2.3. Optimization and Comparison of FFQ Length

MILP was used to minimize the number of food items in the food list for the two different aggregation levels. MILP is an optimization method that combines linear programming and integer programming. This method is suitable for selecting an optimal number of food items for an FFQ, as the decision variables x_n for the food item n must take integer values [14].

Given a set of foods, N, and a set of nutrients, J, the optimization problem can be formulated as follows:

The optimization involves one objective function to minimize the number of food items and two sets of constraints, both comprising 40 nutrient specific constrains, i.e., j ∈ {1, …, 40}. The two sets of constraints reflect two previously mentioned requirements of food items in an FFQ: high nutrient coverage and large interindividual variability of nutrients. The algorithm iteratively searches for the combination of food items that minimizes the number of items while satisfying the two sets of constraints.

The nutrient coverage constraints aim to select food items that account for a large proportion of the intake of the selected nutrients in the target population. The constraints ensure that the percentual coverage for selected nutrients is greater or equal to a specific threshold value b.

The variance coverage constraints aim to select food items that account for a large interindividual variance in nutrient intakes. To determine food items with a high contribution to variance, the conventional approach would involve using R². However, there are two challenges in applying R² in this context. First, R² cannot be formulated as a linear equation, which is necessary for MILP. Second, the contribution of a food item to R² is not explicitly determined but depends on the food items already included in the model. To address these challenges, food items were selected based on their percentual contribution to the sum of variances of a specific nutrient. These variance coverage constraints ensure that the percentual contribution of selected food items to the sum of variances of the overall intake of nutrient j is greater or equal to a specific threshold, b.

To compare optimal food sets across different sets of nutrients, the number of nutrients in the optimization was incrementally increased. Overall, a maximum of 40 nutrients that comprehensively reflect a general diet were considered in the optimization. First, optimization for energy intake was conducted, then the intakes of carbohydrates, protein and fat were added and eventually, the set of nutrients considered was extended with 36 vitamins and minerals. All nutrients included in the optimization are listed in Appendix A Table A1.

To analyze the development of the number of food items, MILP was conducted for different threshold values, b. For both sets of constraints, the same values for b were used. The initially chosen value of b was 0.60, which gradually increased by 0.05 until 0.95 was reached. The final optimization was run with b = 0.99.

Calculations were conducted with the statistical software R version 4.3.0. Optimization was solved using the R package ROI 1.0-1, which provides an interface for various solvers. In this case, the solver GLPK was used.

To test which of the two sets of constraints required more food items, the optimization was additionally performed with only one of the sets of constraints, respectively. The resulting food lists were compared regarding the number and the kind of food items. It was assumed that food items with a large nutrient coverage also have a large coverage of the sum of variances. Therefore, Pearson’s correlation coefficients between nutrient coverage and variance coverage were calculated for energy, carbohydrates, protein, and fat intakes.

To identify the relative change in the number of optimal food items, growth rates were calculated with

where N0 is the number of food items resulting from MILP with a lower proportion of both nutrient coverage and variance coverage. N1 is the number of food items resulting from MILP with a 5% higher proportion of both nutrient coverage and variance coverage.