London's Tree Nursery - Qm Problem

LONDON'S TREE NURSERY

Model Problem Solving

London's Nursery is a business that grows and sells evergreen trees. Here lately London's has been looking into purchasing some new land in order to be able to grow some additional trees. This new land purchase will just be intended for the production of Colorado Blue Spruce trees and Concolor Fir trees. The London's are looking at a section of land that is ten acres big. Before London's decides to buy this land they want to know the amount of profits that they will be able to make off the land with the two different types of trees. They also want to know how many of each type of tree they will be able to plant on this section of land.

The model that will be used to determine the best option for London's will be an Integer Programming Model and this model will be a total integer model. The reason why I selected to use a total integer model is because of the fact that you can not technically grow or sell only part of a tree. The issue of the amount of land can be divided out into fractions of land, but when objects such as these are being dealt with, fractions are not going to be able to properly give you a precise answer. When you use a total integer model, all of the decisions variables are required to have integer solution values. So, instead of coming up with an answer that might say you should produce 125.4 Colorado trees and 136.8 Concolor trees, your answer will not require any guessing or rounding down to try to determine the results. With a total integer model, your answer will be exact and it won't require taking any chances.

The new trees that London's plan on growing on their new land have different requirements. The Colorado Blue Spruce requires about five square feet of room, and the Concolor Fir tree normally needs around seven square feet of room to properly grow and be maintained. The two trees also require different amounts of labor time in maintaining them. During the time that the nursery will have the trees a Colorado tree needs 1.5 hours of labor, but since the Concolor tree is more of an easy adapting tree they only need 1 hour(s) of labor a week. The London's Tree Nursery has an extra 300 hours that they plan on using for these trees. The Profits off of a Colorado Blue Spruce is $41 a tree. The profits off of a Concolor Fir are $44 a tree.

The problem that the London's will be facing on their new land will be to try to decide between the Colorado and Concolor trees. The best way to make the most amount of profit is the deciding factor. The problem is to decide how many of each tree to plant. What is going to make this difficult for London's is the fact that there are several different constraints that they are going to have to take into factor. Normally someone would just assume that to make the most profits you would just plant more of the tree that will bring or makes more money, but this situation is different. In this problem the Colorado Blue Spruce makes the least amount of profits, but at the same time it also requires a less amount of space than does London's other tree; the Concolor Fir. So even though the profits are less, London's would be able to plant more Colorado trees. Now on the other hand, there is another constraint. While the Concolor does require more space than the Colorado it does not require as much labor, and since London's only have a set amount of time that they can spend on labor, this is also going to have a factor in their decision. And last but not least is probably the biggest constraint in London's goals of making the most profits and that is the amount of land that they will be using. The land that London's will probably be buying is a section of ten acres. So, now you see why using something such as management science can begin to come in handy when trying to decide the best approach to take on a problem.

To be able to solve this problem for London's Tree Nursery I will use the software program called QM for Windows. This program allows a user to plug in information into tables once the constraints are determined and get some sort of answer. The program basically does all of the math for you, but it is only as correct as the information and the way that the information was put into the program by the user. First off I have to look at the information that is provided by London's, and try to figure out what exactly the variables, constraints, and parameters are to help to begin putting together a model to solve this problem.

From looking over the information I know that London's wants the maximum amount of profits that they can make, so that easily tells me the objective of the model will be to maximize for profits. Next, I determine what the variables are, and since the profits are being taken to their max, I assume that the variables are linked to whatever is going to be making the money for London's and that would be the Colorado Blue Spruce trees and the Concolor Fir trees. So, now my first line of the model contains my objective and variables.

Maximize Z = 41x1 + 44x2

The next step in forming the model is to determine what the constraints are going to be. With the data, I know that London's has only a certain amount of land, and only a certain amount of labor time, so this would make the labor and land both a constraint. The information for London's states that there are to be no more than 300 hours spent on labor time, and the amount of labor for a Colorado tree is 90 minutes and a Concolor tree is only 60 minutes. The constraint for labor would be:

90x1 + 60x2 < / = 18,000 minutes of labor

The labor constraint was changed or converted over from hours to minutes. This makes things a little bit simpler because you don't have to work with fractions that hours may have caused. The last constraint would have to be the amount of land that the London's are preparing to buy. Again, from the information that London's provided I know that each Colorado tree needs to have at least five square feet, and each Concolor tree needs to have at least seven square feet of land to be able to properly grow and be taken care of. The information also gives us the total amount of land that London's will have to work on. The section of land that London's are looking at to purchase is a section of ten acres. Again, to make things easier I will convert the amount of land into square feet since the information concerning the trees is already using square feet, and it will save a lot of confusion later on into the problem. So, ten acres of land converted into