Hi There all!

my 1st post, i`m in need for some real help! i have this linear programming problem and i`m really struggling to put togather the constraints to solve the problem! any help would be great, i`ve been looking at this for a couple days now and have got nowhere! . What i`ve been trying to do is to formualte contstraints for each area in the form of (Advertising) + (Sales) < or = Profit, but i`ve had 0 luck...

I`ve attached the file to this post as well!

The Effect of Advertising on Sales

In order to investigate the effect of advertising on sales of one of its products (product P), a company has carried out an experiment covering 15 television areas. It has varied the level of spending on television advertising for the product, so that some television areas have more advertising than usual, some have less and some continue at the normal level. The company has measured the levels of sales of the product in each television area resulting from the experimental pattern of advertising expenditure, and has also produced an advertising weight measure, which gives an indication of the cumulative amount of advertising in each area.

T

he results of the experimental variation of the advertising weight are as follows:

TV area Advertising weight (units) Sales (in standardised units)

1 99 100

2 40 59

3 161 121

4 98 99

5 40 60

6 158 118

7 102 102

8 41 60

9 159 120

10 101 98

11 40 61

12 160 118

13 100 101

14 39 60

15 162 123

The advertising weight can vary from the planned level because it is often not

possible to book exactly the numbers and timing of advertising slots specified by the

media plan.

The sales measured in standardised units have been adjusted for the size of the TV area by dividing the annual sales in tonnes for the area by the number of homes in the area that receive terrestrial television to find a sales/home figure, then expressing this sale/home figure as a percentage of the (annual sales/television homes) for the whole country last year.

The company’s total sales of product P for the whole country last year were 26,024 tonnes. The totalvolume of sales is expected to be at the same level this year.The company also has the following information about the media cost of advertising,which can be taken as representing the marginal cost of advertising, and the unit profit contribution of sales, which gives the marginal profitability of each unit of extra sales:

Media cost of one unit of advertising weight (across the whole country): £650,000 Unit profit contribution of sales: £4980 per tonne

The company is planning to develop a model that relates sales to advertising weight, and to use this model to find the optimum level of advertising weight (i.e. the level of advertising weight that maximises the marginal profit from sales less the media cost of advertising).

In developing its model, the company will make the following assumptions about the relationship between advertising and sales:

The level of sales increases with the level of advertising (i.e. there is a positive relationship between advertising and sales)

Diminishing returns to advertising: the positive effect on sales of an increase of one unit in the level of advertising will reduce as the levels of advertising and sales increase. (Putting this another way: the next unit increase in advertising weight will have less effect on sales than the last unit increase.)

There would be some sales even with no advertising.

There is some maximum level of sales that will not be exceeded however high the level of advertising.

The number of homes receiving terrestrial television in each television area is the same this year as last.

The company believes that it is acceptable to carry out the analysis ignoring the effect of advertising that uses other media, including satellite and cable television and the web.

While sales can be affected by factors other than advertising, if the advertising weight was the same in each television area, the level of sales would tend to be the same in each area. For example, if the advertising weight was 100 units in each area, the level of sales measured in standardised units would tend to be the same in each area.

Required

(a) You are required to develop a suitable model based on the information provided (showing how you have found your model and what the model is) and to use this model to advise the company on the optimum level of advertising weight.