Linear Programming Involving Indices

[I Like Pi]

Homework Statement

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?

Homework Equations

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?

Thanks in advance,
I Like Pi

 

[Ray Vickson]

Homework Statement

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?

Homework Equations

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?

Thanks in advance,
I Like Pi

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

 

[I Like Pi]

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?
RGV

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?

 

[Mark44]

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?


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?

These types of problems typically use z for the variable that is maximized or minimized, although C_max is OK, too.

It seems clear to me that z (or C_max) 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.

 

[Ray Vickson]

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?

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

 

[I Like Pi]

These types of problems typically use z for the variable that is maximized or minimized, although C_max is OK, too.

It seems clear to me that z (or C_max) 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.

Okay, so C_max (or z) is the index...

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

I still don't understand...so for every dollar spent, the index for internet is 60 and so on. What if i did this: C_max = (60x+90y)/30 000?
That's all I can figure out...
I Like Pi

 

[Mark44]

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.)

 

[I Like Pi]

The objective function is:
z = 60x + 90y

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

 

[Mark44]

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.

 

[I Like Pi]

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.

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

 

[Ray Vickson]

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

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

 

[I Like Pi]

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

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

 

[Ray Vickson]

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

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

RGV