The Diet Problem: A Brief History
The diet problem is one of the first optimization problems being studied back in the year 1930’s and 40’s. This was first motivated by the Army’s desire to meet the nutritional needs of the field GI’s while minimizing the cost. One of the early researchers to study this problem was George Stigler. He made an educated guess of the optimal solution to linear program by using a heuristic method. His guess for the cost of an optimal diet was $39.93 per year (1939price). In the fall of 1947, Jack Laderman of the Mathematical Tables Project of the National Bureau of Standards undertook solving Stigler’s model with the new simplex method. This was the first”large scale” computation in the optimization. The linear program comprised of nine equations with 77 decision variables. This took nine clerks by using hand-operated desk calculators120 man-days to solve for the optimal solution of $39.69. Stigler’s guess for the optimal solution was o? by only 24 cents per year.
Read the article authored by the George Dantzig form or detail on the diet problem.
Description of the Diet Problem:
The main goal of diet problem is of indan”optimal” combination of foods which will satisfy all the daily nutritional needs of an individual. The problem can be formulated as a linear program with an objective and constraints that require that daily nutritional needs be satisfied. We comprise constraints which regulate the number of calories and amounts of vitamins, minerals, fats, sodium and cholesterol in the diet.
The original mathematical formulation is simple, however you will find out (from Dantzig’s article) by running the simple model, the diet returns is far from optimal. While our nutritional needs can be met, there are serious concerns for taste and variety. Since Dantzig’s diet problem describes, we would never drink gallons of vinegar nor comprise a few boullion cubesinour meals. Though, with careful design of the linear or integer program, healthy optimal meals have been created very successfully.
Formulate the diet problem for your own:
• Choose a variety of 100 favorite types of food which you would eat. Do not use any fast food or pre-processed food. Therefore, as and which will consist of bread and ham, or peanut butter, jelly, etc. If bread is chosen in the solution, then, ham/peanut butter and jelly will as well be selected and so on.
• By using the nutrition information from US Department of:
Agriculturehttp://www.nal.usda.gov/fnic/foodcomp/ to determine the nutritional value and cost (per serving size) of each of the food you select.
• Generate the AMPL data file.
• Choose an objective of your choice.
• Formulate a linear or integer program by using AMPL and hence the feasible solutions give you the food combination which satisfies the daily nutritional requirements for a healthy individual.
• Solve the linear programming CPLEX and LINDO and interpret the outcomes.