Analysis of Diet Choice towards a Proper Nutrition Plan by Linear Programming

Linear Programming is an optimization technique to attain the most effective outcome or optimize the objective function (like maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships called the constraints. In this paper, we have discussed fundamental and detailed techniques of formulating LPs models in various real-life decision problems, decisions, works, etc. In the human body, an unhealthy diet can cause a lot of nutrition-related diseases. Sometimes, having a proper diet costs beyond one’s limit and it affects us to develop a diet based budget-friendly nutrition model. Our goal is to minimize the total cost considering the required amount of nutrition values required. To construct the study we took some standard values of nutrition ingredients to compute the budget-friendly values. It's quite hard to resolve most of the real-life models with a large number of decision variables & constraints by hand calculations implies the use of AMPL (A Mathematical Programming Language) coding to get the optimal result. The number of variables & constraints isn't mattered in any respect for the computer techniques used in this study. This study results in some standard values of diet plan for optimizing the nutrition for a particular person with limited costs.


Introduction
In practical life, we have to decide every step. While decision making we seek to answer the question `what is best?' Always we want the best output with limited resources. A typical example would be taking the limitations of materials and labor and then determining the "best" production levels for maximal profits under those conditions. A linear programming (LP) problem is an optimization model by which we can optimize a measure of effectiveness under conditions of allocating scarce resources and before doing that we have to formulate LP according to the given restrictions.
The problem of solving a system of linear inequalities dates back at least as far as Fourier, after whom the tactic of Fourier-Motzkin elimination is named. Linear Programming (LP) was first developed by Leonid Kantorovich in 1939 [2].
It had been used during World War II to plan expenditures and returns to cut back costs to the military and increase losses to the enemy. The three founding figures within the subject are considered to be Leonid Kantorovich, who developed the earliest LP problems in 1939, George Dantzig, who published the simplex method in 1947, and John mathematician who developed the speculation of the duality in the same year [1,3]. The method was kept secret until 1947 when George B. Dantzig published the simplex method and John mathematician developed the idea of duality as a linear optimization solution and applied it in the field of game theory. Postwar, many industries found their use in their daily planning. The LP problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a much bigger theoretical and practical breakthrough within the field came in 1984 when Narendra Carmaker introduced a replacement idea named, the interior-point method for solving LP problems. Dantzig's original example of finding the most effective assignment of 70 people to 70 jobs exemplifies the usefulness of linear programming [2,4]. The computing power required to check all the permutations to pick the most effective assignment is vast the number of possible configurations exceeds the number of particles in the universe. However, it takes only a rapid to go looking out the optimum solution by posing the matter as a linear program and applying the Simplex algorithm [5]. The idea behind linear programming drastically reduces the number of possible optimal solutions that have got to be checked.
Every person needs nutrients for their sound body. A human cannot live without nutrients. The nutrient helps us to protect our body from different diseases. We can get a required amount of nutrients for our body from various kinds of foods. The amount of nutrient that is required for our body varies from age to age.

Acquaintance with Linear Programming
Linear programming, sometimes known as linear optimization, is the problem of maximizing or minimizing a linear function over a convex polyhedron specified by linear and non-negativity constraints. Simplistically, it is the optimization of an outcome based on some set of constraints using a linear mathematical model. Optimization problems arise in all branches of Economics, Finance, Chemistry, Materials Science, Astronomy, Physics, Structural and Molecular Biology, Engineering, Computer Science, and Medicine [7][8]26].
Linear programming is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships [7]. Linear programming is a specific case of mathematical programming (mathematical optimization).
More formally, (LP) is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polyhedron, which is a set defined as the intersection of finitely many half-spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists [22][23][24][25]27].
Linear programs are problems that can be expressed in canonical form: Where represents the vector of variables (to be determined), and are vectors of (known) coefficients, A is a (known) matrix of coefficients, and (. ) is the matrix transpose. The expression to be maximized or minimized is called the objective function ( in this case). The inequalities ≤ are the constraints that specify a convex polytope over which the objective function is to be optimized.

General Form of Linear Programming
The general mathematical form of an (LP) problem is as follows: Where one and only one of the signs ≤, =, ≥ holds for each constraint in (1) and the sign may vary from one constraint to another [5,6]. Here $ ($, , ,… ….….,!) are called profit (or cost) coefficients and $ ($, , ,… ….….,!) are called decision variables.

Formulation of LP Problem
Problem formulation is the most significant part of solving LP problems. Successful optimization fully depends on the proper formulation of the problem. Formulation refers to the creating of components of the LP inappropriate mathematical relationships or structures in step with the conditions. During this section, we'll discuss how we can formulate an LP problem. The procedure for the mathematical formulation of an LP problem consists of the following major steps: [5,14] Step 1: To Identify Variables We identify the unknown variables to be determined (decision variables) and represent them in terms of algebraic symbols.
Step 2: To seek out the Objective Function We identify the objective or criterion and represent it as a linear function of the decision variables, which is to be maximized or minimized.
Step 3: To Find the Constraints We formulate the other conditions of the problem such as resource limitations, market constraints and interrelation between variables, etc. as linear equations or inequalities in terms of decision variables.
Step 4: To Add the Non-negativity Restriction We add the 'Non-negativity' constraint from the consideration that negative values of the decision variables don't have any valid physical interpretation.
Step 5: To Write Down the Entire Problem The objective function, the set of constraints, and also the non-negative restrictions together form an LP problem.

Problem Definition
Every person needs nutrients for their sound body. They can get the required nutrients from various kinds of foods [10]. In this chapter, we discuss the required amount of nutrients for a person in a week in different range levels of people. We also show a linear program for the diet problem corresponding to the required amount of food and nutrients for different ages level of people.
In this project, we, work on the formulation of real-life diet problems by using the AMPL (A Mathematical Programming Language) programming [31,26]. To establish this project paper we need so much data and information such as the nutrition value of the food, maximum and minimum required amount of nutrients for different ages, people, in a week, food cost per unit, etc. Here we have worked about 30 kinds of foods, corresponding 15 kinds of nutrients. In this project, we work on three age-levels and these are categorized as below 12 years, 12-40 years, and above 40 years. For our limitations, we have shown only the level of ages below 12 years. If any reader is interested to know the three categories you can collect the file from the authors.
We have collected the above data from various sources. Some data are collected from the internet [6,9], some are supplied by the students of medical colleges, and the Department of Food and Nutrition. Based on per week the maximum and minimum required quantity of nutrients for each age-level and nutrition value of each food corresponding to the vitamin are collected from a book which we collect from the department of Food and Nutrition [9,10].
Moreover, we collected the prices of these foods from the local market and converted these prices from taka into the dollar.

A Linear Programming for the Diet Problem
In this section, we will show the linear program for the real-life diet problem. To construct a linear program for the diet problem we consider the 30 foods and their corresponding 15 nutrients. For the age level below 12-years, the required amount of nutrients are given below: The table is given below shows the maximum and minimum amount of required nutrients for a person in a week corresponding to the nutrients [11][12][13][14][15][16][17][18][19][20][21]. Here we calculate the cost of food per unit in the dollar. The table is given below shows the number of nutrients in different kinds of food corresponding to their vitamins. In the same process, we can construct a linear program for the different age levels of people.