# lab10 - ENGRI 115 Engineering Applications of OR Fall 2007...

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ENGRI 115 Engineering Applications of OR Fall 2007 The Diet Problem Lab 10 Name: Objectives: Give the students practice in formulating linear programming problems Introduce students to the idea of linear relaxation Demonstrate the relationship between an integer program and its linear relaxation Demonstrate the idea of sensitivity analysis in linear programming Key Ideas: Objective function Constraints Optimal solution Fractional solution Integer solution Linear program Integer program Linear relaxation Sensitivity analysis Part #1 - The Diet Problem During the past few years, people have become more and more aware of what they eat and how it aﬀects their physical and mental health. We now know that if we expect to live a long and healthy life, we must have a well-balanced diet. Too much fat or sodium in our diet can lead to serious health problems. Similarly, diets deﬁcient in essential vitamins and minerals should be avoided. We will consider an application of linear programming in constructing a healthy well-balanced diet. The ﬁrst part of this lab is aimed at helping you to formulate this problem as a linear programming problem. We can view the diet problem in the following way. We are given a variety of foods that we could buy to achieve a balanced diet. For example, we might consider diet consisting of 2% milk, spaghetti (with sauce), peanut butter, wheat bread, tomato soup, bagels, and oranges. To make things simpler, we will specify the variables of this linear programming formulation. We will use x ( food-type ) to specify the number of daily servings of food-type that you are willing to consume. We wish to write constraints that capture whether one can satisfy certain daily requirements with just these foods. First of all, there is the boredom constraint. Write constraints that ensure that the diet speciﬁes that we eat at most 10 servings/day of each food-type. 1

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Next consider the total number of calories consumed. A proper diet requires that you consume between 2000 and 2250 calories per day. Write two constraints that ensure that the diet speciﬁes that we eat an appropriate number of total calories. To write this constraint you need to know how many calories are in one serving of each of the food-type s. While we will supply this data later, let cal ( food-type ) represent this data, in that it is equal to the number of calories in one serving of food-type . Of course, we could repeat this sort of constraint with any number of (un)nutritional requirements, such as cholesterol, fat, sodium, dietary ﬁber, carbohydrates, protein, vitamin A, vitamin C, calcium, and iron. For each such requirement, what additional data would be needed to incorporate that requirement into our
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## This note was uploaded on 06/01/2008 for the course ENGRI 1101 taught by Professor Trotter during the Spring '05 term at Cornell.

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lab10 - ENGRI 115 Engineering Applications of OR Fall 2007...

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