# Linear Programming Involving Indices

1. Sep 19, 2012

### I Like Pi

1. The problem statement, all variables and given/known data

A marketing team is given a budget of 30,000 to advertise a new show. It wants to promote this show using two methods, internet and posters. The executives have decided that the overall budget used on the internet can't be greater than double the budget used on posters. Furthermore, at the least, a quarter of the total budget must be used on each method. The marketing team uses an index that measures customer contact for every dollar spent on the ad. The index may range from 0 to 100 (bigger meaning more customer contact). The internet has an index of 60 and the posters have an index of 90.

How should the marketing team organize it's budget to maximize contact?

2. Relevant equations

3. The attempt at a solution

Let's statement:
let x = $spent on internet ad let y =$ spent on poster ad

Constraints:
x≤2y
x+y≤30,000
x,y≥7,500

Objective:
This is where I am having a tremendously hard time. I know that we want to max customer contact. So let Cmax be the objective function. What i am unsure of is whether Cmax is an index, or the actual number of contacts, or a dollar value? I believe it should be a dollar value because the problem deals with monetary units. So Cmax = I + P, where I is the internet portion of the contact and P is the poster portion. Now, Cmax = ix + py, where i is internet portion and p is poster portion. Now I am unsure of what i and p are... I can't let i and p be the indices given... So do i have to convert the indices? And how would i go about doing so?

I Like Pi

2. Sep 19, 2012

### Ray Vickson

The formulation so far is OK.

About the objective: what would be the measure of customer contact if $1 is spent on internet ads? What is the contact measure if$x is spent on internet ads?

Note: this is some type of "measurement" of customer contact, not necessarily an actual number of customers contacted. Presumably the company's marketing research department devised these scales, or hired a consultant to do it for them. The only thing that matters is that the company wants to use it.

RGV

3. Sep 19, 2012

That is the part I am unsure of... I don't know what the measure of customer contact for $1 would be... I was thinking of relating the 0-100 to the price constraints? 4. Sep 19, 2012 ### Mark44 ### Staff: Mentor These types of problems typically use z for the variable that is maximized or minimized, although Cmax is OK, too. It seems clear to me that z (or Cmax) is an index (or measure) of customer contact. For this problem it will come out somewhere between 60 (internet ads only) and 90 (posters only). If there were no constraints, the maximum customer contact would be attained by using posters only. 5. Sep 19, 2012 ### Ray Vickson Did you read the question? In part it said "The marketing team uses an index that measures customer contact for every dollar spent on the ad." RGV 6. Sep 19, 2012 ### I Like Pi Okay, so Cmax (or z) is the index... I still don't understand...so for every dollar spent, the index for internet is 60 and so on. What if i did this: Cmax = (60x+90y)/30 000? That's all I can figure out... I Like Pi Last edited: Sep 19, 2012 7. Sep 19, 2012 ### Mark44 ### Staff: Mentor The objective function is: z = 60x + 90y subject to the various constraints on total spent, relative amounts for internet and posters, and so on. Your goal is to maximize the objective function. Since there are only two variables, a graph of the feasible region would be one approach. (As opposed to setting up the equations and inequalities in a tableau.) 8. Sep 19, 2012 ### I Like Pi I had this initially, but I don't see how it makes sense... shouldn't your z be in the 0-100 range? And what unit do you get when you multiply the index by the budget? I Like Pi 9. Sep 19, 2012 ### Mark44 ### Staff: Mentor No, z is not in the range 0 through 100. The index is a per-dollar measure of customer contact - the more dollars, the more contact. 10. Sep 19, 2012 ### I Like Pi So z doesn't matter? I am still having a hard time understanding the logic behind that equation... what does it mean when the index is multiplied by the dollar value? Wouldn't the units of z be in dollars? I Like Pi 11. Sep 19, 2012 ### Ray Vickson For internet ads the measure of exposure is 60 per$ spent. So, if they spend $1 they get exposure 60. If they spend$1,000,000 they get exposure 60,000,000. Obviously, the TOTAL exposure cannot just range from 1 to 100, since they need to know if the exposure is 60 or 6,000 or 60,000 or what; if they just divide by something to put all such figures in the range 0-100, they will have no way to tell the difference. You are just over-thinking the problem!

RGV

12. Sep 19, 2012

### I Like Pi

I see what what you're getting at... so 60x would then be rationalized again to be in the 0 to 100 range (to be an index)?

I Like Pi

13. Sep 19, 2012

### Ray Vickson

Sigh! I said the exact opposite. At his point I give up.

RGV