# Business Mathematics and Mathematical Models for Marketing

Mathematical models are basically the usage of numbers and math to describe a system. This representation of systems can be from any field; biology, physics, management, music, engineering, astrology, and even cooking. The process of actually formulating a mathematical representation is called ‘modeling’ (Aris, 1994). In the world of today, where businesses are competing fiercely for market shares and cutting down their competition, they are making more and more use of mathematical models to make their businesses edgier. If we take a basic example, we see that numbers are involved in businesses anyway, just making a proper set of observations and inferring them to make business decisions is what is being talked about (Solomon, 1999).

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There are a number of mathematical models; linear vs. non-linear, deterministic vs. probabilistic, lumped vs. distributed patterns, and lastly static vs. dynamic. Statistics is one field that is used immensely and extremely useful in businesses; some of its used models are population growth models, probabilities, frequency distributions, and even differential equations. Other more complex examples are the Model of rational behavior for a consumer, the Model of a particle in a potential field, the neighbor-sensing model, and even the Malthusian growth model.

A mathematical model needs a lot of work and understanding; it consists of two or more parts but the basic two parts are the variables and the operators. Variables are the unknowns that the person wishes to work on; these are the ‘abstractions’ that need to be defined and then worked around within the equation. The next part is the operators, which are the signs that join the variables and turn them into an equation; which basically helps to make sense of the relationship between the variables and gives an understanding of the equation. If all operators and variables together represent a linear situation, resulting in a linear equation, the entire model becomes linear in nature. However, the linear model can consist of a few or one maybe non-linear expression; this helps to build the model and complete it as one big model for the working of the business. A simple example is of a statistical linear model which normally has a linear relationship but it does require non-linear predictor variables, while differential equations at times have non-linear expressions as well (Moore, 1992). Basically, if a mathematical model has a function that is the main objective of the model and is non-linear, then the entire model becomes non-linear. The difficult and complex non-linear models can be converted into linear models by a process called linearization because non-linear models are associated with ‘chaos’ and ‘irreversibility’.

The mathematical models mentioned above are very intensive and are meant for higher educational levels. The model that I will be using is the frequency distribution model, which involves histograms, medians, mean and even modes.

The company from the UAE that I have chosen to discuss these mathematical models is Nakheel Properties. Nakheel Properties is a real estate developer that is based in Dubai and has also run a few land reclamation projects in Dubai and surrounding areas. Nakheel actually means Palms or Palm trees in English, being an Arabic word itself. Some of its work involves Palm Islands, Dubai Waterfront, and Universal Islands. Land reclamation is not the only area it deals with; it has even worked in residential schemes and shopping arcades such as Dragon Mart. The world has become so fast and competitive that producing good quality products just is not enough; companies need to make sure that their customers know about their existence, good work and need certain tools for promotion. All these point towards one thing: Marketing. Production is really not the only important task, marketing the product in the right places, right time and to the right people is essential. Marketing strategy is the course of action to follow to market the product in the market; it determines how to use the company’s resources and to divert them in a direction that results in maximum consumer satisfaction, leading to an increase in sales. It is strategic planning that decides upon the target market, market segmentation, positioning, and even the marketing mix and other key decisions. The biggest competitor of Nakheel Properties is Emaar Properties; which means that the marketing strategy of Nakheel Properties should be extremely impacting to make sure that the competitors are driven out of the market and the customers are attracted towards it.

One of the best promotion tools is advertisements. This is direct interaction with the customers using the most basic mediums of communication such as television, newspaper, and outdoor ads. Keeping track of television ads is very important to check how much money is being spent for how much television ad coverage and repeats; how long the ad is for, how many people are viewing them, how many times a particular channel is showing, so on and so forth.

Nakheel Properties use statistical business models for keeping track of their marketing department, and mainly their television commercials. Although, their business model is extremely simple because it is making usage of frequency distribution and basic statistics like mean mode and median, Nakheel Properties is doing an extremely good job in maintaining its records and inferring the data to be able to make decisions (Lange, Little & Taylor, 1989). Nakheel Properties needs to keep a good and extremely accurate track of how many ads are being shown on a particular channel in a day. These statistics help them infer whether the customers are exposed to the ads are not, whether the amount that they are paying for the ads is being justified by showing as many numbers of ads, whether the right people are being targeted or not (channel type) and many other things.

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The data collection for the Marketing department, in this case, is not very difficult; it has one good source and another survey being carried out. This is very different from what the Human Resources department as Nakheel Properties has to go through – they usually need to have discussions with the employees, then have one on one interviews, focus group discussions, a 360-degree feedback program, performance evaluations sheets, and balanced scorecards before deciding whom to give a pay rise to and whom to give a warning or promotion to. Compared to this, the Marketing department does not have much work to do. The data is collected from three sources; firstly, one person/employee is assigned one channel which they watch all day long and count the number of times the ad is put on air; secondly, the signed official document that states how many times the ad is supposed to run and the payment made for it; and thirdly and lastly, a survey carried out by company representatives who ask the general public with an emphasis on the target audience whether they remember seeing Nakheel Properties ads on TV, if yes, how many times and on which channel. This helps figure out whether the money given to the channel for advertisements is worth it or not, should the company change the channel, which channel is the ad viewed on the most, etc.

The frequency distribution table is formulated, with two axes – the x-axis being the horizontal one, representing the channels, with one separate histogram bar for each channel; and the y-axis being the vertical axis represents the frequency which is the number of times the ad was aired on that particular channel per day. This is the data that is collected from the employees who sit and watch one particular channel and record the number of times the ad was seen. The channels that show these ads are Ajman TV, Al-Arabiya News Channel, E-Vision, One TV Dubai, Infinity TV and Sharjah TV. For May, 2009, Nakheel Properties collected the following data: Ajman TV – 10, Al-Arabiya News Channel – 8, E-Vision – 7, One TV Dubai – 6, Infinity TV – 10 and Sharjah TV – 7. This results in a frequency distribution graphs such as below:

The contract with all channels is of 10 ads per day; judging from the data, only Infinity TV and Ajman TV are fulfilling this requirement and taking money for their work; apart from that, none of the other channels are working with honesty and skipping a few times the ad appears on TV and taking money for something they are not doing, with One TV Dubai being the lowest. This is the data that can be of help for the Finance Department as well; they can either cut the payment that they have to make to the channels that are not showing the ads in full, or can either cancel their deals. They can even speak to the company and ask what the reason was. In this case, Nakheel Properties did not make the full payment for this act and cut money for the number of times the ads were not shown on air.

From the data collected from the general public about where the ad was seen, whether they remember the ad or not were collected using surveys in public markets on one on one basis. Mostly, adult males and females were targeted because children and teenagers do not really show interest in property builder’s ads anyway. The data showed that our of a sample size of randomly selected 100 individuals, 23 did not remember seeing any ad of Nakheel Properties and 77 remembered the ad. Of the 77 who did remember the ad: 28 saw it on Ajman TV, 20 on Infinity TV, 21 on Sharjah TV, 8 watched it on Al-Arabiya News Channel, with a few results overlapping.

We have, thus, seen how statistical inference, being a mathematical model has helped this company so much in making decisions. If these models are studied and implemented properly with a good analysis of the results, they can do wonders for businesses and the decisions these businesses make can almost always be accurate and risk free.

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## Works Cited

Aris, R., (1994), Mathematical Modelling Techniques; New York : Dover

Lange, K..; Little, R.; Taylor, J.; (1989); Robust Statistical Modeling Using the t Distribution Journal of the American Statistical Association. Web.

Moore, D., (1992); Teaching Statistics as a Respectable Subject; Statistics for the Twenty-First Century; Washington, DC: The Mathematical Association of America Nakheel Properties.

Solomon, R.C. (1999), A better way to think about business, Oxford University Press.

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