Retention Rates

My senior research colloquium involved0 analyzing the retention rates between the past five academic classes' freshmen and sophomore years through the use of linear programming. Linear programming is a mathematical method for determining a way to achieve the best outcome for some list of requirements. Within the aforementioned years, I have reorganized the students into two categories: academic aptitude and financial need from the data received from the admissions office. From there, I analyzed these years and found the average retention rates. I determined how much we have spent on average from these forty-two different tiers, and the differences between the freshmen and sophomore tiers. I then ran multiple programs maximizing the retention rates by giving each program different restrictions, making sure that the number of retained students did not exceed the class size, or that the school would not spend more than was already allotted. These findings have given me an understanding of the retentions of the different student tiers, and whom the university should be accepting.

In order to fully understand the project, we must first look at what 'operations research' actually is; its history and the history of linear programming. Overall, operations research utilizes techniques from mathematical modeling, statistical analysis, and mathematical optimization. Some other terms used for this field are management science or decision science. My colloquium focuses on mathematical optimization, more specifically; linear programming and its simplex algorithm.

Charles Babbage performed early work similar to operations research by investigating the cost of mail in the United Kingdom. His research in the transportation of mail and its sorting costs led to what was known as the "Penny Post" in 1840 (Sodhi). However, it would not be until 1937 at Bawdsey Research Station in the United Kingdom, and the start of World War II that the modern field of operations research came about. The superintendent at Bawdsey Research Station, A.P Rowe used operations research to analyze and improve the United Kingdom's early warning radar system which helped the British military to locate and track enemy aircraft (Encyclopedia Britannica). During World War II, many scientists from the United Kingdom and the United States used operations research to make optimal decisions in areas such as logistics and training programs.

Additionally, another important person to the field of operations research is Patrick Blackett. His use of operations research greatly assisted many areas of the British military. By working with the Royal Aircraft Establishment, he was able to reduce the amount of rounds needed to shoot down an enemy aircraft from 20,000 rounds at the beginning of the war to 4,000 rounds by 1941 (Rajgopal). With Royal Air Force Costal Command, Blackett performed an analysis for the RAF's bombers to find and destroy submarines (Rajgopal).

Originally the undersides of the planes were painted black for nighttime operations because they thought that color would provide the most coverage. Later, they ran tests to see which color would be best to camouflage planes. The test results showed that the best camouflage did not come from planes painted black, rather from planes painted white. These new white planes were not spotted until they were 20% closer, versus their black alternatives, which then indicated a change that caused 30% more submarines to sink (Rajgopal). This is because the submarines were not able to dive deep enough before the planes were in target range. Other findings in operations research helped bombers double their on target rates. Further research showed that a wolf pack of three United States submarines were the most effective to engage targets. (Milkman, 1968) Operations researchers were even placed among the armed forces in Normandy after its invasion in 1944. During these placements, they analyzed the effectiveness of the allied military strategies (Rajgopal). After the war, with the development of the simplex algorithm in 1947 for linear programming and the advances in technology particularly with the computer, we can now solve problems with millions of variables and constraints in a matter of minutes.

As previously stated, linear programming is a mathematical method for determining a way to achieve the best outcome for some list of requirements. A more formal definition is that in mathematical optimization, linear programming is a technique of an objective function. (Sinha) This is subjected to linear equality and linear inequality constraints. The theory behind linear programming drastically reduces the number of possible optimal solutions that must be checked. Leonid Kantorovich developed the earliest linear program in 1939. His method helped plan expenses during World War II in the USSR from which his program reduced the costs for the army and increased the losses of the enemies (Overton).

Kantorovich's method was kept secret until 1947, when George B. Danzig published his simplex algorithm method. His example of how to find the best assignment of seventy people for seventy jobs exemplifies the usefulness of linear programming in all forms of industry (Overton). Postwar, many industries found its use in daily planning, production, transportation, and technology. It is always in a company's best interest to maximize profits while minimizing costs, so many management issues in business can be classified as linear programming problems.

The standard form for a linear programming function has three parts to describe the problem. The first part is the function that needs to be maximized which will follow the form:

f(x_1,x)=c_1 x_1+c_2 x_2.

The next part of a linear programming problem is the problem's constraints. These constraints will follow the form:

a_1 x_1+a_2 x_2≤ b_1

a_3 x_1+a_4 x_2≤ b_2

a_5 x_1+a_6 x_2≤ b_3

(Sinha)

In this form, the constraints must be less than or equal to some set value. However, this does not have to be the case. Sometimes the inequality can be written in chained notation. The third part of linear programming is to state that there can be no negative variables such that: x_1,x_2≥0. The problem is then transformed into matrix form and written as:

max⁡{c^T x|Ax≤b ⋀▒〖x≥0}〗

So as to complete my research colloquium I needed to use a computer program to solve my linear programs quickly and accurately. Therefore, I used the solver add-in from Microsoft Excel. This program helps to easily perform many different maximization problems utilizing

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