Answer to Question #171447 in Operations Research for mara

Question

A marketing research group conducting a telephone survey must contact at least 150 wives and 120 husbands. It costs P100 to make a daytime call and (because of higher labor costs) P150 to make an evening call. On average, daytime calls reach wives 30% of the time, husbands 10% of the time, and neither of these 60% of the time, whereas evening calls reach wives 30% of the time, husbands 30% of the time, and neither of these 40% of the time. Staffing considerations mean that daytime calls must be less than or equal to half of the total calls made. How many should be interviewed at each period to minimize the cost of completing the survey?

Answer 1

x₁ - daytime calls,

x₂ - evening calls,

$0.3x_1+0.3x_2\geq 150,$

$0.1x_1+0.3x_2\geq 120,$

$x_1\leq 0.5(x_1+x_2),$

$x_1\geq 0,$

$x_2\geq 0,$

$f=100x_1+150x_2\to min,$

$x_1+x_2\geq 500,$

$x_1+3x_2\geq 1200,$

$x_1\leq x_2,$

$x_1\geq0,$

$x_2\geq0,$

$x_1+x_2=500,$

$x_1+3x_2=1200,$

from where

$x_1=150,~x_2=350,$

$f=100\cdot 150+150\cdot 350=67500,$

daytime: 45 wives, 15 husbands,

evening: 105 wives, 105 husbands.