Evaluation and Optimal Design of a Balanced Diet

Abstract

Malnutrition has led to growth retardation in many adolescents and health deterioration in adults all over the world. Recently, there has been increasing attention on balanced diets as a preventive measure. This study evaluates the daily diet of a student, aiming to optimize the amino acid score (AAS) across three meals per day. By using balanced diet criteria as constraints, we established a single-objective nonlinear programming model, maximizing AAS as the objective function. The model was solved by using a simulated annealing algorithm, and we obtained a diet that is both balanced and high in AAS. This study helps to raise awareness about individual nutritional needs and provides guidance for dietary structure improvements, thereby contributing to global public health enhancement.

1. Introduction

Malnutrition is a global health problem, referring to the body’s inability to obtain sufficient nutrition for maintaining normal physiological function and health due to inadequate intake or metabolic disorders. It also includes excessive nutrition due to overeating or the excessive intake of specific nutrients [1]. Malnutrition can result from insufficient protein intake due to disease, hunger, age, and other factors, leading to changes in body composition and, ultimately, damage to physical and mental functioning [2,3].

Malnutrition is primarily categorized into nutritional deficiency and overnutrition [4]. Nutritional deficiency can be divided into two main aspects: energy deficiency (such as emaciation and being underweight) and micronutrient deficiency (such as iron deficiency anemia, vitamin A deficiency, etc.) [5]. Overnutrition can be divided into traditional nutrition surplus (such as micronutrient excess, overweight, obesity, and diet-related non-communicable diseases) and emerging nutrition surplus [6].

In recent years, the prevalence of malnutrition in countries around the world has been increasing. Malnutrition has become a major cause of the spread of global diseases, posing a major pressure on the global health system [7,8,9,10]. The 2016 global nutrition report shows that 44% of the countries with available data are currently facing severe challenges of malnutrition and adult overweight obesity. Although the situation in some countries has improved slightly, the overall global development trend remains concerning. Its main findings show that “one in three people in the world is malnourished, and malnutrition has become a global ‘new normal’ [11]”. According to the report of the World Health Organization (WHO) in 2013, malnutrition is the primary factor leading to the death of children around the world, accounting for 45%. According to the data on the official website of the American Charity Association in 2015, 48.8 million people in the United States are malnourished, which includes 16.2 million children. Because this situation is so severe in a developed country, the United States, the situation in other countries can only be imagined [12].

Currently, the dietary habits and nutritional status of college students are an important issue in the field of public health. Local college students often skip breakfast, consume snacks and late-night meals, prefer heavily flavored foods, and do not drink enough water to meet physical needs, leading to unbalanced nutrition and other poor eating habits [13]. The consequences of dietary changes are related to adverse health outcomes, including weight gain, elevated blood glucose levels, elevated cholesterol levels, hypertension, and mental problems [14].

In this context, it is particularly important to re-examine and optimize the dietary structure. At present, intervention measures for malnutrition, such as the use of nutritional supplements, dietary structure adjustment, and nutrition education, have become the focus of global research [15]. Against this background, we employ a single-objective nonlinear programming model and a simulated annealing algorithm to study daily dietary structure.

The single-objective programming model is widely used in many fields, including computer science [16,17], production logistics, resource allocation, transportation [18], energy control [19,20], and agriculture, aiming to find the best solution by optimizing the single-objective function. A simulated annealing algorithm has also been extensively applied in research, especially in optimization problems. It can help solve complex optimization problems [21,22], informatics [23], and production logistics [24].

This paper aims to develop a more reasonable dietary structure according to the basic principles of a balanced diet through the single-objective nonlinear programming model and simulated annealing algorithm to ensure that people can obtain comprehensive and balanced nutrition. By comprehensively analyzing existing nutritional theories, dietary patterns, and global health data, we will propose a set of strategies to enhance students’ nutritional awareness, promote public health, and prevent malnutrition and related diseases [25]. Simultaneously, more diversified and nutritionally balanced food options are recommended for university canteens. Our goal is to provide policymakers, nutritionists, and the public with a practical guide to promote global nutrition improvements and lay the foundation for a healthier and more sustainable future.

2. Process of Evaluating Diet

We use the Chinese dietary guidelines as a reference and consider the following five aspects (2.1–2.5) to evaluate whether a daily diet meets the balanced diet criteria. For data sources of all food ingredients, please refer to China Food Composition Tables [26].

2.1. Analysis of Food Structure

According to the basic principles of a balanced diet in the guidelines, the average daily intake of food should include more than 12 different types. Therefore, we can first assess whether a student’s daily diet contains 12 kinds of food and five categories of food through simple statistics.

2.2. Calculation of Main Nutrients Content

The intake of nutrients is crucial for a person’s growth. We consider the daily intake of the following major nutrients that are common and important in life: water, protein, fat, carbohydrates, calcium, iron, zinc, vitamin A, vitamin B1, vitamin B2, and vitamin C. Their content in every 100 g of edible food can be found in China Food Composition Tables.

Suppose we know a student’s daily diet , where $G_{\sigma }$ represents the content of a certain nutrient (such as in $\sigma =water$ or $\sigma =protein$) in the diet, $m_{\sigma }$ represents the quality of a certain nutrient in every 100 g of edible food, $m_i$ represents the edible part quality of the major ingredient, i (see Appendix A for the major ingredient contained in a type of food), in the diet, and $n_i$ represents the number of portions of the major ingredient, i, in the diet. We can use the following formula to calculate the content of the main nutrients in the edible part of a student’s daily intake of food.

$$G_{\sigma }=\sum _{i=1}^n{m}_i{n}_i·{\displaystyle \frac{m_{\sigma }}{100}}.$$

(1)

where n represents the number of major ingredients in a daily diet.

According to the basic principles of a balanced diet, a student should intake enough nonproductive major nutrients in a day.

Table 1. Dietary reference intakes of nonproductive major nutrients.

Sex	Calcium	Iron	Zinc	Vitamin A	Vitamin B1	Vitamin B2	Vitamin C
Male	800	12	12.5	800	1.4	1.4	100
Female	800	20	7.5	700	1.2	1.2	100

Except for the unit of vitamin A being $\mu g·d^{-1}$, the other units are in $mg·d^{-1}$.

shows the student’s reference intake of nonproductive major nutrients in a day. By comparing the calculation result of formula (1) with it, we can know whether the student ingests enough nonproductive main nutrients.

2.3. Calculation of Energy Provided by Diet and Meal Ratio

According to the basic principles of a balanced diet, female students should consume 1900 kcal of energy per day, while male students should consume 2400 kcal of energy per day. The reference value of the energy distribution of three meals as a percentage of the total energy (i.e., meal ratio) is 30% for breakfast, 30–40% for lunch, and 30–40% for dinner.

Table 2. Nutrient energy conversion coefficient.

Nutrient	Protein	Fat	Carbohydrate	Dietary Fiber
Conversion coefficient (kcal/g)	4	9	4	2

shows the nutrient energy conversion coefficients.

Now, use $E_{\sigma }$ to represent the total energy provided by a certain nutrient in the daily diet. We can use the following formula to calculate it.

$$E_{\sigma }={\eta }_{\sigma }·G_{\sigma }.$$

(2)

where ${\eta }_{\sigma }$ is the energy conversion coefficient of a certain nutrient, $\sigma$ represents protein, fat, carbohydrate, or dietary fiber.

If the diet is divided into breakfast, lunch, and dinner, the energy intake corresponding to each of the three meals can be calculated by using the same calculation method, and then the ratio with the total energy can be calculated. The rationality of a student’s daily energy intake can be evaluated by comparing the results with the above reference values.

2.4. Calculation of Energy Sources for Diet

The reference values of daily macronutrient energy supply in the total energy of college students are protein 10–15%, fat 20–30%, and carbohydrate 50–65%. According to the following formula, the percentage of macronutrient energy provided by food intake in one day in total energy can be calculated:

$$w_{\sigma }={\displaystyle \frac{E_{\sigma }}{\sum E_{\sigma }}}$$

(3)

where $w_{\sigma }$ is the percentage of energy supplied by a certain nutrient in the total energy in a diet. $\sigma$ represents protein, fat, or carbohydrate.

The rationality of the student’s energy sources can be evaluated by comparing the calculation results with the above reference values.

Calculation of Protein Amino Acid Score of Each Meal

Amino acid score (AAS), also known as protein chemistry score, is the ratio calculated by comparing the essential amino acids of the tested food protein with the amino acid pattern of the reference protein. The lowest ratio is the first limiting amino acid. The score of the first limiting amino acid is the AAS of the food proteins. The AAS is not only suitable for the evaluation of single-food protein but also for the evaluation of mixed-food protein. The reference protein AAS mode is shown in

Table 3. The reference protein AAS mode.

Essential Amino Acid	Isoleucine	Leucine	Lysine	Sulfur Containing Amino Acids	Aromatic Amino Acids	Threonine	Tryptophan	Valine
AAS	40	70	55	35	60	40	10	50

.

One of AAS can be calculated as follows:

$$AA{S}_{amino}={\displaystyle \frac{\sum m_i{n}_i·\frac{{m^{\left(i\right)}}_{amino}}{100}}{\sum m_i{n}_i·\frac{{m^{\left(i\right)}}_{protein}}{100}}}·{\displaystyle \frac{100}{{\epsilon }_{amino}}}$$

(4)

where ${m^{\left(i\right)}}_{amino}$ represents the amino acid quality per 100 g of the edible part of an ingredient, i, in the daily diet, ${m^{\left(i\right)}}_{protein}$ represents the protein mass per 100 g of the edible part of an ingredient, i, in the daily diet, ${\epsilon }_{amino}$ represents the content of essential amino acids per gram of a reference protein.

Calculate the score of all essential amino acids in sequence according to the above formula, and the lowest score is the AAS of the food protein. The higher the AAS of a student, the more food protein the student ingests to meet their amino acid needs, and the higher their nutritional value.

2.5. Numerical Simulation

Let us assume that the one-day diet of a male or female college student is as shown in

Table 4. One-day diet of a male or female college student.

Sex	Breakfast		Lunch		Dinner
	Food Name	Servings	Food Name	Servings	Food Name	Servings
Male	Millet congee	1	Rice	4	Casserole noodles	1
	Deep-fried dough stick	2	Mixed Wood Ear Mushroom	1	Steamed stuffed bun	1
	Fried egg	1	Fried potato green pepper and eggplant	1	Fried chicken nugget	1
	Mixed Seaweed	1	Braised pork	1
Female	Soya milk	1	Omelett	1	Rice	2
	Chicken cutlet noodles	1	Boiled dumpling	1	Stir-fried mushroom and bokchoy	0.5
			Grape	1	Stir-fried meat with garlic sprout	0.5
					Sardines in tomato sauce	0.5
					Apple	1

. Now, let us analyze their diets according to the steps in 2.1–2.5. First, analyze their food structure. Figure 1 shows the classification results of their daily food intake. The observation shows that their daily food intake includes five categories, all of which meet the basic criteria of a balanced diet.

Then, we calculate the main nutrient content of their daily diet, as shown in

Table 5. Content of main nutrients.

Nutrient	Male	Female
Protein	88.21	54.67
Fat	122.68	43.615
Carbohydrate	289.61	171.085
Calcium	760.16	205.55
Iron	24.66	9.135
Zinc	10.55	5.043
Water	583.27	449.605
Vitamin A	260.80	290.75
Vitamin B1	0.95	0.8205
Vitamin B2	0.85	0.5365
Vitamin C	38.70	59

The unit of protein, fat, carbohydrate, and water is g, the unit of vitamin A is $\mu g·d^{-1}$, and the rest is $mg·d^{-1}$.

. From this, we can calculate the energy provided by their daily intake of food, the ratio of meals, and the content of nonproductive main nutrients, as shown in

Table 6. Energy and meal ratio in male and female students’ day diet.

Sex		Breakfast	Lunch	Dinner	Total
Male	Energy (kcal/d)	796.62	976.71	858.13	2631.45
	Proportion	30.27%	37.12%	32.61%	100.00%
Female	Energy (kcal/d)	318.75	569.78	421.31	1309.84
	Proportion	24.33%	43.50%	32.17%	100.00%

and

Table 7. Contents of nonproductive main nutrients in male and female students’ day diet.

Sex	Nutrient	Calcium	Iron	Zinc	Vitamin A	Vitamin B1	Vitamin B2	Vitamin C
Male	Reference quantity	800	12	12.5	800	1.4	1.4	100
	Intake	760.16	24.66	10.55	260.80	0.95	0.85	38.70
Female	Reference quantity	800	20	7.5	700	1.2	1.2	100
	Intake	205.55	9.135	5.043	290.75	0.8205	0.5365	59

Except for the unit of vitamin A being $\mu g·d^{-1}$, the other units are in $mg·d^{-1}$.

. We think it is appropriate that the difference between the actual daily energy intake and the target daily energy intake of men and women is within ±10%. From the data in the table above, we can find that the daily energy intake of male college students is 2631.45 kcal/d, which is moderate. The energy provided by female college students’ daily food intake was 1309.84 kcal/d, which was low.

Secondly, the meal ratio of male students was 30.27%, 37.12%, and 32.61%, and the meal ratio of female students was 24.33%, 43.50%, and 32.17%, respectively. Compared with the daily energy intake target of college students, it was found that female students’ breakfast energy intake was too low, and lunch energy intake was too high.

Then, we compared the content of nonproductive main nutrients provided by their diet with the reference intake. By observing Table 7, we found that the daily intake of vitamins for both male students and female students was low, and the daily intake of calcium for female students was insufficient.

Next, we examine the percentage of energy supplied by macronutrients in the total energy intake. The calculated results are shown in

Table 8. Percentage of total energy supplied by macronutrients.

Nutrients	Protein	Fat	Carbohydrate
Reference value	10–15%	20–30%	50–65%
Proportion of a male students	13.41%	41.96%	44.02%
Proportion of a female students	16.70%	29.97%	52.25%

. It was found that the daily fat intake of male students was higher, and the carbohydrate intake was lower than the reference values, while the daily protein intake of female students was higher than the reference value.

Finally, we get their AAS, as shown in

Table 9. AAS for male and female students.

Time	Break	Lunch	Dinner
AAS for male student	66.33	92.37	25.97
AAS for female student	89.62	86.00	100.92

. According to the rules, a person’s AAS being less than 60 is unreasonable, 60–80 is unreasonable, 80–90 is relatively reasonable, and more than 90 is reasonable. By observing Table 9, it can be concluded that the AAS of female students at three meals is reasonable, while the AAS of male students at dinner is seriously low.

3. Single-Objective Nonlinear Programming Model

In a meal, the larger the AAS, the higher the nutritional value of protein in the food a student eats at this meal. Therefore, we hope that the AAS of each meal is larger.

Suppose that the food provided by the first to $k_1$ in a canteen diet is breakfast, the food provided by $k_1+1$ to $k_2$ is lunch, and the food provided by $k_2+1$ to $k_3$ is dinner. Then, we let the AAS for breakfast, lunch, and dinner be $A_1$ , $A_2$, and $A_3$, respectively, which can be calculated by the following formula:

$$\begin{array}{c}{A}_1=min\left\{\left({\displaystyle \frac{{\sum }_{i=1}^{k_1}{m}_{p_j}^{\left(i\right)}·N_i}{{\sum }_{i=1}^{k_1}{m}_p^{\left(i\right)}·N_i}}/p_{reference}^{\left(j\right)}\right)·100\right\},\\ A_2=min\left\{\left({\displaystyle \frac{{\sum }_{i=k_1+1}^{k_2}{m}_{p_j}^{\left(i\right)}·N_i}{{\sum }_{i=k_1+1}^{k_2}{m}_p^{\left(i\right)}·N_i}}/p_{reference}^{\left(j\right)}\right)·100\right\},\\ A_3=min\left\{\left({\displaystyle \frac{{\sum }_{i=k_2+1}^{k_3}{m}_{p_j}^{\left(i\right)}·N_i}{{\sum }_{i=k_2+1}^{k_3}{m}_p^{\left(i\right)}·N_i}}/p_{reference}^{\left(j\right)}\right)·100\right\}.\\ j=1,2,\dots ,8\end{array}$$

(5)

where $m_{p_j}^{\left(i\right)}$ represents the quality of the essential amino acid j of the food name, i, $N_i$ represents the number of portions of the food name, i , $m_p^{\left(i\right)}$ represents the protein quality of the food name i, and $p_{reference}^{\left(j\right)}$ is the content of the essential amino acid, j, in the reference protein.

Select the smallest of $A_1$, $A_2$, and $A_3$ to make it reach the maximum, which is the objective function Z of the single-objective nonlinear programming model.

$$Z=max\left\{min\left\{A_1,A_2,A_3\right\}\right\}.$$

(6)

Now, consider the constraints of the model. Assume that the difference between a student’s actual daily energy intake and the intake target is within ±10%; thus, the first constraint is as follows:

$$2400\left(1-10\%\right)⩽\sum _{i=1}^{k_3}\left(4m_p^{\left(i\right)}+9m_f^{\left(i\right)}+4m_c^{\left(i\right)}+2m_d^{\left(i\right)}\right)·N_i⩽2400\left(1+10\%\right).$$

(7)

where $m_f^{\left(i\right)}$ represents the fat quality of the food name, i, $m_c^{\left(i\right)}$ represents the carbohydrate quality of the food name, i, and $m_d^{\left(i\right)}$ represents the dietary fiber quality of the food name, i.

Secondly, the percentage of productive nutrients in the total energy should try to meet the requirements of 10–15% protein, 20–30% fat, and 50–65% carbohydrate; thus, the second constraint is as follows:

$$\begin{array}{c}10\%⩽{\displaystyle \frac{{\sum }_{i=1}^{k_3}4m_p^{\left(i\right)}·N_i}{{\sum }_{i=1}^{k_3}\left(4m_p^{\left(i\right)}+9m_f^{\left(i\right)}+4m_c^{\left(i\right)}+2m_d^{\left(i\right)}\right)·N_i}}\times 100\%⩽15\%,\\ 20\%⩽{\displaystyle \frac{{\sum }_{i=1}^{k_3}9m_f^{\left(i\right)}·N_i}{{\sum }_{i=1}^{k_3}\left(4m_p^{\left(i\right)}+9m_f^{\left(i\right)}+4m_c^{\left(i\right)}+2m_d^{\left(i\right)}\right)·N_i}}\times 100\%⩽30\%,\\ 50\%⩽{\displaystyle \frac{{\sum }_{i=1}^{k_3}4m_c^{\left(i\right)}·N_i}{{\sum }_{i=1}^{k_3}\left(4m_p^{\left(i\right)}+9m_f^{\left(i\right)}+4m_c^{\left(i\right)}+2m_d^{\left(i\right)}\right)·N_i}}\times 100\%⩽65\%.\end{array}$$

(8)

Then, the actual intake of the seven main nutrients of production capacity—calcium, iron, zinc, vitamin A, vitamin B1, vitamin B2, and vitamin C—is as close as possible to the reference intake; thus, the third constraint is as follows:

$$w_{reference}^{\left(j\right)}·\left(1-10\%\right)⩽\sum _{i=1}^{k_3}{w}_j^{\left(i\right)}{N}_i⩽w_{reference}^{\left(j\right)}·\left(1+10\%\right),j=1,2,\dots ,7.$$

(9)

where $w_{reference}^{\left(j\right)}$ represents the reference intake of the nutrient j, $w_j^{\left(i\right)}$ represents the quality of the nonproductive nutrient, j, in the food name, i, and $j=1,2,\dots ,7$ represents calcium, iron, zinc, vitamin A, vitamin B1, vitamin B2, and vitamin C, respectively.

Finally, as far as possible, the ratio of meal times should meet the requirements of 25–35% for breakfast, 30–40% for lunch, and 30–40% for dinner; thus, the fourth constraint is as follows:

$$\begin{array}{c}25\%⩽{\displaystyle \frac{{\sum }_{j=1}^{k_1}\left(4m_p^{\left(j\right)}+9m_f^{\left(j\right)}+4m_c^{\left(j\right)}\right)·N_j}{{\sum }_{i=1}^{k_3}\left(4m_p^{\left(i\right)}+9m_f^{\left(i\right)}+4m_c^{\left(i\right)}+2m_d^{\left(i\right)}\right)·N_j}}⩽35\%,\\ 30\%⩽{\displaystyle \frac{{\sum }_{j=k_1+1}^{k_2}\left(4m_p^{\left(j\right)}+9m_f^{\left(j\right)}+4m_c^{\left(j\right)}\right)·N_j}{{\sum }_{i=1}^{k_3}\left(4m_p^{\left(i\right)}+9m_f^{\left(i\right)}+4m_c^{\left(i\right)}+2m_d^{\left(i\right)}\right)·N_j}}⩽40\%,\\ 30\%⩽{\displaystyle \frac{{\sum }_{j=k_2+1}^{k_3}\left(4m_p^{\left(j\right)}+9m_f^{\left(j\right)}+4m_c^{\left(j\right)}\right)·N_j}{{\sum }_{i=1}^{k_3}\left(4m_p^{\left(i\right)}+9m_f^{\left(i\right)}+4m_c^{\left(i\right)}+2m_d^{\left(i\right)}\right)·N_j}}⩽40\%.\end{array}$$

(10)

4. Simulated Annealing Algorithm

4.1. Brief Description of Algorithm

The problem studied in this article belongs to a combinatorial optimization problem. Geng, Xiutang, and others [27] used the classical simulated annealing (SA) algorithm to study the maximum clique problem. Akbulut, Hatice Erdogan, and others [28] have applied SA to solve the faculty-level university course timetabling problem. The inspiration for SA comes from the similarity between the annealing process of solid materials and combinatorial optimization problems. SA is a probabilistic algorithm that simulates the gradual cooling process of solid materials from a high temperature. In this process, the algorithm searches for the optimal solution of the objective function in the solution space and gradually approaches the global optimal solution from the local optimal solution through the iterative process. SA mainly considers the following three processes: the iterative process, the cooling process, and the end condition. The more detailed process is shown in Appendix B.

4.2. Numerical Simulation

This numerical simulation is carried out on the Windows 10 system using MATLAB R2022a. According to the above process, the parameters of the SA we set are shown in

Table 10. Parameter table of SA.

Parameter	Value
$T_0$	1000
$T_{end}$	$7.27\times {10}^{-9}$
$\alpha$	0.95

.

Assuming a university cafeteria provides three meals a day, as shown in Appendix A, based on the single-objective nonlinear programming model and SA established above, we used MATLAB to solve for a daily menu that students can choose from, as shown in

Table 11. Daily available diet for students to choose from.

Sex	Breakfast		Lunch		Dinner		Optimal AAS
	Food Name	Servings	Food Name	Servings	Food Name	Servings
Male	Potato cake	1	Fried chicken nugget	1	Steamed dumpling	1	88.00
	Pan-fried stuffed bun	1	Pumpkin porridge	1	Casserole noodles	1
	Fried egg	1	Double cooked pork slices	1	Watermelon	1
	Orange	1	Dry fried yellow croaker	1	Dry fried yellow croaker	1
Female	Steamed sweet potato	1	Pasty	1	Steamed dumpling	1	105.61
	Grape	1	Braised tofu with pollak	1	Casserole noodles	1
	Pumpkin porridge	1	Honeydew melon	1	Omelett	1
	Mixed seaweed	1	Chicken stewed with potatoe and carrot	1	Dry fried yellow croaker	1

.

5. Conclusions

The largest amino acid score (AAS) for one person for three meals a day being greater than 80 is relatively reasonable. From the simulation results, it was found that the AAS of a male is 8% higher than the reasonable value, and the AAS of a female is 20.8% higher than the reasonable value. Thus, after establishing the model and solving the algorithm, the provided recipe ensures a balanced diet and a reasonable combination of nutrients while maximizing the amino acid score (AAS). Due to the randomness of the simulated annealing algorithm, various daily recipes can be obtained, and even a nutritious weekly recipe can be generated.

Naturally, it is difficult to require that students strictly follow the recipes generated by the algorithm, but students should try to pay attention to their eating habits, refer to these recipes, and prevent malnutrition. Additionally, we aim to provide more diverse and nutritionally balanced meal options for university cafeterias worldwide.

The single-objective nonlinear programming mentioned above focuses solely on AAS. In fact, the selection of diet may be influenced by personal economic factors, seasonal factors, and other factors. By considering other factors, we can add more constraints to the model we establish and even establish different objective functions according to different needs to obtain a diverse diet. For the solution of the model, we used the classic simulated annexing method. There are currently many innovative and advanced intelligent algorithms available (such as the sea lion optimization, grey wolf optimization, and whale optimization algorithms), and we can also use these more efficient algorithms to obtain the corresponding diet.