Can Fast Food Eating Patterns Be Healthy? An Optimization-Based Analysis of Popular US Fast-Food Chains

Abstract

This paper applies linear programming modeling to determine whether USDA dietary guidelines can be satisfied by an optimal selection of items from the current menus of popular fast-food chains. More specifically, optimization models aiming to determine diets that minimize one of the three objectives (calories from sugars, calories from fat, or sodium consumption), while satisfying the key USDA dietary guidelines are proposed. The models are applied to the menu data of 44 popular fast food restaurants available via Fast Food Nutrition database. A detailed study is performed for 13 age-sex groups specified in USDA guidelines. It is found that only one of the 44 restaurants was found acceptable for each of the 13 age-sex groups, and 10 restaurants were found acceptable for at least one group. Moreover, removing only a small fraction of foods from the combined fast-food menu including all the items from the 44 restaurants eliminates the possibility of designing an acceptable diet based on the remaining menu items. Therefore, it is concluded that the available fast-food options provide very limited opportunities for supporting healthy eating patterns that would satisfy the USDA dietary guidelines. Optimization methods are not only useful for determining the acceptability of a menu with respect to given guidelines, but can also be utilized towards developing better dietary guidelines and healthier menus.

1 Introduction

The increasing accessibility of nutrition data provides unprecedented opportunities for detailed quantitative analysis and comparison of various diets. The US Department of Agriculture (USDA) and the US Department of Health and Human Services (HHS) maintain the Dietary Guidelines for Americans [1], updated every 5 years. This document provides guidelines and recommendations for healthy diets based on state-of-the-art research; see Fig. 1 for a brief summary. In particular, it is used as the basis for the Healthy Eating Index (HEI), a quantitative metric developed by Food and Nutrition Service of the US Department of Agriculture, which aims to measure diet quality with respect to the key recommendations of the Dietary Guidelines for Americans [2].

Fig. 1

A brief summary of key guidelines and recommendations provided in [1]

According to the US Census Bureau, estimated fast-food (limited service restaurants) sales in the USA increased from 87.26 billion dollars in 1992 to 392.44 billion dollars in 2021 [3]. During 2013–2016, 36.6% of adults consumed fast food on a given day, and the percentage of consumption of fast food by adults decreased with age and increased with increasing family income [4]. A similar study reported that 36.3% of children and adolescents consumed fast food on a given day [5].

The focus of this paper is on the quality of fast-food menus, a topic that has attracted considerable attention both in the media and in research literature. Unsurprisingly, the narrative focus has been on the adverse effects the fast-food nutrition has on the public heath. More specifically, the published studies have connected fast foods with obesity and increased risk of diabetes [6,7,8]. While major fast-food restaurant chains responded to these widespread criticisms by introducing numerous healthier options to their menus, the improvements have arguably been insignificant [9], and concerns over the low quality of fast food have persisted. For example, Kirkpatrick et al. [10] evaluated the menus of five major fast-food chains (Burger King, McDonald’s, Subway, Taco Bell, and Wendy’s) by determining their Healthy Eating Index-2005 (HEI-2005) scores. They concluded that “fast-food menu offerings vary in dietary quality, but are consistently poor.”

It should be noted, however, that the studies concerning the fast-food menus focused on their overall quality, measured across all the offerings. In particular, according to the authors of [10], “Each unique item was counted once to provide a sense of the overall quality of each chain’s offerings.” Notwithstanding the validity and merits of this approach for evaluating the general quality of a menu, it has obvious limitations. For example, it does not take into account the popularity of each menu item and provides no information on the level of customer demand for the newly introduced healthy options. Also, an arguably more important question is the following: Do fast-food restaurant menus provide options for forming eating patterns that satisfy the recommendations of the Dietary Guidelines for Americans? The objective of this work is to address this question, which naturally falls within the emerging discipline of nutrition informatics, defined as“the effective retrieval, organization, storage and optimum use of information, data and knowledge for food and nutrition related problem solving and decision-making” [11]. This nontrivial task will be approached by taking advantage of mathematical optimization tools.

Nutrition planning has a long and illustrious history in operations research and optimization modeling [12, 13]. In fact, the classic diet problem is universally used as a textbook example of linear optimization modeling. The first known instance of this problem was presented by Stigler in 1945 [14] and was stated as follows:

For a moderately active man weighing 154 pounds, how much of each of 77 foods should be eaten on a daily basis so that the man’s intake of nine nutrients will be at least equal to the recommended dietary allowances (RDAs) suggested by the National Research Council in 1943, with the cost of the diet being minimal?

In his original work, Stigler provided a heuristic solution with total cost of $39.93 in 1939 US dollars. Several years later, the development of Dantzig’s simplex algorithm allowed them to solve the problem to global optimally with the exact solution value of $39.69. This was the first ever “large scale” example of a problem solved by the simplex method; a task that took 120 clerk-days (with nine clerks operating desk calculators) to complete in 1947 [12, 15]. Modern optimization solvers handle problems of this scale in a fraction of a second and routinely solve linear models with millions of variables and constraints.

While the objective of the classic diet problem was to minimize the cost of a diet satisfying the recommended dietary allowances, this study is not concerned with cost. Instead, the aim is to minimize the consumption of three entities specified in the key quantitative recommendations of Dietary Guidelines for Americans [1], as described in Fig. 1: (1) calories from sugars, (2) calories from fats, and (3) sodium. One of these three entities is used in the minimization objective, and the remaining two entities are incorporated in the constraints, along with the recommendations for other intakes, including carbohydrates, dietary fiber, protein, and calories. This approach allows us to not only assess the feasibility of healthy eating patterns based on a fast-food menu, but also provide a comparison and ranking of the menus of different fast-food restaurant chains based on the specific quantitative criteria.

The remainder of this article is organized as follows. Section 2 describes the proposed optimization models and describes the data used. Section 3 reports the results of empirical study based on the menu data of 44 fast-food restaurants, and Section 4 concludes the paper. Details of optimal solutions obtained for each restaurant and age-sex group are reported in Appendix.

2 Methods

This section provides a detailed description of the proposed optimization models based on the recommendations in the 2020–2025 edition of Dietary Guidelines for Americans [1]. Specifically, the focus is on minimizing the consumption of nutrients specified in the key quantitative recommendations (that is, calories from sugars, calories from fats, and sodium), subject to the limits on the most common nutrients described in the daily nutritional goals in [1]. While the document provides intake recommendations for a number of macronutrients, minerals, and vitamins, only those of the entities that are reported in the Fast Food Nutrition website [16] are taken into account in the considered models. To the best of the authors’ knowledge, this is the most comprehensive publicly available resource on fast-food nutrition facts; hence, it is utilized in this study. The website provides nutrition facts for over 40 popular fast-food restaurants operating in the USA. More specifically, for each menu item represented in the online database, the serving size and the per-serving amount of the following nutrients are given:

• Macronutrients: carbohydrates, dietary fiber, protein, fat, cholesterol, and sugars;

• Minerals: calcium, iron, and sodium;

• Vitamins: vitamin A and vitamin C.

Figure 2 provides an illustration of the format of presentation of the nutrition facts on the website. As can be seen from the figure, the data for vitamins, calcium, and iron are listed in terms of percentage of the daily value, without any specific reference. Moreover, many of the items on the website contain no entries for vitamins. Hence, to ensure the consistency of analysis, the vitamin, calcium, and iron requirements will not be included in the proposed mathematical models. Table 1 displays the daily nutritional goals with respect to each of the nutrients included in the study at the specified calorie level for each age-sex group, ages 2 and older. (It should be noted that [1] also provides guidelines for infants, toddlers, pregnant women, and lactating women, as well as recommendations for calorie levels depending on a life style. While this study can be easily extended to the additional groups, it focuses on the groups included in the table.)

Fig. 2

A screenshot of nutrition facts for McDonald’s Big Mac from Fast Food Nutrition website [16]

Next, the details of the mathematical models, stated for a given restaurant and age-sex group, are described.

Sets, indices, and parameters

The following sets, indices, and parameters are employed in the proposed models to describe the nutrients, menu items, and nutritional characteristics and requirements for each meal.

• $i\in N$: the i-th element of the set of n considered nutrients $N=\{1,\ldots ,n\}$. The set of considered nutrients consists of the following elements: (1) fat, (2) sugar, (3) sodium, (4) carbohydrates, (5) dietary fiber, (6) protein, and (7) calories, so $n=7$ in this study.

• $j\in M$: the j-th element of the set of m menu items $M=\{1,\ldots ,m\}$ of the considered restaurant. The set M and the number m of menu items depend on the specific restaurant.

• $k \in C$: the k-th element of the set of food categories $C=\{1,\ldots ,c\}$. The following categories were considered: (1) Appetizers, (2) Beverages, (3) Breakfast, (4) Condiments, (5) Desserts, (6) General, (7) Pizzas, (8) Salads, (9) Sandwiches, (10) Sides, (11) Soups, (12) Seafood, and (13) Tacos. That is, $c=13$ in this study.

• $a_{ij}, i\in N, j \in M:$ the amount of nutrient i contained in one serving of menu item j.

• $c_{jk}, j\in M, k \in C:$ the amount a serving of menu item j contributes to food category k.

• $b_k, k\in C:$ the upper bound on the amount of food from category k in a diet.

• $l_i, i\in N:$ the lower bound on the daily consumption of nutrient i.

• $u_i, i\in N:$ the upper bound on the daily consumption of nutrient i.

• $l_i^c, i\in N:$ the lower bound on the daily consumption of calories yielded by nutrient i.

• $u_i^c, i\in N:$ the upper bound on the daily consumption of calories yielded by nutrient i.

Decision variables

• $x_{j}\ge 0, j\in M:$ a real variable representing the average number of servings of menu item $j \in M$ to be consumed (per day).

Objective function

In each proposed model, one of the three different objective functions will be minimized:

$$\begin{aligned} \text {minimize} \; z_p=\sum \limits _{j\in M} a_{pj}x_j, \end{aligned}$$

(1)

where $p\in \{1,2,3\}\subset N$ defines the nutrient addressed in the objective. Recall that the first three elements of N are (1) fat, (2) sugar, and (3) sodium, respectively; thus, $z_p$ expresses the average daily consumption of the corresponding nutrient. In the proposed models, the consumption of the key nutrient $p\in \{1,2,3\}$ chosen in the objective will be minimized, while the consumption of the other two nutrients will be controlled by the constraints, as described next.

Constraints

The primary nutrient constraints are used to guarantee that the diet produced falls into the healthy-diet ranges as defined by Dietary Guidelines. Note that in Table 1 the requirements for sodium, carbohydrate, fiber, protein, and calories are specified in absolute figures (grams or milligrams), and the requirements for fat, added sugar, carbohydrate, and protein are given in terms of the percentage of consumed calories they are responsible for (% kcal). The first set of constraints addresses the requirements given in absolute terms:

$$\begin{aligned} l_i\le \sum _{j \in M} a_{ij}x_{j} \le u_i, \;\;\; i \in N\setminus \{1,2,p\}. \end{aligned}$$

(2)

Nutrients 1 (fat) and 2 (sugar) are dropped from the set of nutrients providing a constraint of this type as the requirements for these nutrients are only given in percentage of calories in Table 1. Also, nutrient p is dropped since its quantity is minimized in the objective. We set $l_3=0$ and $u_4=u_5=u_6=\infty$ since no corresponding lower or upper bound is given in Table 1.

The calorie requirements ($i=7$) are included in constraints (2) by setting both the lower bound $l_7$ and the upper bound $u_7$ to the “Calorie Level” value from Table 1. For example, $l_7=u_7=1000$ for the M/F 2–3 age-sex group. This will ensure that any feasible solution will provide a 1000-calorie diet plan based on the given restaurant’s menu.

The requirements given in terms of percentage of calories are converted into absolute figures by assuming 9 calories per gram of fat and 4 calories per gram of carbohydrate, sugar, or protein [17]. These values are multiplied by the amount $a_{ij}$ of the respective nutrient i contained in the serving of a given menu item j in order to obtain the calorie yield nutrient i is responsible for in menu item j. In addition, the ranges for % kcal specified in Table 1 are divided by 100% and multiplied by the calorie level $u_7$ to obtain the appropriate bounds on the number of calories yielded by the respective nutrient. Hence, we have the following constraints:

$$\begin{aligned} \frac{u_7}{100}l_i^c\le \sum _{j \in M}9a_{ij}x_{j} \le \frac{u_7}{100}u_i^c, \;\;\; i \in \{1,2\}\setminus \{p\}, \end{aligned}$$

(3)

$$\begin{aligned} \frac{u_7}{100}l_i^c\le \sum _{j \in M}4a_{ij}x_{j} \le \frac{u_7}{100}u_i^c, \;\;\; i \in \{4,6\}. \end{aligned}$$

(4)

Note that no lower bounds are given for the % kcal from fat and sugars in Table 1; hence, $l_1^c=l_2^c=0$. Observe that Table 1 contains requirements for percentage of calories from saturated fat and added sugar. However, the data set we use makes it difficult to differentiate between different types of fat and sugar. Hence, the constraints for these nutrients assume all the fat to be saturated fat and all the sugars to be added sugars. This makes the model somewhat more restrictive. However, we believe this assumption is reasonable in the context of fast foods.

Finally, category constraints can be imposed to ensure that the produced diet plans are balanced and do not include an excessive amount of food from the same category:

$$\begin{aligned} \sum _{j \in M}c_{jk}x_{j} \le b_k, \;\;\; k \in C. \end{aligned}$$

(5)

Constraints (5) were added in response to impracticality of solutions obtained for models (1)–(4), as discussed in the next section.

Table 1 Daily nutritional goals, ages 2 and older, according to Table A1-2 of [1]. The abbreviations in the “Source of goal” column stand for acceptable macronutrient distribution range (AMDR), recommended dietary allowance (RDA), Dietary Guidelines for Americans (DGA), and chronic disease risk reduction level (CDRR). RDAs for vitamin A are given as retinol activity equivalents (RAE)

3 Results and Discussion

Four sets of experiments with the proposed framework are conducted. In the first, constraints (5) (i.e., the category limits) are not considered, and the models (1)–(4) are solved for each individual restaurant. Such an experiment represents the least restricted scenario and can provide valuable insights with respect to the feasibility of each restaurant under the most favorable situation. In the second set of experiments, category limits are imposed in order to provide more reasonable diets. To address the practicality of obtained solutions for short-term diet planning, the third set of experiments imposes integrality constraints on the decision variables. Finally, the fourth set of experiments treats all the considered restaurants as one, with the menu combining all available items across all the restaurants. These experiments are discussed in detail in the following three subsections. All the data and codes used in this study are available online [18]. The models solved in this study contain between 11 and 576 variables and exactly 8 constraints. Each model solves in under one second using Gurobi when run in Windows 10 on an Intel core i5-3570k using 16GB of memory.

Table 2 Results for the models with no category constraints, M 19–30 age-sex group. Highlighted rows indicate the restaurants that were acceptable for the considered group. The infeasible cases are marked with a dash

3.1 Results with No Category Limits

In our preliminary analysis, we consider solutions for the most relaxed situation, in which we do not impose any limits on the number of food items from a given category. This allows for an “optimistic” (and not necessarily realistic) evaluation of the potential feasibility of fast-food diets. To illustrate this point, we report the results of experiments for one of the age-sex groups, M 19–30. Table 2 summarizes the results and reports the optimal objective function value for each of the three objectives as well as the ranking of the restaurant according to each objective. More specifically, models (1)–(4) were solved for each $p\in \{1,2,3\}$ to obtain the optimal value for the corresponding objective. The results reveal several interesting observations. Notably, the optimal objective values are not always within daily nutritional goals. As an example, when minimizing sugars at Arby’s, the least possible sugar level in a menu satisfying other constraints is 361g, whereas the maximum value specified by the daily nutritional goals is 60g. Thus, we can conclude that even though sugar can be minimized at Arby’s while satisfying the other constraints, the objective cannot be under the maximum recommended value while satisfying the other constraints.

However, we observe that any restaurant which has a feasible solution with objective within the allowable range for a given age-sex group (as described in Table 1) is guaranteed to have an optimal solution with the value within allowable range for each of the three objectives for the considered age-sex group. Thus, we can refer to such a restaurant as acceptable for the considered age-sex group. Returning to our example using Arby’s, the problem of optimizing for sugar with guidelines for men between 19 and 30 has a feasible solution, but because this value does not meet the dietary guidelines, Arby’s is not an acceptable restaurant for this group. For example, out of the 44 restaurants, only 15 are acceptable for the M 19–30 group, even though 20 restaurants are feasible for each of the three models. This implies that these 15 restaurants are the healthiest choices amongst all of the restaurants (at least for the M 19–30 group). As such, it is not possible to satisfy the recommended levels of sodium as well as calories from sugar and fat simultaneously in 29 of the restaurants. Lastly, the results show that 14 restaurants (listed at the bottom of the table) are unable to meet the requirement of any of the objectives, implying that these choices are amongst the least healthy, even under the most optimistic outlook.

While this most relaxed model can provide useful observations and high-level insights, it is worth highlighting that it has some practical limitations. For instance, the capability of a human being to consume food is limited by factors such as stomach volume and chemical processes, neither of which are included in our data or addressed by our model. We observed such problematic solutions in the results. As an example, when minimizing sugar at Godfather’s Pizza, the optimal meal consists of 34 servings of Salad Mix and 141.625 servings of Tomatoes (which is a half-cup serving of tomatoes from the salad bar). This solution is impractical from at least two standpoints. Clearly, most humans cannot eat 70 cups of tomatoes in a reasonable amount of time as the average human stomach has a capacity of $\le 20$ cups. In fact, such practical considerations are common and often limit the usability of the standard diet problem. In this paper, we attempt to overcome this drawback by imposing the category constraints (5). As we show next, these additional restrictions result in more reasonable diets and the analysis yields more practical insights and observations.

3.2 Results with the Category Limits

In this analysis, in an effort to provide more practical diets, we consider a restricted variation of the problem in which the number of servings of the food items belonging to the same category (see Section 2 for a list of categories) is limited to at most 3. This additional requirement is motivated by some of the impractical solutions obtained from the analysis in Section 3.1. We attempt to solve each of the three considered models, represented by the different objective functions, for each restaurant and every age-sex group.

Table 3 Optimal objective values for the models with the category limits. The objective values are expressed in grams. Values falling within the allowable ranges are shown in bold

Table 3 summarizes the optimal objective values for all the models that were found feasible. The restaurants that had no feasible model for any age-sex group are not listed in the table. For each included restaurant, the table lists as many rows as the largest number of models (out of three) that were feasible for an age-sex group. The cases that were infeasible for restaurants included in the table are marked with a dash in the corresponding cell. A feasible model is not necessarily acceptable, since the optimal objective value can exceed the allowable limit. Hence, the acceptable cases are shown in bold. Consider, for example, the first row, reporting the results for Arby’s. There is only one row corresponding to Arby’s (fat objective) since the models with the sugar and fat objectives were infeasible for all the age-sex groups. The fat-objective model was feasible for 5 out of 13 age-sex groups, and in each of these five cases the optimal objective value exceeds the allowable limit. In fact, having less than three rows dedicated to a particular restaurant indicates that there is no age-sex group for which that restaurant is acceptable. On the other hand, having at least one bold-faced entry for a given restaurant means that that restaurant is acceptable for at least one age-sex group, which ensures the presence of all three rows dedicated to that restaurant in the table.

Table 4 Feasibility statistics

Table 4 provides the model feasibility statistics for each age-sex group. The first row in this table specifies the number of acceptable restaurants for each group. The next three lines show the number of restaurants for which the model with the sugar, sodium, and fat objective, respectively, is feasible. (This is the same as saying that the model is feasible after dropping the sugar, sodium, and fat limits, respectively.) The last four lines in this table report the number of restaurants for which three, two, one, and no models were feasible, respectively. Note that restaurants for which all three models are feasible for a given group include the acceptable restaurants for that group.

Finally, Table 7 in the Appendix reports the details of optimal solutions (i.e., specific menu items and their quantities) found for each restaurant and every age-sex group.

Fig. 3

The total number of age-sex groups for which a given restaurant is acceptable

The results reveal that the vast majority of restaurants cannot provide an acceptable diet for any age-sex group. In fact, out of the 44 restaurants only 10 can provide an acceptable diet satisfying all the considered intake requirements for at least one age-sex group. Figure 3 shows the number of groups for which each of these 10 restaurants is acceptable. Only three restaurants: Popeyes, Wendy’s, and Blimpie, are acceptable for most age groups, while only one of these restaurants, Popeyes, offers a menu that is capable of satisfying all 3 objectives simultaneously for each of the 13 age-sex groups. In our view, this metric provides a clear and concise measure of the healthiness of restaurants’ respective menus, and restaurants that cannot meet dietary recommendations are simply not ranked.

Fig. 4

The number of acceptable restaurants for each age-sex group

Beyond this ranking, we observed a number of interesting features of the solutions. Figure 4 shows the number of acceptable restaurants across age groups for each gender, and reveals two trends. (This figure is based on the data in the first row of Table 4.) Firstly, men had fewer acceptable restaurants compared to women, implying that it is more difficult for men to meet dietary recommendations at these restaurants than it is for women. Secondly, there is a clear dip in the number of acceptable restaurants around early adulthood and subsequent increase around middle age across both genders. We hypothesize that both of these trends are due to higher calorie recommendations forcing menus to reach category limits and consequently use less healthy foods.

Fig. 5

The number of feasible restaurants for each age-sex group with respect to each of the three models

Figure 5 illustrates the feasibility statistics for individual models, based on the data available in rows 2–4 of Table 4. We observe that across all age-sex groups there were significantly fewer restaurants feasible with respect to the sodium-objective model than the other two. This implies that meeting the recommendations for calories from fat and sugar simultaneously is more difficult than any other combination of the three key constraints.

Looking at the optimal menus qualitatively, we observed some interesting features. For instance, popular menu items (the items for which a restaurant is best known) were not selected at the majority of restaurants. This is not surprising, as popular menu items at fast-food restaurants are generally unhealthy (i.e., high in either sugar, sodium, or fat). However, at some restaurants, such as Burger King, Chipotle, and Red Lobster, the most well-known and popular menu items appeared in their optimal menus. This indicates that, at least for some restaurants, the optimal menu items are representative of an average order. In addition, our model did not account for alcoholic menu items, as these were not identified in our data set and can thus only be assumed by their name. As an example of a problematic menu that arose from this issue, the optimal menu at Red Lobster when minimizing sodium for the 2–3 age group includes 2.48 servings of raspberry Lobsterita (similar to a Margarita). This menu item would be both illegal and dangerous for a 2–3-year-old to consume. There were also two practical issues that we observed in the optimal menus. The first is that some optimal menus include very small portions of a meal. An example of this is the minimization of calories from fat at Blimpie for the female 9–13 group, where the optimal menu includes 0.02 servings of the Grande Chili with Bean and Beef menu item. While a customer could, in theory, eat exactly that portion of the menu item, it is practically difficult to eat a small percentage of a serving of food without measuring its weight. Additionally, many food items are not homogeneous mixtures, and thus extracting an exact percentage of a serving involves separating that percentage out from each part of the menu item. This issue could be addressed in a long-term planning, as one could consume, e.g., 2 servings of the Grande Chili with Bean and Beef during a 100-day period (recall that our decision variables represent the average number of servings per day, and the planning period may vary in duration). Nevertheless, in the next subsection, we experiment with a different way of addressing this problem, by imposing integrality restrictions on the decision variables. The other practical issue is cost. Some of the optimal menus are prohibitively expensive, especially given that some menu items can only be purchased as add-ons to others. This is difficult to measure quantitatively, as our data set does not include prices.

3.3 Results with Integer Restrictions

In order to eliminate fractional portions, we extend the model by restricting the decision variables to integers. The goal of this restriction is to create more practical short-term solutions. This restriction was imposed on the model in addition to the category limits in order to generate the most practical possible solutions. The results of this analysis are presented in Table 5.

As expected, compared to the results without integrality restrictions (Table 4), the number of feasible restaurants decreases for most groups, and the number of acceptable restaurants also decreases for all groups. With integer constraints, 5 age-sex groups have no acceptable restaurants. However, whenever an integer model produces a solution, it is significantly more practical. As an example, we consider the solution obtained for Blimpie for the F 9–13 group, which included 0.02 servings of an item without integer constraints. The new optimal solution is composed of three plain Bluffins, three garden dressings, a classic-style thin pretzel, and two plain soft pretzels. This is significantly more practical from both cost and consumption standpoints and, furthermore, is an acceptable solution.

Although the solutions for this model are more practical, the low level of feasibility and acceptability limits their usefulness. We expect that with more accurate data integer models would be infeasible for most age-sex groups. Furthermore, the goal of this study is to assess the possibility of eating a healthy meal using fast-food restaurants, and studying the practicality of the solutions is of more value once more accurate data is available. The practicality considerations could be further addressed using goal programming methods [19, 20], which could be particularly suitable for creating personalized diets.

Table 5 Feasibility statistics for integer model

3.4 Results for the Combined Menu

Having observed that the options offered by most individual restaurants are too limited for a healthy diet, in the last set of experiments, we combine the menu items of all restaurants into a single “fast food menu” and analyze it using our optimization models for each age-sex group. This experiment is of interest as from a practical standpoint it is possible to combine menu items from different restaurants with the goal of eating healthy.

The combined menu for the 44 considered restaurants contains the total of slightly over 6700 menu items. The detailed results are summarized in Table 8. We compute optimal solutions for each of the 3 objectives and every one of the 13 age-sex groups, yielding a total of 39 models to be solved for the combined menu. The detailed results are reported in Table 8 in the Appendix. Menu items from 19 different restaurants are represented in the solutions found for the 39 considered models. Figure 6 shows the total number of times a menu item from each of the 19 restaurants appeared in an optimal solution.

Fig. 6

Total number of times a menu item from each restaurant is used in the combined menu

Given the large and diverse pool of items to choose from in the combined menu, one would expect to be able to produce a variety of fast-food nutrition plans satisfying the basic requirements reflected in the proposed optimization models. Our final set of experiments aims to find out if this is indeed the case as follows: We focus on the M 19–30 age-sex group, which we identified as the group with the requirements that are particularly difficult to fulfill using fast-food menus. We recursively repeat the following process, until the optimal objective value exceeds the allowable limit.

• 1. Find an optimal solution for each of the three models with different objectives;

• 2. Remove all items that appear in at least one optimal solution from the combined menu.

After performing 10 iterations of this process, we reduced the combined fast-food menu to the extent that no combination of the remaining items could produce an acceptable nutrition plan. Treating different-size portions of the same food as one, we find that removing just 127 foods from 29 restaurants makes the remaining combined fast-food menu unacceptable. This accounts for only 1.9% of the total available menu items, representing 65.9% of the considered restaurants. Table 6 provides the detailed list of the corresponding menu items. This finding further highlights the limitations of the options available for healthy nutrition in fast-food restaurants.

Table 6 Removing these 127 foods from 29 restaurants makes the combined fast-food menu infeasible for the M 19–30 age-sex group. The middle column (“#”) shows the number of items removed from the menu of the corresponding restaurant

4 Conclusion

The results of this study raise several issues and can be useful for various stakeholders. While it had been long established that fast-food restaurants are not healthy in general, it is somewhat surprising to discover that the current menus of most restaurants do not provide options to satisfy even the most basic healthy diet requirements. Incorporating additional criteria, such as the practical limitations discussed toward the end of Section 3.2, which were ignored in this study due to the lack of data, would only exacerbate this observation. The need for introducing more healthy options in fast-food restaurant menus is evident, and optimization methods could be used to help restaurants design such menu items in a cost-effective manner, while taking into account the customers’ preferences. In addition, careful analysis of the differences in the results between the various age-sex categories could be helpful in developing improved dietary guidelines, which are regularly revised by experts. In particular, according to [1] “the U.S. and Canadian Dietary Reference Intake Steering Committees are currently developing plans to re-examine energy, protein, fat, and carbohydrate.” Last but not least, we hope this study will inspire renewed interest in optimization problems related to nutrition in operations research community. The classical diet problem played a historically important role in the field of operations research, but was criticized as impractical due to early results, which had deficiencies similar to those observed in our first set of experiments. We demonstrate that simply adding the category constraints results in solutions that are practically reasonable. The superb performance of modern optimization solvers and constantly improving availability of data open up new opportunities for application of operations research tools in the emerging discipline of nutrition informatics.

Appendix. Detailed results

Table 7 Optimal solutions found for the models with category limits

Table 8 Optimal solutions found for the combined menu