# Linear Programming Involving Indices

• I Like Pi
So, let z = the measure of customer contact for every dollar spent on advertising. Then, the objective function becomes z = 60x + 90y, where x is the amount spent on internet ads and y is the amount spent on poster ads. This means that for every dollar spent on internet ads, the company will have a customer contact measure of 60, and for every dollar spent on poster ads, the company will have a customer contact measure of 90. Now, with the given constraints, the company's budget can be written as 2x + y ≤ 30,000 and x + y

## 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?

## 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

I Like Pi said:

## 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?

## 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

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

Ray Vickson said:
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? I Like Pi said: 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 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. I Like Pi said: 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

Mark44 said:
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.

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

Ray Vickson said:
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: Cmax = (60x+90y)/30 000?
That's all I can figure out...
I Like Pi

Last edited:
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.)

Mark44 said:
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

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.

Mark44 said:
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

I Like Pi said:
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 $1,000,000 they get exposure 60,000,000. Obviously, the TOTAL exposure cannot just range from 1 to 100, since they need to know if the exposure is 60 or 6,000 or 60,000 or what; if they just divide by something to put all such figures in the range 0-100, they will have no way to tell the difference. You are just over-thinking the problem! RGV Ray Vickson said: For internet ads the measure of exposure is 60 per$ 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

I Like Pi said:
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

## 1. What is linear programming involving indices?

Linear programming involving indices is a mathematical technique used to optimize a linear objective function subject to a set of linear constraints. It involves using indices, or exponents, in the objective function and constraints to represent the coefficients of the decision variables.

## 2. How is linear programming involving indices different from traditional linear programming?

The main difference between linear programming involving indices and traditional linear programming is the use of indices in the objective function and constraints. This allows for more flexibility and complexity in representing the coefficients of the decision variables. Traditional linear programming typically uses fixed numerical coefficients.

## 3. What are the applications of linear programming involving indices?

Linear programming involving indices has various applications in fields such as economics, finance, engineering, and management. It is commonly used to solve optimization problems involving resource allocation, production planning, and portfolio optimization.

## 4. What are the steps involved in solving a linear programming problem with indices?

The steps involved in solving a linear programming problem with indices are:

• 1. Formulate the objective function and constraints using indices to represent the coefficients of the decision variables.
• 2. Graph the feasible region, which is the set of all possible solutions that satisfy the constraints.
• 3. Identify the corner points of the feasible region.
• 4. Calculate the objective function value at each corner point.
• 5. Select the corner point with the optimal objective function value as the solution to the linear programming problem.

## 5. Are there any limitations to using linear programming involving indices?

Linear programming involving indices has some limitations, such as the assumption of linearity in the objective function and constraints, and the requirement for all variables to be continuous. It also requires a known and finite set of decision variables. Additionally, the use of indices can increase the complexity of the problem and make it more challenging to solve.