Diet optimization for nutrition per dollar

Providing the most cost-effective, nutritionally-sufficient, food recommendations for Americans Andrew Freiburger March 19, 2025 1 Abstract An increasing percentage of Americans find themselves financially strained in paying for weekly groceries. This often leads to buying the most calorically dense and processed foods, since these are cheaper, over healthy and ostensibly more expensive foods. There have been attempts in literature and at the USDA to create optimizations that find healthful dietary choices while adhering to strict budgets, but these attempts have generally been academic exercises and have failed to impact the demographic that they were trying to help. Motivated by this shortcoming, I leveraged the “official” and open-source USDA data sets pertaining to food prices, food nutritional composition, and human physiological needs to design an optimization model that finds a nutritionally adequate diet for the user, explored for myself herein, at the lowest possible cost. The optimum diet offered a surprisingly diverse set of ingredients and quantities that met my nutritional needs for $9.68 per day at current prices (January 2025). Keywords: Food, nutrition, USDA, optimization, open-source, LP, cost-effective, foodstamps Introduction The Food Stamp Act of 1964, passed as part of President Lyndon Johnson’s “Great Society” legislative acts, ushered in the first continuous program of direct government assistance in the purchasing of groceries (a concept first temporarily employed during the Great Depression). The program initially covered 0.2% of the US population but now subsidizes the groceries of 12.4% of Americans, peaking at 15.1% in 2012 (1), and costs ~$100B in just Federal expenditures (2). The impact of this program and the needs of its growing population of Americans must therefore be an increasing focus of public and private research.. Food choices at a grocery store are exceptionally difficult, despite seeming so ordinary, because there are thousands of options with varying nutritional densities and all 92 of human’s nutritional needs are only known by a tiny fraction of the population. Low-income Americans therefore often make food choices based on some compromise between taste, satiation, and price, which generally results in choosing foods with the most calories and least micronutritional value (3) and will negatively affect their well-being and perhaps the development of their children (4). Given the complexity of this situation and the severe absence of health and nutritional education, however, I believe that even the most health-conscious consumers make suboptimal dietary choices. Background The evident problem of finding nutritional-rich diets on strict budgets has been a topic of academic investigation for a century (5), but has received growing interest presumably because 2 of the advances in computational resources and object-oriented programming languages. A review from 2015 detailed efforts in this direction over the previous decade (6). Unfortunately, these studies generally only examined calories or the aggregate cost of different diet regimes, and their optimizations did not offer specific foods that meet all of the known micronutritional dietary goals. There are several studies that have included micronutritional data in their optimization, although these studies generally examined the affects of dietary choices on nutrition or cost and did not provide targeted dietary guidance (7, 8). The only study I found that offered specific dietary advice to meet nutritional needs at the least possible cost (9) was designed specifically for the Ghanan population, and included local wild foods as a part of the diet that obviously cannot be directly used for American consumers who do not have access to these free foods. The USDA’s Thrifty Food Plan (10) is the government’s attempt to provide dietary guidance through suggesting proportions of food groups (e.g. 10% leafy greens, 20% whole grains, etc) at various cost regimes. Unfortunately, this endeavor and its annual report fail to offer concise, actionable, advice for Americans earnestly looking to make the best dietary choices in the grocery aisles. After reviewing available literature, it appears that there has not been a comprehensive optimization model constructed that provides the Western consumer with suggestions of food selection that most cost-effectively satisfy accepted nutritional guidelines. I believe that this is a huge unaddressed market and would dramatically improve the health of the people in our communities who are perhaps in the most need of aid. Model Formulation Data aggregation Three data sets were acquired for my optimization model: food pricing, physiological needs, and food nutrients. I elected to source these data from the USDA, assuming that sourcing all of these data from the same government entity would create the most robust optimization and streamline connecting the disparate data. These assumptions unfortunately proved false: the pricing data was incredibly sparse (150 foods, and only fruits and vegetables), the physiological data contained less than half of the needs identified in literature, and the data sources used different namespaces so they were not interoperable. The following adjustments were therefore necessary for each of the respective data sources to correct some oversights and standardize the data sources so that they could be integrated into the optimization herein. 3 Pricing data The pricing data was sourced from the Economic Research Service Division of the USDA(11), which defined only 155 food items and had extensive duplication: e.g. fresh raspberries and frozen raspberries, or fresh artichoke and canned artichoke. Frozen foods have minimal differences with fresh foods, and canned foods are much less nutritionally defined in the food composition data than fresh foods, so frozen and canned foods were omitted from consideration to create this optimization. The resulting 69 foods were the basis of the optimization, and are listed in Table A1. After the optimization failed to converge with just these variables, I noticed that it was not possible to meet the fat requirements given these food options. I therefore added flaxseeds to the optimization as a low-cost, whole-food, source of fat and protein. The pricing data was sourced from Amazon (12), the yield was assumed to be 1 (since the whole seed is consumed), and the cupEquivalentSize (defined by the USDA as weight per cup of a food) derived from an analytical website (13). Physiological data Physiological needs for myself – Active, 28 years-old man, 5’10”, 150lb – were sourced from the USDA “DRI [daily required intake] Calculator for Healthcare Professionals” and processed into Table 1. While this is not the most comprehensive list of nutritional needs, containing less than half of the nutrients that are actually required by humans, it was selected because it is an “official” and open-source set of nutritional needs and covers the most important nutrients. The assumption likely made by the USDA is that by following these nutritional guidelines is that by satisfying these nutritional needs through eating whole foods, one would likely also satisfy nutritional needs for the other trace nutrients that are not captured here. Several nutrients – Chromium, Vitamin B12, Vitamin D, Cholesterol, Iodine, Molybdenum, and Phosphorus – were contained in less than 6 foods, and were therefore omitted to prevent the optimization from either becoming infeasible (such as Vitamin D and Iodine which were in no foods) or from forcing the inclusion of the few foods that contain these nutrients (such as Molybdenum in just 3 foods). This is partly based on the belief that some of these nutrients are present in most of whole foods despite them not being explicitly measured in these data: i.e. I believe that unreported nutrient abundance in foods cannot be interpreted as a measured zero abundance of a nutrient in a food. Table 1: The nutritional requirements that are used for the optimization, with changes noted. Notably, all of the “ND” (Not Determined) upper bounds were changed to 10,000 of the respective units, which are not included in the “Changes” column to highlight the more impactful changes to the physiological nutrient needs. All of the units were also standardized – grams for 4 macronutrients and milligrams for micronutrients – to facilitate the direct comparison between nutrients and easier programming. Nutrient Lower bound Upper bound Modifications Carbohydrate 331g 478g Total Fiber 41g ND Protein 54g ND The range was changed to [100,150] to meet my specific exercise needs Fat 65g 114g Saturated fatty acids “As low as possible” Upper bound of 10% total calories (33.3g) based on recent literature Linolenic Acid 1.6g ND Linoleic Acid 17g ND Dietary Cholesterol “As low as possible” Upper bound of 800mg, as a feasible upper limit (~4 eggs worth). Total Water NA 3.7L A low of 0.37 (10% o max) was created since food might be too dry below this. Energy 2400kcal 3200kcal Added this term because it was NOT included in the USDA recommendations Vitamins Vitamin A 900 mcg 3,000 mcg Vitamin C 90 mg 2,000 mg Vitamin D 15 mcg 100 mcg Vitamin B6 1.3 mg 100 mg Vitamin E 15 mg 1,000 mg Vitamin K 120 mcg ND Thiamin 1.2 mg ND Vitamin B12 2.4 mcg ND Riboflavin 1.3 mg ND Folate 400 mcg 1,000 mcg The upper bound was changed to 3000mcg to reflect more recent literature Niacin 16 mg 35 mg Choline 0.55 g 3.5 g Pantothenic Acid 5 mg ND Biotin 30 mcg ND 5 Carotenoids NA ND The range was arbitrarily defined as [0,1E5], since carotenoids are generally not nutritional Minerals Calcium 1,000 mg 2,500 mg Chloride 2.3 g 3.6 g Chromium 35 mcg ND Copper 900 mcg 10,000 mcg Fluoride 4 mg 10 mg Iodine 150 mcg 1,100 mcg Iron 8 mg 45 mg Magnesium 400 mg 350 mg The upper bound here is amazingly lower than the lower bound, so the upper bound was created to be 10,000mg Manganese 2.3 mg 11 mg Molybdenum 45 mcg 2,000 mcg Phosphorus 0.7 g 4 g Potassium 3,400 mg ND The ND here was replaced with 15,000mg because potassium is generally not a concern in excess and was too close to the lower bound at the 1E4mg default. Selenium 55 mcg 400 mcg Sodium 1,500 mg 2,300 mg Zinc 11 mg 40 mg Food composition data The nutritional composition of various foods was sourced from the 10/24 release of the USDA’s “FoodData Central” collection of food data (14). The resulting zip file contained ~50 files, of both data and metadata. The metadata, in addition to documented terminology (15), were used to decypher the IDs from the data files. The resulting data files contained 18.1M nutritional data points and 1.9M foods. Given the limited scope of pricing data, the array of food options were filtered for just foods containing “ raw” to match the exclusive list of fruits and vegetables 6 from the pricing data, which resulted in 2254 foods. These foods included raw meats as well as fruits and vegetables, but these were filtered later in the optimization code. Model construction The model consists of a variable for each food item – ultimately 70 including flaxseeds – that were bound [0,5] to limit consolidation of the solver to just a few nutritionally dense and cheap foods. This is also designed to make the dietary suggestions more attractive for people who like to eat some diversity of foods over the course of a week. The primary constraints in the system is 𝑙𝑏𝑛≤ 𝑛 𝑁 ∑(𝑞𝑛*𝑓𝑛 ∀ 𝑓∈𝐹) ≤ 𝑢𝑏𝑛 where is the quantity of the nutrient in food for the foods that contain nutrient . 𝑞𝑛𝑛 𝑓𝑛𝐹 𝑛∈𝑁 Each sum for each nutrient is bounded by the and USDA physiological requirements for 𝑙𝑏𝑛𝑢𝑏𝑛 each nutrient displayed in Table 1. Another constraint was added for volume 5 [𝑐𝑢𝑝𝑠]≤ 𝑓𝐹 ∑(𝑣𝑜𝑙𝑓*𝑓𝑛) ≤ 20 [𝑐𝑢𝑝𝑠] to ensure that the volume of food is reasonable. The objective function 𝑚𝑖𝑛 𝑓𝐹 ∑(𝑝𝑓*𝑓) finds the total minimal yielded-weight price for 100-gram units of all food . 𝑝𝑓𝑓∈𝐹 Solution Algorithm The optimization was solved through the Simplex algorithm via the GLPK (GNU Linear Programming Kit) solver (16), through the Optlang and OptlangHelper Python package. All variables were continuous, so the problem was strictly a linear program and not a MILP. Results The optimized diet was $9.68 per day and consisted of the foods in Table 2. The diet was highly skewed towards pinto beans and carrots, presumably as sources of protein & vitamins 7 and minerals, respectively. Interestingly, only 38 grams (~¼ of a cup) of flax seeds were needed to balance the diet, and specifically meet the fatty acid requirements, from the otherwise completely whole-food vegan diet. The respective nutrient quantities of this optimized diet were visualized between lower and upper bounds in Figure 1, which, importantly, indicates which nutrient constraints were limiting the optimization. Table 2: The gram amount of each food that satisfies all of the constraints for nutritional needs and volume requirements. Food Carrots Pinto beans Blueberries Collards Corn Celery Raspberries Cucumber Flax seeds Amount (g) 500 500 249 197 173 155 118 75 38 The optimum ingredients from Table 2 were passed into ChatGPT to create an actionable dietary regime that could be followed. The prompt read: “create a day of meals from the following ingredients, without adding any calories. The numbers are denominated in grams of the given food.” The meal regime proposed by ChatGPT is contained in Figure 2. The optimization could conceivably be run iteratively to simulate an entire week, and then the cooking instructions for the full week would be specified by ChatGPT, effectively emulating the role of a personal nutritionist. 8 9 Okra 3.70475 0.8345 0.36375 1.6513 Onions 1.1062 0.9 0.3527 0.4335 Oranges 1.4624 0.68 0.4079 0.8771 Papaya 4.41145 0.81 0.23145 0.9105 Peaches 2.89215 0.98 0.3362 0.98395 Pears 1.8472 0.9 0.3638 0.7466 Pineapple 3.81515 0.755 0.25905 0.7612 Pinto beans 1.4173 2.4692 0.3858 0.2215 Plum 4.4276 0.97 0.2756 1.0818 Pomegranate 2.4672 0.56 0.3417 1.5055 Potatoes 0.8166 0.8113 0.2646 0.2663 Radish 1.8126 0.9 0.2756 0.555 Raspberries 6.9464 0.98 0.3252 2.30605 Red peppers 1.8742 0.82 0.2646 0.6047 Spinach 3.5074667 0.8486666 0.2866 1.169 Strawberries 3.15515 0.97 0.3252 1.0573 Sweet potatoes 1.1565 0.8818 0.4409 0.5782 Tomatoes 2.435 0.91 0.3748 1.0028333 Turnip greens 2.72095 0.763 0.3362 1.19535 Watermelon 0.382 0.52 0.3307 0.2429 Zucchini 1.6359 0.7695 0.3968 0.8437 All data processing, optimization design, simulation, and visualization occurred in a single Python Notebook, displayed in the following pages. Python was used because it connects to workflows developed for my research and allows this project to more easily scale into a proper tool with a user interface. 16