Question: Multiple channel marketing campaign. Potential customers are divided into \(m\) market segments, which are groups of customers with similar demographics, e.g., college educated women aged \(25-29 .\) A company markets its products by purchasing advertising in a set of \(n\) channels, i.e., specific TV or radio shows, magazines, web sites, blogs, direct mail, and so on. The ability of each channel to deliver impressions or views by potential customers is characterized by the reach matrix, the \(m \times n\) matrix \(R\), where \(R_{i j}\) is the number of views of customers in segment \(i\) for each dollar spent on channel

\(j\). (We assume that the total number of views in each market segment is the sum of the views from each channel, and that the views from each channel scale linearly with spending.) The \(n\) -vector \(c\) will denote the company's purchases of advertising, in dollars, in the \(n\) channels. The \(m\) -vector \(v\) gives the total number of impressions in the \(m\) market segments due to the advertising in all channels. Finally, we introduce the \(m\) -vector \(a\), where \(a_{i}\) gives the profit in dollars per impression in market segment

1. The entries of R, c, v, and \(a\) are all nonnegative.

1. Express the total amount of money the company spends on advertising using vector/matrix notation.
2. Express \(v\) using vector/matrix notation, in terms of the other vectors and matrices.
3. Express the total profit from all market segments using vector/matrix notation.
4. How would you find the single channel most effective at reaching market segment 3 , in terms of impressions per dollar spent?
5. What does it mean if \(R_{35}\) is very small (compared to other entries of \(\left.R\right)\) ?

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