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| Reference | Objective function (s) | Decision variable (s) | Constraint (s) | Focus | Mathematical approach used |
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| Darmon et al. [47] | Minimise total departure from the mean food intake | Amount/weight of food | Constraints of energy content, food constraints to ensure compatibility with observed dietary patterns, cost constraints set at a maximum level | | Linear programming |
| Ferguson et al. [38] | Minimise difference in the mean percentage of energy contributed by different food groups between modelled and observed diets | Weights of different foods | Nutrient (energy constraint, RNIs for macro and micronutrients), acceptability (portion size, percentiles on food groups) | To develop a rigorous, reproducible, and objective approach based on linear programming analysis, which can be used to formulate practical FBDGs for high-risk populations | Linear programming |
| Darmon et al. [32] | Minimised diet changes needed to meet nutritional requirements | | Palatability, nutritional, cost | Isoenergetic diets that meet the current nutrition requirements | Linear programming |
| Masset et al. [10] | Minimise departure of optimised diet from the observed quantity of food eaten by the reference population | Quantity of food | Nutritional (energy, macro, and micronutrient constraints) | To use mathematical optimisation tools for dietary guidelines to prevent cancer | Linear programming |
| Maillot et al. [67] | Optimise diet close to observed diet | Quantity of foods | Dietary energy, nutritional targets, maximal quantities of foods, diet weight | Describe dietary changes needed to achieve nutritional recommendations | Linear programming |
| Clerfeuille et al. [68] | Minimise deviations between optimised and observed diet | Weight of foods | Nutritional adequacy (selected nutrients), the constraint on food quantities | To estimate the number of portions of the different milk-based food categories that fit into nutritionally adequate diets | Linear programming—BASAL model |
| Metzgar et al. [69] | Minimise the sum of differences in food intake | Amount of food | Cost (does not exceed a maximum level), the constraint on some food categories (kept at a maximum of 0), nutrient (minimum and maximum levels, DRIs) | This dietary optimisation program uses common food choices to build a suitable diet (Paleolithic diet) | Linear programming |
| Okubo et al. [16] | Minimise the deviation in food intake between the observed and optimised food intake patterns | Quantity of food | Nutritional (meet DRIs), upper limits of each food | To translate nutrient-based recommendations into realistic nutritionally optimum combinations of food by integrating local and culture-specific foods | Linear programming (infused goal programming) |
| Perignon et al. [70] | Minimise departure from the observed diet | Food price | Nutrition | To evaluate the compatibility among the affordability dimensions of diet sustainability | Linear programming |
| Horgan et al. [71] | Minimising dietary changes from their current reported intake (to meet dietary recommendations and GHGE targets) | Weight of food | Nutrient constraints based on dietary reference intake, constraint set on meat and fish, 25% reduction constraint on GHGE, the lower and upper limit on individual foods | To determine the range of dietary changes that achieve dietary recommendations and reduce GHGE (making little changes to current dietary intakes) | Linear programming |
| Scarborough et al. [72] | Minimise deviation between the cost of observed and modelled diets | Amount of food | Dietary recommendations | To model food group consumption and price of diet associated with meeting dietary recommendations with minimum deviation from current diet to redevelop FBDGs (setting, UK) | Nonlinear generalised reduced gradient algorithm |
| Maillot et al. [33] | A nutritionally adequate isocaloric diet that stayed close to the observed diet | The amount of food available | Acceptability (food most frequently eaten), nutrient constraint (based on dietary reference) | Nutritionally adequate isocaloric diet (a modelled diet that came as close as possible to the corresponding observed diet) | Linear programming |
| Raymond et al. [73] | An affordable diet that achieves DRIs for selected nutrients (objective function was to minimise deviations between populations’ food groups and dietary standards) | Grams of food | Amount of food used by the population, so it does not exceed nutrient constraint, acceptability (constraints were set on grams on each food group) | The primary objective of this study was to ascertain if a practical and affordable diet that meets DRIs for some selected nutrients can be developed for rural 6–23-month-old children in Tanzania | Linear goal programming |
| Kramer et al. [74] | Minimisation of changes to the current average diet | Weight of food | Nutritional requirements, environmental targets | The model was done to mimic the current consumer behaviour | Linear programming |
| Raymond et al. [75] | Minimise deviation between modelled and observed diet patterns while meeting dietary standards | Quantity of food | Nutritional (RNIs) | To establish if a realistic and inexpensive diet that meets set nutritional goals for rural women (pregnant and lactating) can be formulated from locally available foods in Tanzania | Linear programming (using goal programming) |
| Barré et al. [76] | Minimum deviation from the observed diet | Quantities of food | Environmental, constraint on dietary macronutrients and RDA, acceptability (a constraint on quantities of food subgroups, bovine meat, and dairy products co-constrained), cost (to remain lower or equal to observed current cost) | The objective was to evaluate the impact of nutrient bioavailability and co-production links considerations on the dietary changes needed—especially regarding meat—to improve diet sustainability | Linear and nonlinear programming |
| Brink et al. [77] | Sustainable diets that are close to the observed patterns | Amount of food groups | Constraints on food groups, environmental considerations, the minimum and maximum constraints on nutrients and energy, acceptability (closeness of modelled diets to current pattern) | To obtain healthy and sustainable food-based dietary guidelines (FBDGs) for different target groups in the Netherlands | Linear programming |
| Kim and Kim [27] | Optimal nutrient levels (minimise deviations) | Amount of food | The upper and lower limits of calories, consumption, and amount of nutrient | An absolute optimal decision that provides the best possible nutrient combination | Linear programming |
| Johnson-Down et al. [78] | The deviation between modelled and observed diet | Gram of food | Cost, nutrient requirement (EAR) | Satisfy medicine macro and micronutrient requirements in healthy individuals based on available foods consumed by the defined population (at minimum cost) | Linear programming |
| Verly-Jr et al. [39] | Optimised diets with food quantities at the lowest deviation from the observed diets | The amount of food | Nutrient (WHO guidelines for NCDs, recommended requirements), acceptability (boundaries limiting changes in food quantities, STRICT and FLEX models on foods), GHGE (stepwise reduction from 10%) | Identify the dietary changes to improve nutrition and reduce diet-related greenhouse gas emissions (GHGE) in Brazil, with consideration given to food habits and prices | Linear programming |
| Gazan et al. [79] | Minimise deviation between observed and optimised diets | Quantity of foods | Constraint on recommended intakes, 30% reduction of carbon impact constraint | Explore the feasibility of plant-based “dairy-like” products in achieving sustainable diets | Linear programming |
| Rocabois et al. [80] | Minimise deviation between observed and modelled diet | Quantity of foods | Constraints on nutritional requirements and environmental impact targets | Develop an approach (INDIGOO) to design sustainable diets with nutrient requirements and achieve set environmental targets | Linear programming |
| Vasilogou et al. [81] | To minimise the deviation between modelled and typical diet consumed in America | Quantity of food | Nutrient and food group constraints | Assess the quality of simulated food patterns that have reduced animal protein using NHANES data from 2017-2018 | Mixed integer linear programming |
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