Using multiobjective optimization models to establish healthy diets in Spain following Mediterranean standards

Abstract

Last reviews show how the Spanish consumption patterns have become away from the Mediterranean diet, traditionally consumed in Spain and widely supported by the nutritional expert community. Hence, the aim of this study is to explore and provide different alternatives to the current Spanish diet. The idea is to obtain a set of palatable diets fulfilling the nutritional requirements and conform the Mediterranean standards, while staying as close as possible to the current population pattern, under a budget constraint. In this context, different models are developed using multiobjective techniques. Additionally, this work defines an alternative diet more stable in comparison with the diets on the boundary of the feasible set. The consumption data used in this study is taken from Ministerio de Agricultura y Pesca, Alimentación y Medio Ambiente (in Spanish MAPAMA) that contains relevant information about the foods consumed in Spain. Using this data, each model has been solved with Matlab Software, obtaining different feasible diets, whose composition corresponds to the suggested daily intake for a Spanish adult. In any case, the budget constraint reduces the current cost and fulfills the nutritional requirements, attending to the Mediterranean standards. Results show different food baskets to guide the current Spanish diet towards the consumption of healthy foods in the appropriate proportions, going back to the diet traditionally consumed.

Appendices

Appendix 1: Complete models

EGP model

$$\begin{array}{*{20}l} {Min\quad \lambda D + \left( {1 -\lambda } \right)\mathop \sum \limits_{i = 1}^{101} w_{i} \left( {n_{i} + p_{i} } \right)} \hfill \\ {subject \, to} \hfill \\ {x_{i} + n_{i} - p_{i} = x_{i}^{0} \quad i = 1,2, \ldots .101} \hfill \\ {w_{i} \left( {n_{i} + p_{i} } \right) \le D\quad i = 1,2, \ldots .101} \hfill \\ \end{array}$$

Hard constraints

$$\begin{aligned} & \mathop \sum \limits_{i = 1}^{101} c_{i} x_{i} \le C \\ & \quad \alpha \cdot E \le \mathop \sum \limits_{i = 1}^{101} E_{i} x_{i} \le E \\ & \quad \alpha_{j}^{L} \le \frac{{\mathop \sum \nolimits_{i = 1}^{101} N_{ji}^{c} x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} E_{i} x_{i} }} \le \alpha_{j}^{U} \quad j = 1,2,3 \\ & \quad RN_{k}^{L} \le \mathop \sum \limits_{i = 1}^{101} N_{ki} x_{i} \le RN_{k}^{U} \quad k = 1,2, \ldots ,13 \\ & \quad \mathop \sum \limits_{i = 1}^{101} N_{Fi} x_{i} \ge 25 \\ & \quad \mathop \sum \limits_{i = 1}^{101} N_{Hi } x_{i} \le 300 \\ & \quad \frac{{\mathop \sum \nolimits_{i = 1}^{101} N_{Si}^{c} x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} E_{i} x_{i} }} \le 0.1 \\ & \quad \frac{{\mathop \sum \nolimits_{i = 1}^{101} PUFA_{i} x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} SFA_{i} x_{i} }} \ge 0.5 \\ & \quad \frac{{ \mathop \sum \nolimits_{i = 1}^{101} (PUFA_{i} + MUFA_{i} )x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} SFA_{i} x_{i} }} \ge 2 \\ & \quad TX_{i}^{L} \le x_{i } \le TX_{i}^{U} \quad i \in \left\{ {1,2, \ldots ,101} \right\} \\ & \quad TG_{r}^{L} \le \mathop \sum \limits_{{i \in G_{r} }} x_{i} \le TG_{r}^{U} \quad r \in \left\{ {1,2, \ldots ,10} \right\} \\ & \quad \mathop \sum \limits_{{i \in S_{1}^{j} }}^{{}} x_{i} \le \mathop \sum \limits_{{i \in S_{2}^{j} }}^{{}} x_{i} \quad j = 1,2, \ldots ,J \\ & \quad where\,S_{1}^{j} , S_{2}^{j} \subset \left\{ {x_{1} ,x_{2} , \ldots ,x_{101} } \right\} \\ & \quad PG_{r}^{L} \le \frac{{\mathop \sum \nolimits_{{i \in G_{r} }} x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} x_{i} }} \le PG_{r}^{U} \quad r = 1,2, \ldots ,10 \\ & \quad x_{i} ,n_{i} ,p_{i} \ge 0\quad i = 1,2, \ldots , 101 \\ \end{aligned}$$

TGP model

$$\begin{array}{*{20}l} {Min\,d} \hfill \\ {\begin{array}{*{20}l} {subject \, to} \hfill & {x_{i} + n_{i} - p_{i} = x_{i}^{0} } \hfill & {i = 1,2, \ldots .101} \hfill \\ {} \hfill & {\mu_{i}^{j} w_{i} \left( {n_{i} + p_{i} } \right) \le d} \hfill & {i = 1,2, \ldots .101} \hfill \\ \end{array} } \hfill \\ \end{array}$$

Hard constraints

$$\begin{aligned} &\mathop \sum \limits_{i = 1}^{101} c_{i} x_{i} \le C \hfill \\ & \quad \alpha \cdot E \le \mathop \sum \limits_{i = 1}^{101} E_{i} x_{i} \le E \hfill \\& \quad \alpha_{j}^{L} \le \frac{{\mathop \sum \nolimits_{i = 1}^{101} N_{ji}^{c} x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} E_{i} x_{i} }} \le \alpha_{j}^{U} \quad j = 1,2,3 \hfill \\& \quad RN_{k}^{L} \le \mathop \sum \limits_{i = 1}^{101} N_{ki} x_{i} \le RN_{k}^{U} \quad k = 1,2, \ldots ,13 \hfill \\& \quad \mathop \sum \limits_{i = 1}^{101} N_{Fi} x_{i} \ge 25 \hfill \\& \quad \mathop \sum \limits_{i = 1}^{101} N_{Hi } x_{i} \le 300 \hfill \\& \quad \frac{{\mathop \sum \nolimits_{i = 1}^{101} N_{Si}^{c} x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} E_{i} x_{i} }} \le 0.1 \hfill \\& \quad \frac{{\mathop \sum \nolimits_{i = 1}^{101} PUFA_{i} x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} SFA_{i} x_{i} }} \ge 0.5 \hfill \\& \quad \frac{{ \mathop \sum \nolimits_{i = 1}^{101} (PUFA_{i} + MUFA_{i} )x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} SFA_{i} x_{i} }} \ge 2 \hfill \\& \quad TX_{i}^{L} \le x_{i } \le TX_{i}^{U} \quad i \in \left\{ {1,2, \ldots ,101} \right\} \hfill \\& \quad TG_{r}^{L} \le \mathop \sum \limits_{{i \in G_{r} }} x_{i} \le TG_{r}^{U} \quad r \in \left\{ {1,2, \ldots ,10} \right\} \hfill \\& \quad \mathop \sum \limits_{{i \in S_{1}^{j} }}^{{}} x_{i} \le \mathop \sum \limits_{{i \in S_{2}^{j} }}^{{}} x_{i} \quad j = 1,2, \ldots ,J \hfill \\& \quad {\text{where}}\,S_{1}^{j} , S_{2}^{j} \left\{ {x_{1} ,x_{2} , \ldots ,x_{101} } \right\} \hfill \\& \quad PG_{r}^{L} \le \frac{{\mathop \sum \nolimits_{{i \in G_{r} }} x_{i} }}{{\mathop \sum \nolimits_{i = 1}^{101} x_{i} }} \le PG_{r}^{U} \quad r = 1,2, \ldots ,10 \hfill \\& \quad x_{i} ,n_{i} ,p_{i} \ge 0\quad i = 1,2, \ldots ,101 \hfill \\ \end{aligned}$$

Appendix 2: Supplementary material

See Tables 5, 6, 7, 8, 9, 10 and Fig. 5.

Table 5 Bounds for particular foods in expression (16)

Table 6 Bounds for particular food groups in expression (17)

Table 7 Food Sets in expression (18)

Table 8 Bounds for the percentage of intake in expression (19)

Table 9 Food sample, divided into groups, cost and consumption per capita in Spain 2013, $\overline{TGP\_1000}$ and $\overline{TGP\_1000noMD}$

Table 10 Average diets for TGP_10, TGP_100, TGP_1000

Fig. 5

Food groups consumption for different Average diets and the current diet