Multiple objective linear programming for environmental farm planning

Multiple objective linear programming for environmental farm planning J. E. Annetts and E. Audsley Silsoe Research Institute, UK We present a multiple objective linear programming model developed to consider a wide range of farming situations, which allows optimization of profit or environmental outcome(s) or both. The modelling considers the problem of planning a farming system within a world where environmental considerations are increasing. The objective is to identify the best cropping and machinery options which are both profitable and result in improvements to the environment, depending upon the farm situation of market prices, potential crop yields, soil and weather characteristics. In particular, the model uses a flexible approach to choosing the machinery, timing of operations, crop rotations and levels of inputs. We show for a UK scenario, that large reductions in environmental impact can be achieved for reductions in farm profit which are insignificant relative to the annual variation due to yields and prices. Keywords: Agriculture, Environmental impacts, Linear programming, Multi-objective, Optimization. Introduction The Silsoe Whole Farm Model is a multiple objective linear programming model developed for a variety of farming scenarios including UK and European arable and mixed arable and livestock farms. In particular the model determines the best cropping and machinery options for given farm, economic and climate details. The crop rotation, timing of operations, and machinery systems used will affect the profitability in terms of potential crop yields, cost of machinery, fuel use, machinery repairs, inputs to apply (eg. fertiliser and herbicides), etc. Strategic farm planning can be considered as choosing the best systems, associated cropping and level of manpower and inputs, with which to carry out the set of tasks required every year, subject to constraints. The plan must be sustainable over the long-term not just for a single year. The main physical constraints on a farm are the particular climate and the soil characteristics of the location. In addition, a farmer has preferences developed by experience, for the sort of crops to be grown, suitable crop rotations (which crop should follow another), the systems of cultivation, sowing and harvesting, level of application of fertilisers, pesticides and other chemicals, and the timing of each of the operations or tasks. Any human activity has effects on the environment, but different choices in farming have a different effects. A particular example is the choice of when an operation should be carried out. A farmer has a window of time in which for example, winter wheat can be planted. There will be an optimum or best time which will give the farmer the highest yield from that crop. The crop could be planted a few weeks later for a loss of yield and a decreased need for chemicals (such as applying herbicide to control grass weeds), a small saving in cost and benefit to the environment. However later sowing will also increase the risk of nitrate that the crop would otherwise have taken up, leaching from the soil to water courses. There are two critical periods of planning on an arable farm in the UK. These are autumn (August to October) and spring (March/April). In autumn, crops must be harvested, land cultivated and crops sown which can survive the winter, before the increasing wetness of the soil and decreasing temperatures prevents further work until spring. In general the earlier the crops can be sown the higher the yield will be. The harvesting of root crops done in the late autumn conflicts with this objective. In spring, it is again generally true that the earlier the spring crops are sown the higher the yield will be, although some crops are damaged by late frosts. Early spring planting conflicts with the spring tasks required on the winter sown crops. The analysis must therefore incorporate the effects of changes in crop, method or timing on yield, costs and environmental outcomes during these periods. Linear programming (LP) is a conventional method for modelling the problem of farm planning, largely because it captures the important constraint of workable days which varies with time of year, location and soil type and is a major cause of differences between farms. Different applications emphasise different aspects. Rehman 1 and Piech 2 use multiple criteria decision making, Boisvert3 considers methods of including risk in the analysis, Mendoza 4 use ranges for the objective function values, Etyang 5 uses

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chance-constrained programming for the workable days, as does Bouzaher 6 in considering ethanol production. Other applications link other models to the LP framework. Moxey 7 links an ecological vegetation model to consider conservation, Nevo 8 uses crop yield formulae derived from experts, England 9 uses experimental data to fit yield nitrogen curves and determine the effect of nitrogen taxes and quotas, Donaldson 10 linked the EPIC crop models to his LP. In using the LP framework to consider the environment, Wossink 11 specifies thousands of variants of a cropping plan in studying how policy options changed the plans selected. Zekri12 uses a multi-objective method to study different management practices which they then simulated. In this application, we require the model to be fully able to adjust the timing of operations, to adjust crop rotations and to adjust the level of labour and machinery to obtain a steady state annual plan. It must also have models relating yield and environmental outcome to variables such as soil type and timing, workrate to machine size and soil type, and workable days to the type of operation being carried out. In particular, it is designed to allow analyses of interactions between environmental outcomes, profitability and farm management. Model formulation Problem description The information available for farm planning can be divided into farm, crops, operations, machinery systems, machines, livestock, feed and wastes . The farm has a cropping area of known soil type(s) and features such as buildings and crop stores and an available amount of irrigation. The year is divided into discrete periods each at least one week, for example 26 two week periods. For each period the number of workable hours for different operation types can be calculated as a function of the climate, rainfall and soil type. Each crop can be described in terms of a gross margin (yield, inputs and prices), base environmental outcomes (a function of yield, inputs, soil type, etc. depending upon the effect being modelled), a sequence of operations, other non-sequential operations and the possible crop rotations and their effects on the crop costs, yield, inputs and environmental outcomes. Other information may also be relevant such as whether the crop is part of a subsidy scheme (such as the Arable Area Payment Scheme, in which the farmer must also set aside a percentage of the land as uncropped) or if specialist storage is required (e.g. a potato store). An operation is defined by a machinery system, number of passes over the field, with start and end points to define the time window in which the operation can be carried out. For each period within the time window there are associated additional effects on costs, yield, inputs and environmental outcomes for the crop being operated on. A machinery system is defined by the combination of machines and labour required, a workrate (a function of machine sizes, soil type, crop yields or inputs) and an additional environmental outcome. Alternative machinery systems can be defined for each operation, for example a farmer can choose between contract harvesting, different sizes of farm machines, or alternative slurry application methods. Machinery and labour are defined by either capital or annual costs, power requirements, minimum and maximum numbers available. Each type of livestock is described in terms of a gross margin, labour needed, requirements for feed and bedding and the amount of wastes produced. Feed and bedding can be purchased or are a product of crops, such as silage from grass or straw from cereals. Feeds have different nutrient contents. Feed intake is limited by the dry matter intake capacity of the animal. The major feed requirements are expressed as metabolisable energy and protein. Feed also can be stored for winter use with associated dry matter losses. Livestock produce wastes which can be in different forms and must be applied to the crops at appropriate times. They provide amounts of the major fertiliser nutrients nitrogen (N), phosphorus (P) and potassium (K) which reduce the need for purchased fertilisers. The method and time of the application of wastes generate environmental effects. This information is all available in a database from which the model structure is filled with appropriate values in order to model a specific farm or situation. The model optimizes a weighted sum of component objective functions which calculate annual net profit and each environmental outcome, subject to a set of constraints. There are basically four types of constraints:1. 2. 3.

Limits on the amount of machinery time available for each period of the year. Restrictions on the total amounts or areas of activities. The requirement for crop operations to be sequenced, including crop rotation sequencing.

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Livestock grazing and feed constraints.

The multi-objective function and component objective functions (Note that the Appendix contains a list of all notation used.) Annual net profit (z 0 ) is calculated as the sum of expected crop gross margins (that is, expected income minus seed and chemical costs) minus the annual cost of machinery and the cost of operations and crop rotations. Environmental outcomes (z e) are calculated similarly, as an expected base outcome for each defined activity (be it crop or livestock) with additive changes as functions of operation timings, crop rotations and machinery systems.

(1) The multi-objective (OBJ) to be maximised is given by Eqn. 1. We assume that annual net profit is maximised and environmental outcomes are defined in such a way as to be minimised. The weights "0 and "e are defined by the user to represent the balance between the relative importance the farmer or decision maker attaches to profitability and environmental outcomes. For the initial solution of the model "0 =1 and "e=0. For further solutions the alphas can be varied between zero and one. Due to the very different units of profitability and environmental outcomes, e.g. £, kgN, kg a.i., it is necessary to normalise these " weights. The normal procedure of optimising each objective is not applicable as the majority optimise to zero. Therefore normalising is done in terms of the amount which occurs for the unconstrained profit objective, i.e. the initial solution of the model where the function values are . The normalised objective function is given by Eqn 2, which is used for further optimisations.

(2) The formulation of the model also allows constraints to be placed on any of the component objectives thus one can solve for a minimum environmental objective given a lower bound on net profit, or maximising net profit for an upper bound on one or more environmental outcomes. The component objective functions (z e) are calculated from values in a database associated with the crops, operations and operation timings, crop rotation, machinery system alternatives and machines. Associated against each of these dependent variables are costs in terms of money (unit of £ for the UK) and environmental outcomes (units such as kgN for nitrate leaching or “10 th of dose” for herbicide use). Each crop has a gross margin (calculated from an expected yield, price and variable costs) and base expected environmental outcomes/effects (calculated from data and other models). The values of these depend upon the farming systems and environmental effects being modelled, as well as the location of the farm. Since this paper is concerned with defining the LP model the details of these values are excluded for simplicity. Details of the data used can be obtained from the authors. These values are proportional to the amount of activity which the LP will calculate. Eqn. 3 gives the partial objective equations for annual net profit (z 0 ) and environmental outcomes (z e), in terms of these costs. The net profit calculation takes account of prices paid for expected crop yields and inputs costs, rotational costs due to changes in yield and inputs, costs of crop operations and timeliness costs due to changes in yield and inputs, etc. The environmental outcome objectives are similar.

(3)

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Crops require a certain amount of inputs such as fertiliser, herbicides, pesticides or irrigation. The amount required is relative to the yield expected to be obtained. Thus high levels of irrigation giving high yields will require more fertiliser, etc. W ithin this target level, operations affect the amount of various inputs required by the crop (e.g. the later the crop is drilled, the less herbicide need be applied, and slurry application provides nitrogen reducing the need for other applications). The amount of input h required is given by Eqn. 4 as the target amount used for each crop minus the effects of operation timing and crop rotation.

(4)

Machinery time availability constraint The primary equipment constraint is that the total hours required from equipment by operations must be less than the hours available. The hours required from a particular machine (note that labour can be considered in the same way as a machine) is the product of the number of machines required by each operation carried out in the particular period, the workrate (in h/ha) and the area of crop requiring that operation (in ha) (the left hand side of Eqn. 5).

(5) Since nearly all operations take place outdoors, the time available is dependent on the soil conditions and the weather, as much as labour availability. The weather conditions vary widely from year to year so that in some years it can seem possible to work all the time, and in others none of the time. It is also possible to work extremely long hours including overnight, to complete tasks when only a short time window is suitable. This and other adjustments to the working procedure make it difficult to use robust planning techniques which minimise an absolute deviation. The Silsoe Whole Farm Model is a planning model and thus it is necessary to make estimates of a reasonable expected level of workable hours available each period which represents a farmer’s views of future years. Over a 10 year period of weather variation from seemingly being able to work all hours to working seemingly none, we choose the 7 th best year in 10, which means that the cropping and consequent operations in a model solution will be achievable 7 years out of 10. Given the nature of the model other options for available workable hours can be introduced without altering the nature of the LP model, which would be necessary if modelling a European situation. The hours available in each period are also operation dependent, since the weather and soil conditions impact the amount of time the farmer can plough, drill, harvest etc, in differing amounts. For example spraying requires good weather and low wind speeds, whereas ploughing can be done in wind, rain or shine, provided the soil is workable. Thus some operations are more restricted in terms of time available than others. Thus, define sets of available hours such that for different operation types (e.g. plough, drill, harvest, spray) each set is strictly inclusive within the other sets (i.e if the conditions are right for harvesting then they are right for ploughing, but not the reverse). Let O n be the set of operations such that if operations in n are workable then operations in n+1 are workable, but not vice-versa (Eqn. 6). For example, if it is possible to harvest, it is possible to plough but not vice-versa. The right hand side of Eqn 5 defines the product of the available hours for operations in set n for one machine m (H n ) and the numbers of machine m (n m ). Thus the left hand side in summed over all operations which can be carried in the available hours set n, as defined in Eqn 6.

(6) Machinery represent a capital cost replaced after a number of years. A procedure was developed 13 to convert these costs to equivalent annual costs taking into account interest and inflation rates, maintenance costs and resale values. This annual equivalent cost E m is deducted from the profit objective as

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... - E m n m .... Since the numbers of machines calculated by the model are non-integer it is possible for an optimal solution to suggest using part of a machine. Interpretation of such a solution will depend upon the farmer scenario. For example, if half a 10t/h combine harvester is required for a 250ha farm, this could suggest the need for one smaller capacity machine or that with the larger harvester, contact work for other farms is an option. Further runs of the model can be done by defining a smaller machine or by setting the minimum and maximum constraints on the numbers of machinery to force a whole harvester to be used. Thus allowing the analysis of various machinery choices. This can be useful when looking at the impact of new machinery. Activity constraints and cropped area For each operation on an activity, the total amount of that operation must equal the amount of the activity or crop, which in turn for each type of activity must be less than or equal to the maximum amount (e.g. crop area or livestock numbers) of the activity (Eqn. 7). Amounts of each activity a i can also be bounded by a minimum and/or maximum value, due to farmer preference.

(7)

In addition the cropped area must be less than or equal to the cropping area on the farm. This constraint must take into account the possibilities of more than one crop per year and more than one year per crop. Crops include permanent crops such as pasture for grazing, perennial crops such as grass for forage, annual crops such as wheat, rape and land set aside where no actual crop is grown but for which the farmer receives a subsidy payment (known as setaside), and catch crops (an extra crop grown between two main crops of the rotation without interfering with the normal farm system). The area of land occupied by a crop or between crops, at any time, must be less than or equal to the area of land available for crops (Eqn. 8).

(8) Where

is the total area of land transferring from crop i to crop c, with the start of

transfer being the first period of the last operation of crop i and the end of transfer, the last period of the first operation of crop c. Taking a particular period of the year, the land must either be in a crop or being transferred; there is no overlap. An arbitrary, but useful, period is the year end. Thus 0i is the number of year ends that an annual, perennial or catch crop i crosses, Z is the set of permanent crops and )ic has value one if the transfer from crop i to c crosses the year end, otherwise zero. Constraints to sequenced crop operations and restrict crop rotations Two types of crop operations are defined: sequenced (one must be done before another) and nonsequenced operations (carried out depending upon the stage of the crop). A farmer has a choice of different machinery systems for the same operation, for example different sizes of tractor. Each machinery system will consist of numbers and a combination of machines and labour used at a particular work rate. The constraint for sequenced crop operations is that the amount of a sequential operation carried out, must be less than or equal to the amount of the preceding operation carried out to date. In addition, there may also be limits on the time allowable between successive operations. This can be zero (they must be done immediately one after the other) or several weeks (time must be allowed for straw to dry after combining or seeds to germinate). Eqn. 9 gives the crop operation sequencing constraints, where 8j and :j are the minimum and maximum times allowed since the previous operation j-1 and P ij is the time over which the jth operation in activity i can be carried out. Note that if :j is effectively infinite then the second equations are not required.

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(9)

To allow for optional operations, such as subsoiling before ploughing or baling after harvesting the grain, null machinery systems are defined as alternatives to the actual systems, with appropriate adjustments in profit and environmental outcomes. A null machinery system requires no labour or machinery. Thus this allows the LP model to choose the optimum system to use or not. Similar constraints are formulated for activity sequencing (i.e. crop rotation) where the final operation of crop i must be followed by the first operation of crops c. Let r ic(d) be the area of crop c following crop i where the final operation (J) of crop i is carried out in period d. Thus the first line of Eqn. 10 holds. The following two lines of Eqn. 10 then constrain the area of the first operation of crop c to be less than the previous operation done, represented by the variables r ic(t).

(10)

Rotational Diseases In addition to Eqn. 10, this part of the model defines crop rotation restrictions. So with data defining which other crops a crop can follow based on the operational timing used in Eqn. 10, we also define each crop to be a particular disease class. Then we define a minimum number of years between crops with the same (or associated) disease class. Diseases are an important consideration in planning crop rotation. These are diseases which may be difficult, very expensive or impossible to control chemically and which build up in the soil when a particular crop or type of crop is grown. If such a crop is grown for two years running, a disease will reduce the yield, possibly very severely, and may make the crop products unmarketable. The soil may then be unsuitable for this crop for very many years. Some diseases are more severe than others, ranging from those which reduce yield by 10-15%, to those which prevent the same crop being grown for 4 or 5 years. There are also some diseases which are encouraged by another disease being around. The crops with an “associated” disease should also be restricted in the rotation. Each annual, perennial or catch crop has a particular disease class. Each disease class is defined by the minimum number of years necessary between crops with the same (or associated) disease class. For each crop following a particular disease class there are yield penalties (from zero to not allowed) and costs, input changes and additional environmental outcomes . These associated penalties are subtracted in the appropriate component objective row (Eqn. 3) against the crop transfer variables, r ic(d). The restrictions on number of years between crops require separate constraints. The annual build up of a disease, k, is offset by growing crops not encouraging that disease or its associated disease, k A . In the simple case where a disease k has no associated disease then once a crop with that disease class has been grown another or the same crop with this disease cannot be grown for B k years. This translates into crop areas on the assumption that the same rotation will apply to all areas of the same soil typ. Thus a crop grown one year in 6 (i.e. B k = 5) can take up to one sixth of the land area. Put another way, the crop with disease k has caused a 100% build up of that disease. The other crops grown will decline that disease by 20% each year. If another crop is a partial host to disease k and requires 2-year gap for its (different) disease, it is assumed (worst case assumption) that it causes a 40%

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build up of the rotational disease k.

This translates the general constraint given in Eqn. 11 to for this particular disease k.

(11) Livestock: grazing and feeding constraints For livestock, we need to consider their food and bedding requirements and their waste production and subsequent disposal to land. The major environmental outcomes are associated with waste disposal, which can result in large emissions of ammonia to the atmosphere and nitrate leaching to water courses, depending on the method and timing. A gross margin (value of output less associated variable costs) for the animals and any other costs are part of the profit objective. Environmental outcomes associated with the animal systems are included in the environmental objectives in the model. Constraints are necessary to allow for feeding the animals. There is a vast range of potential feeds but the major alternatives are straw, grazed grass, grass silage, maize silage, and various manufactured concentrates. Some of these can be grown on the farm as part of the crop rotation, some are by-products of growing the crops (e.g. straw from barley grown for grain) and some are bought in. Each feed is defined in terms of its content of dry matter (DM), metabolisable energy, MJ/kgDM (ME) and crude protein, g/kgDM (CP). For this model it is assumed that animals have a limited intake of dry matter, and require certain amounts of metabolisable energy and crude protein throughout the year (per period) to sustain life and production. There are other food factors which could be considered when formulating a specific ration and deciding what additives to purchase, but DM, ME and CP are the major factors determining the feeding strategy. The animals’ requirements will vary depending upon the stage of the animal’s life, whether it is young, old, lactating, in calf etc. Input requirements per period of DM, ME and CP (and bedding) are provided in the database per animal. Other than fresh grazed grass, farm produced feed can also be stored although there will be a loss of dry matter and quality due to storage. To allow flexibility within the management of the farm, upper and lower bounds are applied to the food requirements per period. The requirement per animal per period of F u (d) (where u is DM, ME & CP) is given by Eqn. 12.

(12) Over the whole year the total intake must be no more than the total intake limit and the total ME must be no less than the total requirement. Thus Eqns 13 and 14 for each component u= DM , ME and CP, determine the amount of each feed f f(d), where are the contents, and and are the upper and lower bounds shown in Eqn 12.

(13) and

(14) where ( = 0.84 if the feed is a concentrate, else = 1. This is because the dry matter of concentrates does not reduce dry matter intake as much as forage. For bedding, the requirement is simply that the amount of bedding used per period is less than or equal to that supplied by crop products or bought in (Eqn. 15).

(15)

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All feed except grass and straw for bedding can be stored with some loss of DM each period, "f. Some feeds, in particular concentrates, can be purchased, some can be sold. Thus the amount in store after each period is the amount left from the previous period less the amount fed, plus the amount bought or produced (Eqn. 16). 2f is the amount of feed f produced from each crop and must be expressed in terms of the crop operation variables of the model (Eqn. 17). The yield harvested in any period depends on the timing and is expressed in terms of an expected crop yield from harvesting at the optimum time reduced by the losses from harvesting at other times. However reductions in yield can also occur due to the timing of other operations, for example late planting of maize for silage. Strictly, in order to maintain a linear formulation, one should therefore define a separate harvest operation for every possible combination of non-harvest losses. As in the majority of cases this is unnecessary because the operations or at the worst most of the operation, are carried out at the optimum time, a simplification is introduced. This deducts the yield losses from the amount harvested at the optimum harvest period. Thus the storage variable will underestimate the amount available until the end of harvest, which is unlikely to be a problem when the storage is to cover the whole winter period. The linearisation of the losses also introduces a small error as strictly they should be applied as cumulative percentage losses (10% and 10%, instead of 10%+10%), but as they are small the error is also small and errs on the safe side.

(16) where h is the harvest operation for crop i and *dh = 1 if d is the optimum period for harvest operation h, else 0.

(17) Grazing For the crop grass where grazing occurs, constraints are necessary to partition grazing and silage making. Fields selected for silage making are closed for a number of periods (L) to allow the grass growth necessary before making silage. Thus in any period d, the grass crop area a g can be partitioned into that for grazing x gg (d) or that for silage in some future period x gs(t) as given in Eqn. 18.

(18) Grazed grass is managed to make best use of the grass. Grazing of a field can be delayed provided one ensures that the quality does not fall too far due to over-mature grass. Thus there is a limited slack in the amount that can be carried over from one period to the next. The amount of grazed grass fed (fg (t)) in period t is thus constrained within two bounds to be close to the amount available, as given in Eqns. 19 and 20.

(19)

(20) Livestock Waste Several different types of waste (e.g. manure, slurry) can be produced from an animal by different management methods and the method of storage can produce other products (e.g. separated liquid, separated solid). Each type of waste has defined amounts of nutrients; nitrogen (N), phosphorus (P) and potassium (K). Waste is applied to the crops grown on the farm, either instead of or as well as chemical fertilisers to make up the crops’ required amounts of nutrients. The amount of waste w in store at the end of each period is reduced by the amount applied in the period and increased by the amount produced by the animals as given in Eqn. 21.

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(21)

where a is the number of animals, and the amount of waste w applied to a crop is calculated in terms of a defined application rate Tw . There are two types of waste application operations. The first is part of the sequence of operations on a crop and is likely to be followed by ploughing (e.g. applying solids). The second type of application is onto the growing crop (e.g. applying a liquid or slurry). The area of waste application on a crop does not need to be the total area of that crop, thus the applications are optional operations modelled as described above. The fertiliser nutrient amounts (i.e. N, P and K) applied to a crop (area a i) must not exceed the amount necessary for that crop (P ih ). Thus Eqn 22 applies for the amount of nutrient in waste, $w h .

(22) An example analysis of nitrate leaching and herbicide use on arable farms. As an illustration of the capability of the Silsoe Whole Farm Model, we present in this section some results from an analysis 14 . The questions to be explored relate to whether reductions in environmental outcomes are possible under different scenarios: • • • •

“Herbicide taxes”, achieved by increasing the herbicide prices by 100%, 200%, 300% “Goal driven reduction of herbicide use” by weighting the farmer’s objective towards herbicide minimisation in addition to profit maximisation. “Restricting nitrogen use” by setting a maximum of 100 kgN/ha across the farm. “Goal driven reduction of nitrate leaching” by weighting the farmer’s objectives towards nitrate leaching minimisation in addition to profit maximisation.

To achieve this analysis, data for typical East Anglian arable farms were used. The farmer can choose to grow winter wheat (WW), winter barley (WB), winter oilseed rape (WR), spring oilseed rape (SR), spring barley (SB), spring beans (Sbn), winter beans (Wbn), sugar beet (Sbe), potatoes (P), peas (Peas) and rotational setaside (RS). Average data from farm studies are used for labour and machinery requirements, availability and costs. Economic data are taken from Nix 15 . Three environmental outcomes were defined: nitrate leaching and herbicide use for blackgrass and wild oat herbicides on cereals. Data were derived using the nitrate leaching model SUNDIAL 16 and a weed control model17 and attached to appropriate crops, rotations and operations. Further information about the data used can be obtained from the authors of the paper. Figure 1 shows results from modelling a typical arable and root crops farm on sandy loam soil texture. The solid lines represent the results where the price paid for herbicides is increased by 100%, 200% and 300% and the LP model is optimised for steady-state annual profit only. Each increase in herbicide price (which could be considered as a tax) reduces the optimum net profit of the farm system, and makes a small decrease in the amounts of herbicides applied to the crops, achieved by small changes in cropping. The dotted lines represent the results of the second scenario, where there is an increasing weight applied to minimising the herbicides’ use as well as maximising net profit. This scenario has a much bigger impact on the use of herbicide relative to decreasing profit. For the same loss (5%) of profit as the 200% herbicide price scenario, where wild oat and blackgrass herbicide use decreases by 10% and 16% respectively, this goal driven scenario achieves reductions in use of more than twice as much, i.e. 37% and 100%. This shows there are a number of near optimal solutions which have lower environmental burdens and tax is not an efficient means of control.

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Figure 2 shows the outputs from three model runs to assess the third and fourth scenarios. The first is a base run with which to compare the other two. In this run the net profit is optimised for a clay soil arable and root crops farm, allowing the model to freely choose the cropping. In the second run a restriction on the amount of nitrogen input allowed across the farm is imposed, resulting in 8 crops being grown as opposed to 6 in the base run, for a much reduced profit. However, the nitrate leaching has increased from the base case, even though less nitrogen is being applied. This is because there are many nitrogen fixing crops which lose nitrogen in winter. The third run optimises a weighted objective of net profit and nitrate leaching, with no nitrogen restriction. The result shows a set of 5 crops are grown, for a small (6%) decrease in net profit. There is an increase in the amount of N fertiliser applied (by 22%), but the crops selected use it more efficiently thus reducing nitrate leaching (by 23%). This shows that as nitrate leaching is not a simple function of nitrogen applied, such restrictions are not appropriate controls. Conclusions and discussion Allowing the optimisation of joint objectives of profitability and environmental outcomes is beneficial in assessing whether reductions in environmental pollution are possible for small reductions in profitability. The conclusions from the analysis presented are that 1. 2. 3. 4.

Herbicide tax will reduce herbicide use but by small amounts at a cost, i.e. about 14% reduction for a 5% reduction in net profit at 200% herbicide tax. Alternative farm management can achieve much higher reductions in herbicide use for a 5% profit loss, e.g. by growing a higher percentage of spring crops. A nitrogen restricting policy can increase nitrate leaching: more spring crops increasing overwinter leaching It is possible to decrease nitrate leaching by growing crops which use the nitrogen applied efficiently, even when larger amounts of nitrogen fertiliser are applied.

In analysing the scenarios shown in the results, it is possible to draw one other conclusion from (2) and (3). (2) states that to reduce herbicide use more spring crops are grown, however growing more spring crops increases nitrate leaching from the over-winter leaching (3). Thus a conclusion from this is that improving one environmental impact can lead to the detriment of another. As discussed in the results applying herbicide tax or imposing a limit on the amount of nitrogen to be used may not have the desired effects of reducing environmental concerns of using too much herbicide or producing nitrate leaching. In the case of herbicide use the tax will reduce the use by a small amount, but the farmer is still concerned with maximising profit, and thus makes changes that will compensate for the additional costs. If instead of imposing a tax, farmers can be encouraged to reduce their herbicide use by making small changes to their cropping or the timing of applications, the modelling shows that larger reductions can be made, with only small reductions in profitability. This is equally true for nitrate leaching. Perhaps this could be achieved by showing farmers how alternative cropping and machinery options affect their profitability and their environmental burdens, using this kind of modelling. The LP model presented allows integration of environmental outcomes with profitability and farm management. The flexibility of the model allows choices of cropping, operations and machinery based upon logical planning constraints, including time availability, allowing rotational and time yield penalties for alternative operation timings and alternative machinery systems. Incorporating environmental outcomes alongside profitability, and allowing optimisation of multiple objectives, allows a user to explore other possible outcomes in terms of farm management solutions to reducing the environmental burdens of farming in a cost-effective way. In using the Silsoe W hole Farm Model, we have created a substantial database of cropping and machinery options. We have been able to model farms in other European countries such as Denmark and France, with changes to the database to reflect alternative cropping, machinery and cultural practices, such as rye, not applying manure at certain times of the year, or shared machinery. The model in particularly useful for looking at future climate and economic scenarios and their impact on land use. This has led us to consider some of the issues of uncertainty which are lacking from this current model. At each location

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the soil type is fixed and with the climate this allows us to predict crop yields, harvest and sowing dates and soil workability. Hence using the Silsoe Whole Farm Model we can determine a future cropping scenario at the location. This has implications for water use. In terms of the current model described here the development of appropriate data to describe the environmental burdens associated with each possible crop, operation and machinery option has been challenging. In particular further considerations of the steady state soil state in terms of fertiliser nutrients and weeds, to prevent solutions which are not sustainable over the long-term. Further work is expected in this area, to study interactions between various environmental effects and profitability, and thus to explore the possibilities for planning farming systems within a world of ever increasing environmental considerations. References 1

Rehman T and Romero C (1993). The application of the MCDM paradigm to the management of agricultural systems: some basic considerations. Agricultural Systems 41: 239-255.

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Piech B and Rehman T (1992). Application of multiple criteria decision making methods to farm planning: a case study. Agricultural Systems 41: 305-319.

3

Boisvert RN and McCarl B (1990). Agricultural risk modelling using mathematical programming. Southern Cooperative Series Bulletin 356, Cornell University, Ithaca NY.

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Figure 1. Optimal results for a typical sandy loam soil, arable and roots farm under alternative strategies to reduce herbicide use. Note each point on the solid lines represents 100%, 200%, 300% herbicide price increases. Each point on the dotted line represents an increase of the weighting of herbicide use relative to profitability.

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Figure 2. Cropping, net profit and environmental outcomes for three model runs to assess the impacts of nitrate leaching strategies for a typical arable and root crops’ farm on clay soil texture. Crop key clockwise from the top: winter wheat (WW), winter barley (WB), spring barley (SB), winter oilseed rape (W R), rotational setaside (RS), peas (Peas), spring oilseed rape (SR), spring beans (Sbn).

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Appendix: Notation All variables to be calculated by the model are lower case letters and all input values are upper case letters. Where reference is made to machines required for a task, this is taken to also include the option of labour. Subscripts/Superscripts b bedding d, t time period e environmental outcome (e = 1,...,E) f feed/fodder g grazed grass h crop input eg fertiliser, herbicide, water i, c activity, e.g. crop, livestock j operation, j=1,.., J k disease class kA disease class associated with disease class k m machine n workability type s machinery system of carrying out an operation u type of nutrient, e.g. metabolisable energy, crude protein, dry matter w product eg grain, straw, feed, waste, manure 0 net profit Dependent Variables Note that all dependent units ai ha, number b f(d) tonnes f f(d) tonnes m f(d) nm ph r ic(d) s f(d) s w (d) x sij(d)

tonnes

z0, ze

units

ha tonnes tonnes

units

Input data Av Bk C ij(d) C ije (d)

£/ha units/ha

variables are non-integer. description amount of activity i; area if crop activity, number if animals amount of fodder (or bedding) f purchased in period d amount of fodder (or bedding) f fed to (used for) animals in period d (where f = g fodder is fresh grass) amount of fodder (or bedding) f sold in period d the number of machines of type m total amount of input h required for all crops area of crop c following crop i completed in period d amount of fodder (or bedding) f in store at the end of period d amount of waste w in store at the end of period d the amount of operation j on activity i in period d using machinery system s, where the amount is an area if the activity is a crop, a number if the activity is livestock and a weight if the activity is a product the objective function, where 0 is net profit and e > 0 are environmental outcomes. Units are currency (£ for UK) for net profit, and as appropriate for environmental outcomes, e.g. “kg Nitrogen” for nitrate leaching or “10 th of dose” for herbicide use. objective function values for LP when solved for optimum net profit only ( "0=1 and "e=0)

maximum amount of activity v, e.g. capacity of livestock housing, sugar beet quota number of years necessary between crops with disease class k for disease to have no effect the cost of operation j on crop i in period d the increase of environmental outcome e from base amount due to operation j on crop i in period d

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C ijw (d) C ijh (d) Em F u (d) F b (d) G(t) H n (d) J K KA L M w (d) N jm On P ih R ich(d) R ice(d) R icw (d) S ij T Vw Vh W j(d)

Y iw Y ie Z

"e "f $u f (f )ic *dh *umin *umax 0i 2fi(d) 8j :j Tw

tonnes/ha

the loss of yield of product w from base yield due to operation j on crop i in period d units/ha the reduction in base input h required due to operation j on crop i in period d £/machine annual equivalent cost of machine m units/animal amount of feed nutrient u required by livestock in period d (units are MJ for metabolisable energy, kg for crude protein, kgDM for dry matter) tonnes/animal amount of bedding required by livestock in period d the proportion of yield produced in each period t over the grazing season. hours the hours available for operation type n on day d the last operation on an activity the set of crops with main disease class k the set of crops with disease class k A number of periods of grass growth before cutting silage tonnes/animal amount of waste w produced in period d per animal the number of machines of type m required by operation j the set of operations such that if operations in n are workable then operations in n+1 are also workable, but not vice-versa. units/ha standard base amount of input h required by activity i for yield Y iw units/ha the reduction in input h due to crop c following crop i in period d units/ha the increase in environmental outcome e due to crop c following crop i in period d tonnes/ha the loss of yield of product w due to crop c following crop i in period d set of time periods d when operation j of activity i can be done ha cropping area on the farm £/tonnes cost or value of product w £/unit cost of input h, units as appropriate for input hours/unit the work rate of operation j for given sizes of machines in period d. The unit depends on whether the activity is crop (ha), livestock (number) or product (tDM) units/ha expected yield of product w (tonnes for arable crops, tDM for grass and silage) units/ha base environmental outcome e from crop i (units depend upon definition of environmental outcome - see objective function z e) the set of permanent crops objective function weights proportion loss of dry matter of feed f in storage content of nutrient u in fodder f or waste w (MJ/kgDM for metabolisable energy, g/kgDM for crude protein, kgN/t for nitrogen fertiliser) kgDM/kgDM effect on dry matter intake of feed f, ((f = 0.84 for concentrates, else 1) = 1 if the transfer from crop i to c crosses the year end, otherwise = 0 = 1 if d is the optimum period for harvest operation h, else 0 units lower bound for feed nutrient u (units as for F u (d)) units upper bound for feed nutrient u (units as for F u (d)) the number of year ends the (annual, perennial or catch) crop i crosses tonnes amount of feed/fodder f produced from crop i in period d the minimum time between operations j-1 and j maximum time between operations j-1 and j tonnes/ha rate of waste application

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