X i∈S X p∈P epi,p,fxi,p ∀ f ∈ F (2.6)
The demand is determined by α, which controls the share of food category/product de- mand in the model, given by,
α Dk,i≤
X
p∈P
bp,kzi,p ∀ k ∈ K, i ∈ C (2.7)
and the nutrient demand, di,n≤
X
p∈P
ap,nzi,p ∀ n ∈ N, i ∈ C (2.8)
with the consumption of product categories constrained by portion sizes: portk,i≥
X
p∈P
bp,kzi,p ∀ k ∈ K, i ∈ C (2.9)
The quantity of products consumed has to be transported to the consumer: zj,p≤ X i∈S X m∈M yi,j,p,m ∀ p ∈ P, j ∈ C (2.10)
The quantity of products transported to the consumer has to be less or equal to the quantity produced: X j∈C X m∈M yi,j,p,m≤ xi,p ∀ p ∈ P, i ∈ S (2.11)
The blending and resource constraints ensure that all the required resources needed for production of a product are available at the production location:
vp,i,r= convr,pxi,p ∀ p ∈ P0, r ∈ P, i ∈ S (2.12)
X p∈P relr,pvp,j,r= X m∈M X i∈S ui,j,r,m ∀ r ∈ P, j ∈ S (2.13)
Produced products have to be transported to the place where they are needed as resources or consumed and otherwise are classified as waste:
X m∈M X j∈S ui,j,p,m+ X m∈M X l∈C
yi,l,p,m+ wi,p= xi,p ∀ p ∈ P, i ∈ S (2.14)
The by-product constraints link the production of a product to its by-products:
xi,p= ratq,pxi,q ∀ p ∈ P, q ∈ P, i ∈ S (2.15)
Defining some of the variables as dynamic in order to reduce the size of the model, the final model includes over 156 thousand rows (constraints) and more than 830 thousand variables. However, given its linear and continuous nature the model can be solved quickly by any standard LP solver.
2.5
Illustrative Case Description and Data Input
2.5.1 Illustrative case
The applicability of the model is illustrated based on a real life case study related to cur- rent consumption patterns in the Netherlands. In connection to the dietary contribution of the beef and dairy system, 5 key nutrients are identified: protein, iron, zinc, calcium and vitamin B12. To allow for dietary replacement, the plant-based alternatives are se- lected based on their nutrient profile with respect to these 5 key nutrients. It should be
consumption. Together with the resources and feed materials needed to supply these products, this results in a network of 72 different products. Examples of such products are feed ingredients for the beef and dairy system or different grains and flour for the production of bread.
The case furthermore includes a total of 39 locations for the sourcing and production of products. These locations are product-dependent, and hence the amount and composi- tion of feasible locations varies per product. Seven different modes of transport have been included in the model, namely air, inland waterways, rail, truck as well as three distance related sea shipping modes. The availability of a mode choice is dependent on the in- frastructure and geographical position of a country, while distances between countries are location and mode dependent. Distances within a country are set to be the same for all available modes and all countries (10km). On a global scale, where products are sourced from all over the world, the transportation within a country is assumed to have a minor impact in comparison to the travel distances between countries. Moreover, we assume that agricultural production locations and the facilities for further processing are likely to be located in close proximity to one another, in order to avoid product decay, save time and delivery costs. While this might be a strong assumption, it has been shown in the scientific literature that within agricultural systems the impact associated with transportation plays only a minor role (Weber and Matthews, 2008; Garnett, 2011). 2.5.2 Data input
The nutritional data for the selected food products were retrieved from the NEVO database (RIVM, 2013). Environmental impact values are based on country specific life cycle inventory data, containing all the inputs and outputs for each specific production and processing step. We apply the ReCiPe impact assessment method (Goedkoop et al., 2013) to translate the emissions and resource extractions into the four environmental in- dicators selected in section 2.3.1. The functional unit, referring to the quantification of the product to which the inputs and outputs relate, is expressed per kilogram of product. In the case of multiple products resulting from the same production step, an allocation method needs to be selected in order to divide the process inputs and outputs among the different products. The chosen allocation method in this study is economic allocation and thus based on the economic value of the products. Note, that no specific system boundaries for the scope of the LCA are applied as each step of the life cycle is included as a separate choice in the model by using life cycle inventory data, instead of using ag- gregated LCA impact values. All the life cycle inventory data as well as the (by-)product relations and conversion factors are extracted from the Agri-footprint database (Blonk Agri-footprint BV, 2015a,b). The environmental impact data for the different transport modes are taken from the same database. The cost figures for the different products,
transportation and processing steps were obtained as part of an extensive data collection from the literature and other sources. In case cost figures were not readily available, the data was calculated based on information from other sources such as labour and energy requirements and the country specific labour and energy costs.
2.5.3 Scenarios
Using the data described above, the model is tested for different scenarios that are then compared with respect to their results. Further specifications of the individual scenarios are detailed in the following.
• Base Case: The demand for different food groups (α = 1) is taken from cur- rent daily consumption data based on the Dutch dietary consumption survey (Van Rossum et al., 2016) and scaled to the population level. The model is op- timised for cost which seems to be most aligned with current consumer choice, thus representing a kind of status quo. Dietary supplements in the form of vitamin B12 supplements are not considered in this case, as no demand for this food category is included. The base case presents a reference case for the other scenarios and will be the basis for comparison.
• Status Quo Scenarios: Given the same setting as in the base case the model is optimised also with respect to different environmental objectives: climate change, land use, water use and fossil fuel depletion.
• Supplement Scenarios: In these scenarios the demand is no longer expressed in the form of food groups but in the form of a nutrient demand (α = 0). Therefore, the food category demand Dkiof the base case is converted via data on nutrient content
(anp) to obtain the nutrient demand din. Vitamin B12 supplementation is possible
to provide the required amount while palatability constraints in relation to portion sizes are added to ensure acceptability and feasibility of the dietary consumption. Provided that a plant-based diet requires a different dietary consumption, portion sizes are assumed to be higher than current standard portion sizes for plant-based food products. Given these specifications, the model is optimised for the different objectives: cost, climate change, land use, water use and fossil fuel depletion. Furthermore, a multi-objective analysis is carried out using compromise programming (Zeleny, 1973) for the different demand scenarios and the epsilon-method (Ehrgott, 2009) for selected indicators in the case of nutrient demand.