Question $1$: Optimizing the Marketing Mix for Maximum Brand Awareness $(50$ points$)$

Background:

You are a marketing manager at a company called EcoTech, which specializes in eco$-$friendly

consumer electronics. The company is planning its marketing campaign for the upcoming quarter

and has identified three marketing channels: Social Media Advertising, Email Campaigns, and

Influencer Partnerships. Your objective is to allocate the marketing budget in a way that maximizes

brand awareness while adhering to budget and channel$-$specific constraints.

Marketing Channels:

$$ Social Media Advertising:

o Expected Brand Awareness Contribution: $40$ units per $$1,000$ spent.

o Cost per Campaign: $$1,000.$

o Maximum campaigns: $50$ per quarter.

$$ Email Campaigns:

o Expected Brand Awareness Contribution: $50$ units per $$1,000$ spent.

o Cost per Campaign: $$1,000.$

o Maximum campaigns: $30$ per quarter.

$$ Influencer Partnerships:

o Expected Brand Awareness Contribution: $90$ units per $$1,000$ spent.

o Cost per Partnership: $$2,000.$

o Maximum partnerships: $20$ per quarter.

The total budget for the quarter is $$100,000.$

Formulate this problem as a linear programming model and solve it using Excel's Solver to

determine the optimal number of Social Media Advertising campaigns, Email Campaigns,

and Influencer Partnerships that should be undertaken to maximize brand awareness.

Question $2$: Optimal Sampling Plan $(50$ points$)$

Background:

A political polling organization is to conduct a poll of likely voters prior to an upcoming election.

Each voter is to be interviewed in person. It is known that the costs of interviewing different types

of voters vary due to the differences in proportion within the population. The costs to interview

males, for example, are $$10$ per Democrat, $$9$ per Republican, and $$13\mathrm{.}50$ per Independent voter.

The costs to interview females are $$12,$ $$11$ and $$13\mathrm{.}50$ for Democrat, Republican, and

Independent voters, respectively. The polling service has been given certain criteria to which it

must adhere:

$$ There must be at least $4,500$ total interviews.

$$ At least $1,000$ independent voters must be polled.

$$ At least $2,000$ males must be polled.

$$ At least $1,750$ females must be polled.

$$ No more than $40\%$ of those polled may be Democrats.

$$ No more than $40\%$ of those polled may be Republicans.

$$ No more than one$-$quarter of those polled may be Republican males.

$$ Each of the six categories of voters must be represented in the poll by at least $10\%$ of the

total interviews.

Using a linear programming model, determine the least expensive sampling plan and the

total cost to carry out the plan.