IE 405 CS 5.docx

IE 405 Case Study 5

May 1, 2014

George Monninger

Steven Nguyen

Shane Davis

Kusumaker Gupta

Introduction

A diet plan is an essential tool for healthy and comfortable living. Throughout the course of the semester, a diet plan has been constructed and analyzed using various linear programing techniques learned throughout the semester. Linear programming can be used in many ways to optimize and solve problems in many industries. Some common examples of situations involving linear programming analysis include, but are not limited to transportation, manufacturing, and the cost of living.

Constructing the diet plan required a list of at least 20 items. These items are all from various food groups and were selected at the discretion of the team members associated with the project. A number of nutritional constraints were added to the food list, consisting of calories, fat, protein, carbohydrates, sodium, fiber, and cholesterol. Initially, the objective of the case study sequence was to determine the optimal choice of foods from the list based on the nutritional constraints while minimizing cost. Using the lists of foods and nutritional constraints in case study one, it was possible to conduct a series of experiments analyzing additional constraints, the theory of duality, and variety maximization using integer programing.

Problem Statement

In case study one, an issue is presented that exists every day for almost everyone, which involves the development of a diet plan that is financially feasible and healthy. However, at the same time this diet plan should also meet certain nutrient requirements such as calorie intake, sodium, protein, fats, and carbohydrates. Using the website www.fatsecret.com, it was possible to develop a list of foods from various groups of the food pyramid that would act as a diet plan. The food list developed consisted of twenty one foods chosen by group members based on popularity and taste. A list of nutritional constraints used through the course of these case studies is shown in table 1 of the appendix.

Although excel solver failed to select an appealing and adequate list of foods in case study one, it was possible to achieve a more appealing diet plan in case study two by adding additional constraints. It can be seen by in answer report 1 that the solver only selected swordfish, bread, and bananas instead. This selection was made due to the lack of constraints on the current diet plan. Case study two required solver to select at least one item from each food group at least once, with a limit of three items per food group. The food groups present in the diet plan were meats and poultry, fish, fast food, drinks, bread, and vegetables.

Case study three beings to focus on the concepts of sensitivity analysis, shadow prices and dual linear programs. The first part of the case study dealt with analyzing the sensitivity report from case study one. We were tasked with determining what would happen to results on the meal plan if we were to adjust values slightly.

The second half of Case Study three involved testing the theory of duality by modeling the dual linear program for the original meal plan from case study one in Excel. The theory of duality states that the solutions for both the primal and dual linear programs. The meal plan was able to be converted to its dual linear program by manipulating the constraints according to duality theorems.

Case study four focused on variety maximization rather than cost minimization of the food list, as seen in case studies one and two. The first part was hard on cost minimization with nutrition constraints but with help on factors and variety maximization it would give more items on the dietary plan and secondly also make it more realistic by given whole number. To achieve a more realistic dietary plan based on nutrients, we combined the factors from case study one with case study two, keeping nutrient requirements from the given table in case study four. To get a more realistic number on whole number we used Integer function in the constraints of excel solver for are linear programming problem.

Solution Procedure

In the first case study, a linear program was designed using Microsoft Excel in order to minimize cost while selecting a group of foods. In the Excel spreadsheet attached in the appendix, it can be seen that the objective function consists of minimizing costs, while fulfilling the necessary nutritional constraints. Excel solver was used in order to set up the nutritional constraints and solve the diet plan for the optimal solution.

In order to solve case study two, a number of additional constraints had to be created. Since the requirement of this case study was to select an item from each food group at least once, but no more than three times, individual constraints were added to the excel spreadsheet for each item, represented by 1s. In order to further balance the diet plan, nutritional constraints from the first case study regarding calories, fat, protein, carbohydrates, sodium, fiber, and cholesterol were implemented into this spreadsheet.

The first half of case study 3 can be solved by having a strong grasp of sensitivity analysis concepts. It is important to understand how the slightest of changes to values in the linear program can change the optimal value. Shadow prices can affect the optimum solution of a problem.

The next section of case study 3 involved using the constraint and variable rules that can be found in Table 2. It is very important to properly convert the constraints and variables to their proper duality form for the linear program to produce an optimal value.

Computational Results

Using the excel solver, it was possible to calculate which items we needed to purchase in order to optimize our dietary plan. As seen in answer report 1, Excel Solver computed a diet plan adhering to the nutritional constraints while minimizing costs. Unfortunately the diet plan was nothing near to realistically because the diet plan only consisted of bananas, swordfish, and bread, which would be difficult to survive on.

In reference to the answer report for case study 2, it can be seen that a total of eight items were selected from the list of foods. The answer report generated provides an optimal solution that minimizes cost, selects the appropriate amount of foods from each food group, and adheres to the nutritional requirements outlined in the first case study. Although the foods selected in our list may not be the most optimal for healthy living, it is a step up from case study 1, which only selected three items at random to minimize cost while meeting the nutritional restrictions imposed upon it.

The sensitivity analysis of case study 3 reacted to the subtle changes made as expected. For example, when one gram of fat was added to the meal plan, the price of the meal plan did not change because fat has a zero value as a shadow price. On the other hand, since calories has a shadow price of 1, when we add one extra calorie to our diet, it ends up costing us an extra $0.0005486 to our meal plan. In this section of the case study, the basic variables are bananas, swordfish, and white bread. Only these three variables are able to fluctuate in the linear program as long as it does not affect the optimal solution. As with any dual solution, the spreadsheet selected the same foods as the primal linear program, and also suggested selecting the same amount of the basic variable foods.

In case study four we made the dietary plan more realistic and appealing. To achieve this we used Integer in Constraints in the solver. Using combination of constraints from previous case study we were able to achieve the goal given to us. This resulted in altogether new solution for our original problem from case study one has it gave us more variety. This can be easily noticed by comparing numbers from case study one to four for example in case study one it impossible to quantitatively analyses 14.8 bananas compared to 3 Bananas in Case study 4. It seems clear to that there was real maximization of variety in case study 4 as compared to previous one as in case study 4 solving with solver gave 8 different food and drinking items as compared to 3 in the first one.

Conclusion

Through the analysis of this series of case studies, it was possible to develop a realistic diet plan using excel solver that appeared to be realistic and appealing. Although the constraints in the first case study failed to produce a realistic diet plan, a variety of additional methods were tested in order to observe the effect on the solution of the diet plan. Additional methods included choosing one of each food type, testing the theory of duality, and using integer programing to maximize variety. It can be seen throughout the attached answer reports that each theory produced the desired results successfully. This proves that linear programming and Excel Solver are tools necessary and useful to solve problems such as diet plans with quite a degree of accuracy, as long as the desired constraints are entered correctly. Excel solver is just like any other calculator, in that it is only as intelligent as the user.

Appendix

Table 1: Nutrition Constraints:

Nutrient

Calorie

Total Fat (g)

Saturated Fat (g)

Protein (g)

Carbohydrate (g)

Sodium (mg)

Dietary Fiber (g)

Cholesterol (mg)

Minimum Requirement (Daily)

2500

50

375

30

Maximum Requirement (Daily)

80

25

2400

300

Recommended Daily Allowance (RDA)

Table 2: Duality Constraints

Max Problem

Min Problem

Constraints

Variables

=

Free (unconstrained in sign)

Variables

Constraints

Free (unconstrained in sign)

=

Works Cited:

"Foods." Foods. N.p., n.d. Web. 29 Apr. 2014. .