Jeffrey Salazar
Linear programming is a mathematical technique used to solve complex problems with multiple constraints. One of the earliest applications of linear programming was in solving The Diet Problem, which aimed to find a low-cost diet that met the nutritional needs of US Army soldiers during World War II. The problem was first formulated by mathematician Jerry Cornfield, and later worked on by economist George Stigler, who made an early attempt at a solution using a heuristic method, guessing the cost of an optimal diet to be $39.93 per year (in 1939 prices). Linear programming has since been applied to a variety of diet-related problems, from food aid and national food programs to individual dietary plans, allowing for the generation of optimal solutions that satisfy nutritional requirements while minimizing costs.
| Characteristics | Values |
|---|---|
| History | The "Diet Problem" was first formulated by Jerry Cornfield during World War II to meet the nutritional needs of US Army soldiers at a low cost. |
| Objective | To select a set of foods that satisfy daily nutritional requirements at a minimum cost. |
| Nutritional Requirements | Calories, vitamins, minerals, proteins, fats, sodium, cholesterol, macronutrients, and micronutrients. |
| Methods | Simplex method, Quadratic Programming (QP), Excel Solver, NEOS Server solvers, and graphical method. |
| Tools | SAS, Microsoft Excel, LINGO Hyper, GNU Linear Programming Kit, Nutritionist Pro, SPSS, and Wolfram Demonstrations Project. |
| Applications | Food aid, national food programs, dietary guidelines, individual issues, and cancer prevention. |
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What You'll Learn
The simplex method
To use the simplex method, one must first identify and set up a linear program in standard maximisation form. Next, convert inequality constraints to equations using slack variables. Then, set up the initial simplex tableau using the objective function and slack equations. The next step is to find the optimal simplex tableau by performing pivoting operations. Finally, identify the optimal solution from the optimal simplex tableau.
An example of the simplex method in action can be seen in the following problem:
> Suppose there are three foods available, corn, milk, and bread, and there are restrictions on the number of calories (between 2000 and 2250) and the amount of Vitamin A (between 5000 and 50,000). The first table lists, for each food, the cost per serving, the amount of Vitamin A per serving, and the number of calories per serving. Suppose that the maximum number of servings is 10. Then, the optimal solution for the problem is 1.94 servings of corn, 10 servings of milk, and 10 servings of bread with a total cost of $3.15. The total amount of Vitamin A is 5208 and the total number of calories is 2000.
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$14.72 $24.95
Linear programming models
The history of linear programming models for dietary problems can be traced back to World War II when mathematician Jerry Cornfield formulated "The Diet Problem" for the US Army. The goal was to find a low-cost diet that fulfilled the nutritional requirements of soldiers. Economist George Stigler, who made an early attempt at a solution with an estimated cost of $39.93 per year (in 1939 prices), played a pivotal role in this initial phase. In 1947, Jack Laderman and a team of nine clerks used the simplex method, a standard approach to maximizing a linear function with multiple variables, to solve Stigler's model. This effort took 120 man-days and resulted in a nearly identical cost estimate to Stigler's, differing by only $0.24 per year.
The diet problem, as it came to be known, was one of the earliest optimization challenges studied in the 1930s and 1940s. The primary objective was to minimize feeding costs while ensuring soldiers received a healthy diet. This problem can be mathematically formulated as a linear program, where the goal is to minimize cost while satisfying nutritional requirements such as calories, vitamins, minerals, fats, sodium, and cholesterol. However, it's important to note that the mathematical solution may not always result in a palatable menu, as taste and variety are not typically considered in these models.
To solve linear programming problems related to diet, various tools and solvers are available, such as the NEOS Server solvers and the Simplex Method Tool on the Finite Mathematics and Applied Calculus website. These tools enable users to input their specific dietary constraints and requirements to obtain optimal solutions. For instance, a simple example in a NEOS Guide involves selecting from three foods (corn, milk, and bread) to meet calorie and Vitamin A restrictions while minimizing cost. The solution includes specific serving sizes for each food, staying within the given constraints and providing a balanced diet at a minimal cost.
In recent years, the application of linear programming in diet optimization has evolved with the advent of computerized programming models and quadratic mathematical functions. Software such as Nutritionist Pro™ and statistical analysis tools like SPSS have been used to analyze dietary intake and plan diets that align with nutritional guidelines. Additionally, Microsoft Excel and its Solver feature have been leveraged to model and solve linear programming problems related to diet optimization, allowing users to create customized food plans that meet their nutritional goals.
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Nutritional requirements
Linear programming models for dietary problems often include constraints on the number of calories and the amounts of vitamins, minerals, fats, sodium, and cholesterol in the diet. These constraints ensure that the selected foods meet the minimum and maximum recommended intake levels for each nutrient. For example, a diet may need to provide between 2000 and 2250 calories per day and include between 5000 and 50,000 units of Vitamin A.
The nutritional requirements used in these models can come from various sources, such as dietary guidelines, recommended nutrient intakes, or specific nutritional goals, such as cancer prevention. For instance, in a cancer prevention diet, it is important to include adequate amounts of fibre, calcium, potassium, iron, B12, folate, vitamin A, vitamin E, vitamin K, and beta-carotene, while limiting sugar and sodium intake.
Linear programming can be a valuable tool for optimizing diets to meet specific nutritional goals. It allows for the conversion of precise nutrient constraints into food combinations, ensuring that the resulting diet meets all the necessary nutritional requirements. This can be especially useful for individuals with specific health needs, such as cancer prevention, or for populations with limited financial resources who need to achieve nutritional requirements within a restricted budget.
Additionally, linear programming can be applied beyond individual diets to develop food-based dietary guidelines for larger groups, such as national food programs or dietary guidelines for specific populations, like low-income women in Malaysia. By using linear programming, these guidelines can be optimized to ensure they are achievable and affordable for the target population while still meeting their nutritional needs. This helps to address the issue that, in practice, not everyone following dietary guidelines may be receiving all the recommended nutrients.
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Cost constraints
The "Diet Problem" involves finding a low-cost diet that meets all nutritional needs. It is a question of resource optimization or, in mathematical terms, of minimization of a linear function subject to multiple linear constraints. The goal is to select a set of foods that will satisfy a set of daily nutritional requirements at a minimum cost.
The diet problem was first posed in the 1930s and 1940s, with the objective of determining the lowest-cost subsistence diet for a U.S. soldier. The problem was motivated by the Army's desire to minimize the cost of feeding soldiers while still providing a healthy diet. The "Stigler Diet Problem", posed by economist and 1982 Nobel Laureate in Economics, George Stigler, endeavoured to establish the cheapest diet that would deliver enough energy, proteins, vitamins, and minerals.
Linear programming can be used to formulate minimum-cost menus while ensuring that all nutritional criteria are met. It is a useful tool for converting precise nutrient constraints into food combinations. For example, Raffensperger used LP to study the lowest available cost of a low-carbohydrate diet in New Zealand. Introducing constraints for carbohydrates and fats, they found that energy, calcium, and fibre were the most expensive nutrients.
In another example, linear programming was used to develop a low-cost cancer prevention food plan for selected adults in Kuala Lumpur, Malaysia. The study aimed to develop a healthy and balanced menu at a minimal cost to help reduce cancer risk.
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Individual dietary needs
Linear programming is a mathematical technique that helps find the best solution among multiple constraints. In the context of dietary needs, it can be used to create a personalized nutrition plan that meets all the necessary nutritional requirements while minimizing costs. This is especially beneficial for those who have specific dietary restrictions or health goals, such as weight management or preventing chronic diseases like cancer. By using linear programming, individuals can identify the optimal combination of foods that satisfy their daily nutritional needs within their budget.
To apply linear programming to individual dietary needs, several steps are involved. Firstly, data collection is necessary to gather information about the individual's socio-demographic characteristics, anthropometric measurements, and current dietary habits. This provides a baseline understanding of the person's lifestyle and eating patterns. Then, nutritional requirements are determined based on factors such as age, gender, and activity level, considering any specific health conditions or restrictions.
The next step is to construct a linear programming model that incorporates the individual's nutritional needs, cost constraints, and food preferences. This model typically includes a set of equations that represent the nutritional requirements, with variables representing the different food options and their corresponding costs and nutritional content. By solving this system of equations, an optimal solution can be found, indicating the quantities of each food to consume to meet the specified goals.
For example, consider an individual who wants to follow a vegetarian diet while ensuring they meet their protein requirements. Using linear programming, they can optimize their food choices to include plant-based proteins like pulses, legumes, soy, and quinoa, ensuring they get enough protein while adhering to their dietary preferences. Similarly, linear programming can be adapted to various dietary restrictions, such as kosher or keto diets, to ensure nutritional needs are met within those constraints.
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Frequently asked questions
The "Diet Problem" is the search for a low-cost diet that meets the nutritional needs of a US Army soldier. The problem was first formulated by mathematician Jerry Cornfield for the Army during World War II (1941-1945).
Linear Programming (LP) is a mathematical technique that allows the generation of optimal solutions that satisfy multiple constraints at once. LP can be used to formulate minimum-cost menus while ensuring that all the nutritional criteria set by dietary guidelines are met.
Some examples of tools used to solve the "Diet Problem" include the simplex method, IBM 701 computers, Nutritionist Pro™ software, SPSS, Microsoft Excel, and NEOS Server solvers.
