Download IE 405 Case Study 5

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 1’s. 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
Minimum
Requirement
(Daily)
Maximum
Requirement
(Daily)
Calorie
Total
Fat (g)
Saturated
Fat (g)
2500
Protein
(g)
50
80
Carbohydrate Sodium
(g)
(mg)
375
25
Recommended Daily Allowance (RDA)
Dietary
Fiber (g)
Cholesterol
(mg)
30
2400
300
Table 2: Duality Constraints
Max Problem
Min Problem
Constraints
Variables
≥
↔
≤0
≤
↔
≥0
=
↔
Free (unconstrained in sign)
Variables
Constraints
≥0
↔
≥
≤0
↔
≤
Free (unconstrained in sign)
↔
=
Works Cited:
"Foods." Foods. N.p., n.d. Web. 29 Apr. 2014. <http://fatsecret.com/calories-nutrition/>.