Question

M3_A5. A political polling organization has been hired to conduct a poll of likely voters prior to an upcoming election. Each voter is to be interviewed in person. It is known that he costs of interviewing different types of voters vary due to the difference in proportion within the population. The costs to interview males, for example, are $20 per Democrat, $18 per Republican, and $27 per Independent voter. The costs to interview female are $24, $22 and $28 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 7,500 total interviews.  At least 2,500 independent voters must be polled  At least 3,800 males must be polled.  At least 3,250 females must be polled  No more than 35% of those polled may be Democrats  No more than 35% of those polled may be Republicans  No more than 15% of those polled may be Republican males.  Each of the six types of voters must be represented in the poll by at least 13% of the total interviews Formulate and solve this problem in Excel to determine the number of each type of voter that meets the requirements and minimize the cost to carry out the interviews. a) How many decision variables does this problem have? b) Not counting the non-negativity constraint - how many constraints does this problem have? c) What is the minimum cost in your optimal solution (the value of the objective function)? d) Based on your solution – how many Democrat Males should be interviewed? e) Based on your solution – how many Republican Males should be interviewed? f) Based on your solution – how many Independent Males should be interviewed? g) Based on your solution – how many Democrat Females should be interviewed? h) Based on your solution – how many Republican Females should be interviewed? i) Based on your solution – how many Independent Females should be interviewed?

Answer 1

There are 6 types of voters that need to be interviewed

Let

These are the 6 decision variables

The total cost of interviewing these many is

We want to minimize this cost. Hence this is the objective function.

Finally we have these constraints

There must be at least 7,500 total interviews.

• 

At least 2,500 independent voters must be polled

• 

At least 3,800 males must be polled.

• 

At least 3,250 females must be polled

• 

No more than 35% of those polled may be Democrats

• 

No more than 35% of those polled may be Republicans

• 

No more than 15% of those polled may be Republican males.

• 

Each of the six types of voters must be represented in the poll by at least 13% of the total interviews

• 

The Linear Programming Model that we want to solve is

Minimize

s.t.

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We will create the following excel sheet

get these values

set up the solver using data-->solver

get this solution

a) How many decision variables does this problem have?

ans: 6

b) Not counting the non-negativity constraint - how many constraints does this problem have?

ans: 13

c) What is the minimum cost in your optimal solution (the value of the objective function)?

ans: $172,675

d) Based on your solution – how many Democrat Males should be interviewed?

ans:1,650

e) Based on your solution – how many Republican Males should be interviewed?

ans: 1,125

f) Based on your solution – how many Independent Males should be interviewed?

ans:1,475

g) Based on your solution – how many Democrat Females should be interviewed?

ans: 975

h) Based on your solution – how many Republican Females should be interviewed?

ans: 1,250

i) Based on your solution – how many Independent Females should be interviewed?

ans: 1,025