# Linear Programming Problem

### Linear Programming Problem

An interview is to be conducted and there are three family types that have been identified for the interview; 1-Child Family, 2-Child Family, and 3-Child Family.

At least 2200 families have to be interviewed for the entire interview exercise.

There are 3 interview timings; Weekdays Morning, Weekdays Evening, and Saturday Morning

The costs of the interviews are as follows:
Family Type Weekdays Morning Weekdays Evening Saturdays
One-Child $20$22 $20 Two-child$23 $25$23
Three-child $26$28 \$26

The total number of respondents during weekdays evening interviews must be at least equals to the total number of respondents during weekdays morning interviews.

The minimum number of respondents for each family types for weekdays interview are:
One child - 650
Two-child - 600
Three-child - 700

The combined total of two and three child households being interviewed on Saturday has to be at least 110.

Below is the allocation ratio for weekdays evening interviews to weekdays morning interviews per family type are:
One child - 55:45
Two child - 60:40
Three child - 60:40

Question - Do up a Linear Programming model based on the information given above to minimize the cost of the interview exercise.
I've tried formulating the LP model for the question above and would like to check if I'm doing anything incorrectly.

Below is my LP model.

Let X1 be the number of one-child family interviewed on weekdays morning
Let X2 be the number of one-child family interviewed on weekdays evening
Let X3 be the number of one-child family interviewed on saturday
Let X4 be the number of two-child family interviewed on weekdays morning
Let X5 be the number of two-child family interviewed on weekdays evening
Let X6 be the number of two-child family interviewed on saturday
Let X7 be the number of three-child family interviewed on weekdays morning
Let X8 be the number of three-child family interviewed on weekdays evening
Let X9 be the number of three-child family interviewed on saturday
Let Z be the total cost of the interviews

Z = 20X1 + 22X2 + 20X3 + 23X4 + 25X5 + 23X6 + 26X7 + 28X8 + 26X9

Constraints
X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 =>2200
X1 + X2 => 650
X4 + X5 => 600
X7 + X8 => 700
X6 + X9 => 110
X2 + X5 + X8 => X1 + X4 + X7
X1 = 0.45(X1 + X2)
X2 = 0.55(X1 + X2)
X4 = 0.4(X4 + X5)
X5 = 0.6(X4 + X5)
X7 = 0.4(X7 + X8)
X8 = 0.6(X7 + X8)
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