The initial impetus for this project was the mathematical optimization methods that were used to generate Obaminoes. As we worked on the project we uncovered a great diversity of links to the mathematics curriculum.
Students used dominoes in their mathematics classes throughout the spring as we worked on our projects. There were many great activities that the dominoes were used for. Some of these activities are described in the attached .pdf:
Obaminoes is a striking piece of art, yet determining how to best arrange dominoes does not seem to be illustrative of a pressing “real world” problem where mathematics’ great power can be used. In fact, the creation of Obaminoes uses a mathematical algorithm known as linear programming – which has been called “the most important real world mathematics problem.” Essential to every major shipping, distribution, transportation, telecommunication, large retail, and production company, this tool’s importance cannot be understated.
So what is linear programming? Linear programming is a method of determining how to allocate scarce resources among competing needs to determine an optimum outcome.
Example:
- On a warm spring day two friends decide to open a lemonade stand. They have an idea to make and sell chocolate chip cookies too. They decide on prices, make a sign, and have everything planned. But then they realize that they have a limited amount of sugar and limited amount of time to get their stand open while people are out and about. The amount of sugar and time they need both depend linearly on how many cookies and how much lemonade they make. The cookies take much more time to make but require less sugar than the lemonade. What the friends need to decide on is how many cookies and how much lemonade they should make to earn the highest revenue from their lemonade/cookie stand.
This problem is easily understood by even the youngest of students. It is a wonderful problem because it can be considered on so many levels: using hands-on manipulatives with grades K - 3, for collecting data with grades 2 - 5, for graphing with grades 2 - 6, as pre-algebra for grades 4 - 8, graphs of linear equations for grades 8 - 10, for solving systems of simultaneous equations in grades 8 - 11, and all the way on to the study of linear programming in college and beyond.
A detailed handout with information on brining this problem into K - 8 is available as a pdf:
Linear programming was developed during the Second World War by George B. Danzig and others to solve optimization problems very much like this. Today this tool and its variants are a central part of the field known as operations research. In their classic textbook Introduction to Operations Research, Hillier and Lieberman tell us:
- The development of linear programming has been ranked among the most important scientific advances of the mid-20thcentury... Today it is a standard tool that has saved many thousands or millions of dollars for most companies or businesses of even moderate size in the various industrialized countries of the world... A major proportion of all scientific computation on computers is devoted to the use of linear programming. (1995; p. 25)
Hillier and Lieberman go on to describe how during the mid-1980's a linear programming team develop a "refinery linear programming system" for Citgo Petroleum Corporation to help determine the efficiency of their supply and distribution of petroleum. This system determined that at any given time Citgo had $116.5 million in excess inventory resulting in a $14 million per year loss in interest expenses.
The employment statistics in this area also give us some idea of the importance of these areas. In 1995 Hillier and Liberman reported:
- The U.S. Bureau of Labor Statistics predicted that OR [Operations Research] will be the third fastest- growing career area for U.S. college graduates from 1990 to 2005. It is also predicted that 100,000 people will be employed as operations research analysts in the United States by the year 2005.
While there was no category for Operations Research Analysts, in the study by Careers.com that was reported in the Wall Street Journal on 26 Jaunary, 2009 (available online at:http://online.wsj.com/article/SB123119236117055127.html) the top three ranked jobs are:
- Mathematician
- Actuary
- Statistician
The importance of mathematical tools like linear programming and job prospects for employment in related areas cannot be understated.
So how is the creation of Obaminoes a linear program? Well, our dominoes are the scarce resources - we have only a fixed number of each different tile configuration. Our objective is to find the optimal arrangement of dominoes so the difference between the grayscale of the domino arrangement and the pixilated image of Obama is minimized. Omitting quite a few of the details, the link to other problems described above should be apparent.