A marketing research professor is conducting a telephone survey and needs to con

Question

A marketing research professor is conducting a telephone survey and needs to contact at least 160 wives, 140 husbands, 110 single adult males, and 120 single adult females. It costs $2 to make a daytime call and $4 (because of higher labor costs) to make an evening call. The table shown below lists the expected results. For example, 10% of all daytime calls are answered by a single male, and 15% of all evening calls are answered by a single female. Because of a limited staff, at most half of all phone calls can be evening calls.

Write down the linear programming formulation to minimize the total cost of the survey (DO NOT SOLVE)

Percentages Daytime Evening Wife 25% 25% Husband 15% 30% Single Male 10% 25% Single Female 15% 15% None 35% 5%

Explanation / Answer

Let w1 and w2 are the number of daytime and evening calls are made to wives
Let h1 and h2 are the number of daytime and evening calls are made to husbands
Let m1 and m2 are the number of daytime and evening calls are made to single males
Let f1 and f2 are the number of daytime and evening calls are made to single females
Let n1 and n2 are the number of daytime and evening calls are made to none

Objective function is to minimize the cost
minimize Z = 2w1 + 2h1 + 2m1 + 2f1 + 2n1 + 4w2 + 4h2 + 4m2 + 4f2 + 4n2

Constraints are
1) At least 160 wives are to be contacted
0.25w1 + 0.25w2 >= 160

2) At least 140 husbands to be contacted
0.15h1 + 0.3h2 >= 140

3) At least 110 single adult males to be contacted
0.1m1 + 0.25m2 >= 110

4) At least 120 single adult females to be contacted
0.15f1 + 0.15f2 >= 120

5) At most half of all phone calls can be evening calls
w2 + h2 + m2 + f2 + n2 <= 0.5*(w1 + w2 + h1 + h2 + m1 + m2 + f1 + f2 + n1 + n2)
i.e. 0.5w2 - 0.5w1 + 0.5h2 - 0.5h1 + 0.5m2 - 0.5m1 + 0.5f2 - 0.5f1 + 0.5n2 - 0.5n1 <= 0

Hence LP formulation is
Min Z = 2w1 + 2h1 + 2m1 + 2f1 + 2n1 + 4w2 + 4h2 + 4m2 + 4f2 + 4n2

subject to,
0.25w1 + 0.25w2 >= 160
0.15h1 + 0.3h2 >= 140
0.1m1 + 0.25m2 >= 110
0.15f1 + 0.15f2 >= 120
0.5w2 - 0.5w1 + 0.5h2 - 0.5h1 + 0.5m2 - 0.5m1 + 0.5f2 - 0.5f1 + 0.5n2 - 0.5n1 <= 0

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