The results are presented as a set of trade-offs between management actions, serving as discussion tools between scientists, farmers and policy makers.

Key Words

Evolutionary Algorithms; Agricultural Systems Modelling.

Introduction

Although performance of cropping systems is primarily assessed in economic terms, farmers and legislators increasingly consider alternative performance criteria such as ecological impact and efficiency of resource use. To optimise these systems, indicators are assigned to each criterion as objectives to be maximised or minimised, with management actions or rules as the variables to be optimised.

In classical, single objective optimisation problems, the optimal (highest ranking) solution is clearly defined – the ‘top of the list’. This does not hold for multi-objective problems: instead of a single optimum, there exists a set of equivalent solutions, known as a pareto-optimal solution after the Italian economist Vilfredo Pareto (1848-1923). This set represents the set of best solutions across all criteria as no other solutions are superior when all objectives are considered. Figure 1 shows a 2 dimensional problem as a series of “peels”, with each peel a pareto-optimal frontier. Choice between members of the Pareto-optimal set is a trade-off – a reward in one dimension against a penalty in another.

Figure 1: Pareto ranking. The top 3 Pareto-Optimal solution frontiers are joined by lines. Objectives were to maximise Gross Margin and minimise Erosion.

The concept of maintaining a set of problem solutions is central to the methodology of Evolutionary Algorithms (EA), which keep a ‘population’ which is ‘evolved’. A class of EA, Differential Evolution (DE, Storn & Price, 1997), has shown advantages in speed and complexity (Mayer et al., 2003) over ‘genetic’ algorithms such as Genesis (Grefenstette, 1994), and Genial (Widell, 1998).

Methods

Cropping Systems

Dryland cropping in the Northern Australian grain growing region is characterised by highly opportunistic production of a range of cereal, pulse, oilseed, forage and fibre crops. A sub-tropical climate allows both summer and winter crops to be grown, with yields largely determined by water supply from either in-season rainfall or water stored in the soil prior to planting.

However, seasonal rainfall in this region is highly variable, such that the prospects for any one crop are risky (Hammer et al., 1996). To reduce dependence on in-season rainfall, fallowing the paddock between crops is a recommended technique to store moisture for subsequent crops. The reduction in risk from fallowing has consequences: financial returns are lower (1 crop instead of 2), and resource degradation through increased soil erosion and solute leaching is possible. This issue of cropping vs fallowing demonstrates the trade-offs that exist between economic and ecological objectives.

These management options can be reduced to an optimisation problem. The traditional approach to formulating such a problem is to assign costs to the ecological objectives and maximise the single profit objective – an approach which a) hides the fact that there are tradeoffs to be made, and b) promotes the falsity that such equivalences can be made a priori.
The number of combinations of decisions on land cultivation, crop choice and crop management is extremely large, and restrictions have to be made to define a relevant yet feasible optimisation problem: in practise some decisions are set at agronomically plausible levels, while others, considered less obvious, are candidates for optimisation.

Differential Evolution

The DE algorithm is a simplification of traditional ‘Genetic’ algorithms: the search process is a statistical operation, which contrasts against how biological operators are mimicked in traditional evolutionary algorithms. The essential differences are that:

  • To avoid stagnation, there is no concept of ‘elites’ – each individual is challenged in each generation. However, the only challenger to an individual is that individuals progeny, not the entire population.
  • New individuals are evolved from parameter differences, not from within a static range. This attempts to make the algorithm self adapting, giving more rapid convergence.

There is choice as to evolutionary ‘strategy’: Price (1999) describes many strategies that define the method of parent selection. The experiment in this paper uses the DE/rand/1/bin strategy described in the same paper as “standard and robust”, however experimentation with the choice of strategy is recommended for each optimisation problem.

After the choice of strategy, there are 3 basic parameters to the algorithm: population size N, the crossover rate C, and mutation rate F. The algorithm can be adapted for parallel computation.

Pareto Ranking

A non-dominated sorting procedure determines ranks (or peels) of equivalent individuals. The procedure starts by finding the set of individuals that are Pareto-optimal in the current population, to which the highest rank is assigned. This set is called the first Pareto stratum. The individuals that are Pareto optimal in the remaining population are assigned the next highest rank. These individuals comprise the second Pareto stratum. This process continues until all the individuals are ranked. Figuratively speaking, the subsequent non-dominated fronts are peeled off step by step as illustrated in figure 1.

A complication can arise when a population ‘collapses’ to the same value in one or more dimensions – the algorithm must re-rank these individuals via recursion over the remaining dimensions.

Problem Representation

This experiment presents the EAs operation in a case in which the full pareto optimal set is known. The variables optimised are a discrete choice of land use in four seasons, and discrete threshold levels of plant available water to consider planting (one of four) crops. The cropping system is designed as a two-year rotation, with cotton and sorghum as potential summer crops, and wheat and chickpea as winter crops. Fallowing may take place in either season. Decisions to be optimised comprise which crop to plant at what stage in the rotation, and the timing of planting in relation to current soil moisture conditions.

APSIM (McCown, et al, 1995) version 2.00 was configured to simulate crop rotations grown on a Brigalow soil type of 5% slope at Dalby (27S, 153E) over a 40 year historical weather record. At the start of each cycle within the rotation, soil water level was set to 260mm available soil water, representing a 100% full profile to 1800mm depth. Sowing rules for each crop consisted of a fixed sowing window, i.e. a period during which sowing was possible, the requirement of 30 mm rainfall over a 3 day period prior to sowing, and a minimum necessary level of plant available soil water. Sowing windows were 1 October to 30 November for cotton, 15 September to 15 January for sorghum, and 15 May to 1 August for both chickpea and wheat. At sowing, cotton was fertilized with 100 kg N/ha, wheat and sorghum with 50 kg N/ha and chickpea was left unfertilized. Prices of inputs and outputs are assumed fixed over the period of the simulation, reflecting levels prevalent in 1999.

Each individual within the EA encodes land use type and requirements of plant available soil water at sowing for each crop into 8 ‘genes’, as described in Table 1.

Table 1: Functions and values for ‘genes’ used in the optimisation procedure.

Gene

Range

Function

Summer Crop 1,

Summer Crop 2

Cotton, Sorghum, or Fallow

Land use in 1st and 2nd summer

Winter Crop 1,

Winter Crop 2

Wheat, Chickpea, or Fallow

Land use in 1st and 2nd winter

Cotton Esw

100, 160, 220mm

mm extractable soil water required for a cotton planting

Sorghum Esw

20, 120, 220mm

mm extractable soil water required for a sorghum planting

Wheat Esw

20, 120, 220mm

mm extractable soil water required for a wheat planting

Chickpea Esw

20, 120, 220mm

mm extractable soil water required for a chickpea planting

The simulations expose each system to 40 years of climate, evaluated in similar terms as in Carberry et al. (2000):

1) Average yearly gross margin ($/ha), and

2) Soil loss, defined as the simulated average annual soil erosion (t/ha).

These two criteria, while not exhaustive, serve to demonstrate the tradeoffs made between financial returns and long-term sustainability of the system in question. A more extensive elaboration of the cropping systems design issue is in preparation.

Results

Figure 2 demonstrates how the algorithm, while sampling parameter space, evolves through the solution space and converges to the (known) pareto frontier. In this case, a population of 100 individuals is evolved at a cross-over probability of C=0.5 and a mutation probability of F=0.5 (Mayer, pers. comm).

Figure 2. 60 iterations of the EA. 1st ,2nd and 3rd pareto rankings are joined by lines.

The Pareto-optimal set described in Table 2 shows that increased intensity of cropping, and the inclusion of summer cropping of sorghum, is needed to achieve high gross margins. However, this increases erosion. Increased winter cropping of wheat was required to minimise erosion. While these trade-offs are specific to the conditions specified for this analysis, they provide an ideal basis to inform discussion about design of cropping systems with practitioners. Our experience with discussion support for decisions associated with single crop management (Nelson et al., 2002) has shown this to be an effective means to interact with decision makers and their key advisers.

Table 2: The Pareto-optimal set of parameter values and outcomes from the entire population. The first 4 columns describe the crop sequence, ‘S’:Sorghum; ‘W’:Wheat; ‘F’:Fallow; ‘W’:Wheat; ’Cp’:Chickpea, and the soil water criterion for planting. The last two show outcomes for each management system over 40 years of simulation. Note Cotton does not appear in this set.


Summer Crop 1

Winter Crop 1

Summer Crop 2

Winter Crop 2

Gross margin
($/ha)

Erosion
(t/ha)

S120

W20

F

W20

332

1.09

S20

W120

S20

W120

711

1.13

S20

W20

S20

W20

746

1.16

S120

W120

S120

W120

762

1.41

S20

F

S20

W20

810

1.92

S120

F

S120

W20

829

2.04

S20

Cp120

S20

W20

840

2.14

S20

Cp20

S20

W20

840

2.33

S20

Cp120

S20

W120

862

2.44

S20

W120

S20

Cp20

937

2.66

S20

W220

S20

Cp20

948

3.34

Conclusion

The Differential Evolution algorithm can be successfully combined with the Pareto ranking technique to search the feasible solution space for a complex cropping systems design problem that involves multiple criteria. The concept of Pareto-optimal frontier has proved useful for exploring trade-offs between important conflicting criteria (such as economic and environmental consequences) associated with the system design problem The results enable a costing of erosion on the basis of best available agronomic practices. The quantification of attributes of cropping systems that represent Pareto-optimal combinations of economic and environmental objectives provides highly interesting material for informed discussions on strategic decision making with decision makers.

References

Carberry PS, Hammer GL, Meinke H, and Bange M, (2000). The potential value of seasonal climate forecasting in managing cropping systems. in ‘Applications of Seasonal Climate Forecasting in Agricultural and Natural Ecosystems’, 167-181. (Kluwer Academic).
Hammer GL, Holzworth DP, and Stone R, (1996). The value of skill in seasonal forecasting to wheat crop management in a region with high climatic variability. Australian Journal of Agricultural Research 47, 717-737.

McCown et al., (1995). APSIM: an agricultural production system simulation model for operational research. Mathematics and Computers in Simulation 39, 225-234.

Grefenstette JJ, (1994). http://www.aic.nrl.navy.mil/galist/src/genesis.tar.Z

Widell H, (1998). http://hjem.get2net.dk/widell/genial.htm

Mayer DG, Kinghorn, BP, Archer AA. (2003). ‘Simple’ differential evolution for beef model optimisation. Proceedings of MODSIM 2003 International Congress on Modelling and Simulation, Townsville, Australia.

Nelson RA, Holzworth DP, Hammer GL, and Hayman PT (2002). Infusing the use of seasonal climate forecasting into crop management practice in North East Australia using discussion support software. Agricultural Systems 74, 393-414.

Price KV, (1999). In ‘New Ideas in Optimization’. (Eds. Corne D, Dorigo M, and Glover F.) 79-108. (McGraw-Hill, London).

Storn R. and Price K, (1997). Differential evolution – A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization 11, 341-359.

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