[Chapter 12] Linear Optimization Models Florence Marketing Group is being hired by a client firm to assist in ascertaining consumer reaction to a recently marketed product in Chicagoland. Florence decided to conduct personal interviews to obtain responses from consumers. From experience, Florence learns that interview cost varies due to differences in gender and location, as shown in the table below.

The contract called for Florence to conduct interviews under the requirements (as detailed in the Constraints section) given below.

Please help Florence develop an interview plan by formulating a linear optimization model.

I. Define the decision variables.

(Note: This step is done for you. Please use these decision variables hereafter.)

X₁ = Number of interviews with males from the city;

X₂ = Number of interviews with males from the suburbs;

X₃ = Number of interviews with females from the city;

X₄ = Number of interviews with females from the suburbs.

II. Write the objective (mathematically).

Question 38 (6 points)

What is the objective?

Question 38 options:

	Max Z = 11 X1 + 9 X2 + 10 X3 + 12 X4
	Min Z = 11 X1 + 9 X2 + 10 X3 + 12 X4
	Max Z = 11 X1 + 10 X2 + 9 X3 + 12 X4
	Min Z = X1 + X2 + X3 + X4
	Max Z = X1 + X2 + X3 + X4
	Min Z = 11 X1 + 10 X2 + 9 X3 + 12 X4

III. Write the constraints (mathematically), which are the contract requirements.

Question 39 (6 points)

There should be at most 1,200 total interviews.

Question 39 options:

	11 X1 + 9 X2 + 10 X3 + 12 X4 > 1,200
	11 X1 + 9 X2 + 10 X3 + 12 X4
	X1 + X2 + X3 + X4 > 1,200
	X1 + X2 + X3 + X4
	X1 + X2 + X3 + X4 1,200
	11 X1 + 9 X2 + 10 X3 + 12 X4 1,200
	X1 + X2 + X3 + X4 1,200
	11 X1 + 9 X2 + 10 X3 + 12 X4 1,200

Question 40 (6 points)

At most 40% of the total interviews may be from the suburbs.

Question 40 options:

	X2 + X4 0.4 X1 + X2 + X3 + X4
	X2 + X4 0.4 (X1 + X2 + X3 + X4)
	X2 + X4 0.4
	X2 + X4 1 + X2 + X3 + X4)
	X2 + X4 0.4
	X2 + X4 0.4 (X1 + X2 + X3 + X4)
	X2 + X4 > 0.4 (X1 + X2 + X3 + X4)
	X2 + X4 0.4 X1 + X2 + X3 + X4

Question 41 (6 points)

The interviews with consumers who are females and from the suburbs should account for at least 20% of the total interviews.

Question 41 options:

	X4 0.2
	X4 0.2 X1 + X2 + X3 + X4
	X4 0.2 (X1 + X2 + X3 + X4)
	X4 0.2 (X1 + X2 + X3 + X4)
	X4 0.2
	X2 0.2 (X1 + X2 + X3 + X4)
	X4 0.2 X1 + X2 + X3 + X4

Question 42 (6 points)

At least 60% of the interviews with males must be from the city.

Question 42 options:

	X1 0.6 (X1 + X2)
	X1 0.6 (X1 + X2 + X3 + X4)
	X1 > 0.6 (X1 + X2)
	X1 0.6 (X1 + X2 + X3 + X4)
	X1 + X3 0.6 (X1 + X2)
	X1 1 + X2)
	X1 + X2 0.6 (X1 + X3)
	X1 0.6 (X1 + X2)

Question 43 (6 points)

At least 600 females must be interviewed.

Question 43 options:

	10 X3 + 12 X4 600
	10 X3 + 12 X4 > 600
	X3 + X4 600
	10 X3 + 12 X4 600
	10 X3 + 12 X4
	X3 + X4 > 600
	X3 + X4
	X3 + X4 600

Question 44

At least 500 males must be interviewed.

Question 44 options:

	X1 + X2 > 500
	11 X1 + 9 X2
	11 X1 + 9 X2 > 500
	X1 + X2 500
	X1 + X2
	11 X1 + 9 X2 500
	X1 + X2 500
	11 X1 + 9 X2 500

Constraints on decision variables.

Question 45 options:

	X1, X2, X3, X4 > 0
	X1 + X2 + X3 + X4 0
	X1 + X2 + X3 + X4 > 0
	X1, X2, X3, X4
	X1, X2, X3, X4 0
	X1 + X2 + X3 + X4 0
	X1, X2, X3, X4 0
	X1 + X2 + X3 + X4 = 0
	X1, X2, X3, X4 = 0 or 1

\begin{tabular}{|l|c|c|} \hline & \multicolumn{2}{|c|}{ Cost Per Interview } \\ \hline Gender & City & Suburbs \\ \hline Male & $11 & $9 \\ \hline Female & $10 & $12 \\ \hline \end{tabular}