This section describes the IDEAL (Integrated Dairy Enterprise Analysis Tool) model used to examine the value of nitrification inhibitors and the sources of the model coefficients.

#### 3.1. Farm Model

The value of nitrification inhibitors is investigated using an optimisation framework because such innovations can impact many facets of a farming system, understanding of such changes is currently limited, and simulation can fail to identify valuable management plans. To improve the clarity of exposition, lower case letters typically represent coefficients in the model description, while upper case letters typically denote decision variables.

The static model represents a single year consisting of 26 fortnightly periods ($i=[1,2,\mathrm{...},26]$), beginning on 1 July. The first time period follows the last time period in a cyclical fashion, such that time occurs in a continuous and repeatable fashion. The description of the grazing strategy is complicated because New Zealand dairy herds are typically rotated between individual fields to improve pasture growth and utilisation. Excess pasture growth can be stored as winter fodder through silage production.

The farm consists of $a$ hectares. This is segregated into two areas, that which is treated with DCD and that which is not. The optimal relative size of these two areas is determined in the model. The area of pasture grazed at time t that has not been grazed during period i and has received no DCD application is defined ${A}_{i,t}^{G}$. Similarly, ${A}_{i,t}^{SM}$ denotes the area harvested for silage production (i.e. ensiled) at time t that has not been grazed since period i on areas without DCD. Additionally, ${A}_{i,t}^{X}$ represents the area of pasture grazed at time t that was ensiled in period i on this untreated portion of farm. In contrast, ${A}_{i,t}^{G-NI}$, ${A}_{i,t}^{SM-NI}$, and ${A}_{i,t}^{X-NI}$ respectively represent their equivalents on land that has been treated with DCD application. These activities collectively describe the rotational land-use system.

Total land use at time

t is therefore defined:

The first term on the right hand side (RHS) describes land use at time t. The second and third terms describe land from which stock is excluded at time t ($i\ne t$) and thus is rested for future use.

The area on which nitrification inhibitors is applied (

NI) is a key decision variable in the model and provides an upper bound to the grazing and ensilement activities used on this land. This is described through:

Pasture is subject to minimum (${m}_{t}$) and maximum (${n}_{t}$) biomass levels before it is grazed or ensiled, set in accordance with standard practice. Additionally, grazing or ensilement ceases at a residual biomass (${r}_{t}$). These are agronomic limits defined to ensure adequate rates of regrowth and cow intake.

Total feed production in period

t (

${P}_{t}^{j}$) for

j = {

G,

SM,

X} is represented as:

where

${b}_{g}$ is pasture biomass growth in period

g. In comparison, total feed production in period

t (

${P}_{t}^{j}$) for

j = {

G-NI,

SM-NI,

X-NI} is represented as:

where ${b}_{g}^{NI}$ is pasture biomass growth with DCD application. Pasture growth is defined over individual growth periods since pasture response varies according to the time of year.

The feasibility of production activities defined in Equation 3 is constrained by:

Similarly, Equation 4 is limited by:

Pasture supply may be promoted using N fertiliser. This is described through:

where

${P}_{t}^{N}$ is the pasture biomass (t ha

^{−1}) produced through N fertilisation in period

t,

${N}_{i}$ is the amount of N fertiliser (t ha

^{−1}) applied during period

i, and

${f}_{i,t}$ is the yield response (t DM) in time

t following application of one tonne of N fertiliser in period

i.

The cow herd consists of individuals that vary by calving date, lactation length, genetic status, and productivity level. Calving begins on 1 July, 15 July, and 1 August. There are five possible lactation lengths: 180, 210, 240, 270, and 300 days. There are two herd classifications: cull or standard. Cull herds can be milked for any of the five lactation lengths, with all cows culled at the end of lactation. In contrast, standard herds can only be milked for 240, 270, and 300 days. There are nine general productivity levels representing genetic diversity in milk production. There are thus 216 possible attribute configurations for each individual cow. Temporal demand for energy depends on the characteristics of the herd. Milk production increases with productivity level and lactation length for a given initial calving date.

Feed supply is represented as a pool of metabolisable energy to be allocated among cows. Energy may be obtained from grazed pasture, grass silage, maize silage, and palm kernel extract. Grass silage is produced on-farm, but maize silage and palm kernel extract are purchased.

The demand and supply of energy is calculated for each fortnightly period through the equation:

where ${D}_{h}$ represents the number of cows with attribute combination h, ${E}_{h,t}$ represents the energy requirement (MJ of ME per fortnightly period) of a cow with attribute combination h at time t, $u$ represents the proportion of the feed that is consumed by livestock (e.g., ${u}^{P}$ represents pasture utilisation), ${q}_{t}$ is the energy content of each feed at time t specified in MJ of ME per tonne of DM, ${P}_{t}^{SF}$ (${P}_{t}^{SF-NI}$) is the total amount of silage fed to cows on the area without (with) DCD application, ${P}_{t}^{SM}\ge {P}_{t}^{SF}$, ${P}_{t}^{SM-NI}\ge {P}_{t}^{SF-NI}$, ${F}_{t}$ is the amount of maize silage (t DM) fed to cows at time t, and ${K}_{t}$ is the amount of palm kernel extract (t DM) fed to cows at time t.

The feed intake constraints of cows is represented by:

where ${V}_{t}^{P}$ is the maximum per cow intake of pasture dry matter at time t (t DM cow^{−1}), ${V}^{S}$ is the substitution rate of pasture to forage supplements (grass and maize silage), and ${V}^{K}$ is the substitution rate of pasture to grain.

Total nitrate leaching is defined:

where

M is the proportion of nitrate leaching decreased through additional mitigation strategies (see Equation 11),

$\chi $ is a constant term,

${z}_{h}$ is annual milk production (t cow

^{−1}) of a cow in herd

h, and

$\{\varphi ,\eta ,\tau ,\upsilon \}$ are slope coefficients describing the correlation between nitrate leaching and N fertiliser application, cow number, milk production, and maize silage feeding, respectively. The term in square brackets calculates the nitrate leaching arising from relevant decision variables within the model. This is modified through the use of additional mitigations, described through:

where ${e}_{\vartheta}$ for $\vartheta =[1,2,\mathrm{...},5]$ is the proportional decrease in nitrate leaching achieved with mitigation $\vartheta $, ${E}_{1}$ is the extent to which low-rate effluent application is used, ${E}_{2}$ is the extent to which dairy shed innovations are used to reduce effluent volumes, ${E}_{3}$ is the extent to which deferred effluent application is used, ${P}_{h}$ is the number of cows in herd h maintained on a self-feeding pad for 10 weeks (70 days) from 21 April to 31 June, and NI is the number of hectares over which the nitrification inhibitor is used. The extent of the first four mitigations is computed per-cow for ease of computation. Cows on the feed pad may only be fed supplementary feed (i.e., concentrates, grass silage, and maize silage). Equation 10 and Equation 11 allow nitrate loads on the representative farm to be lowered through reducing N fertiliser application, stocking rate, or per-cow milk production or through the use of low N feed, low-rate effluent application, dairy shed innovations, deferred effluent application, a feed pad, and nitrification inhibitors.

The objective function is:

where ${p}^{milk}$ is the price received for milk solids (MS) ($ t^{−1}), ${p}^{cull}$ is the price received for one cull cow ($ cow^{−1}), there are 135 cull herds, ${p}^{calf}$ is the price received for one calf ($ calf^{−1}), $\psi $ is the calving rate, $\omega $ is the replacement rate, ${c}^{D}$ is the variable cost associated with a single cow ($ cow^{−1}), ${c}^{S}$ is the cost of conserving grass silage ($ per t DM), ${c}^{V}$ is the cost of maize silage ($ per t DM), ${c}^{K}$ is the cost of palm kernel extract ($ per t DM), ${c}^{F}$ is the cost of N fertiliser ($ t^{−1}), and ${c}^{FC}$ is the fixed cost of production ($ ha^{−1}).

#### 3.2. Parameter Values

The model represents a standard farm with allophanic soils in the Waikato region in the 2008/09 milking season. Allophanic soils are appropriate given that they are the predominant dairy farming soil in the region and also experience significant rates of nitrification. The farm is assumed to be 109 ha in size, in line with the typical farm reported by [

31]. All monetary values reported throughout the paper are stated in New Zealand dollars.

N fertiliser responses and minimum, maximum, and residual pasture masses are taken from [

32]. Feed energy, substitution, and utilisation rates are taken from [

22] and [

33]. Average pasture production is taken from [

34].

Predicting the impact of inhibitors on nutrient leaching and pasture growth is complex, as it depends on both climatic and soil factors. This study utilises realistic estimates based on expert opinion of likely responses achieved on typical farms. Increases in pasture production associated with the use of nitrification inhibitors are provided by [

35]. It is assumed that nitrification inhibitors increase pasture growth by 10 percent overall, with two-thirds of this increase experienced over July–December and one-third occurring over January–June. There is substantial debate surrounding the impact of nitrification inhibitor application on the rate of additional pasture production ([

21]). Thus, an extensive sensitivity analysis pertaining to this characteristic of this technology is presented in

Section 4.5. It is demonstrated that the magnitude of this factor has little to no bearing on the key findings of this study. The efficacy of inhibitors for reducing nitrate leaching (15 percent reduction) is taken from the midpoint of the range computed for the representative farm in the BMP toolbox [

20].

Energy demand for each cow attribute combination as a function of grazing, milk production, and pregnancy is computed using a simulation model constructed using information from [

33].

Leachate burdens are calculated for numerous combinations of maize silage use, milk production, N fertiliser use, and stocking rate using the OVERSEER model [

36]. The metamodel for Equation 10 is generated through linear regression of this data using SHAZAM econometric software [

37]. The efficacy of the alternative mitigations described in Equation 11 is taken as the midpoint from those ranges computed for the representative farm in the BMP toolbox [

20].

The OVERSEER model [

36] is the leading software used to identify the implications of alternative management strategies for nitrate leaching loads in New Zealand farming systems. It calculates the level of nitrate leached for a given farm based on the computation of an N balance that quantifies all N inputs and outputs. Key factors in the N balance for a farm are the stocking rate, animal type, level of supplementary feeding, level of N fertiliser application, clover incidence, annual rainfall level, the adoption of mitigation practices, and soil group. The processes represented in the model are based on annual averages. Thus, variability in leaching response across a series of years is not considered. This approach is common in hydrological models (e.g., [

12,

38]) because it simplifies use and parameterisation of the model. Extensive validation has occurred (e.g., [

39,

40]). For example, Whistler

et al. [

36] reported that OVERSEER had a 99 percent accuracy rate when predicting N leaching loads from a sample of dairy farms.

The milk price for 2008/09 ($5140 t

^{−1} MS) is taken from [

23]. Production costs are drawn from [

41,

42,

43,

44]. The costs of the mitigations described in Equation 11 are taken from [

20,

41,

44]. The standard cost of nitrification inhibitors is $160 ha

^{−1} [

44].

The IDEAL model consists of 13,605 equations and 8,656 decision variables. It is solved using NLP in the CONOPT3 solver in the Generalised Algebraic Modelling System (GAMS) [

45]. NLP is a useful technique for farm-level modelling given its capacity to incorporate multiple nonlinearities, its rapid solution time (e.g., the base model solves within 8 seconds), and economic interpretation.

Model verification is an important part of model development. Optimisation models, such as IDEAL, are difficult to validate [

46]. Nevertheless, the IDEAL model applied in the paper has undergone rigourous verification:

The structure of the model is based on that of an optimisation model that was subject to peer review in [

32].

The input values and model structure applied in this study have been validated through peer review [

22,

47].

The capacity of the model to report useful results compared with expected output has been investigated using near-optimal solution space analysis [

47,

48].

The ability of the model to report results that are consistent with real-world observations is presented in

Section 4.1.