How Do You Formulate Decision Variables in a Linear Programming Problem?

[Erenjaeger]

I was given the problem attached in the photo below and the first question is to define the decision variables and formulate the problem as a linear program. There are no solutions online, so it would be helpful if someone on the mighty PF could check them to see if they are correct, thanks.
https://scontent.fhlz2-1.fna.fbcdn.net/v/t34.0-12/23113532_1505762442836859_1149946816_n.png?oh=f7a855d08561239a24b48af9741cbfbf&oe=59FAF30B
decision variables are..
let x_ig be the amount of time the investor spends gardening
let x_wg be the amount of time the worker spends gardening
let x_ir be the amount of time the investor spends rewiring the house
let x_ii be the amount of time the investor spends insulating the house
let x_cv be the amount of time the contractor spends checking ventilation
Max: 400x_ig + 400x_wg +350x_ir +200x_ii + 300x_cv - 100x_ig - 100x_ir - 60x_wg - 150x_cv
subject to:
x_ig + x_ir + x_ii ≤ 50
x_ig ≤ 10
x_ir ≥ 6
x_ii / (x_ig + x_wg ) ≥ 2
Note: I'm not that sure about the last constraint, It is meant to be the one that says for 1 hour gardening there has to be at least 30 minutes insulating.

 

[mfb]

I interpret the last constraint in the same way as you, but it should be 1/2, not 2.
There are some costs missing and i can check the ventilation as well. The maximal hours for each task are missing as well. The i gardening constraint is wrong (although it won’t change the result here).

A more careful check of your statements would have helped.

The problem has a maximum that is easy to find manually step by step, that gives a nice cross check.

 

[Erenjaeger]

I interpret the last constraint in the same way as you, but it should be 1/2, not 2.
There are some costs missing and i can check the ventilation as well. The maximal hours for each task are missing as well. The i gardening constraint is wrong (although it won’t change the result here).

A more careful check of your statements would have helped.

The problem has a maximum that is easy to find manually step by step, that gives a nice cross check.

if the last constraint had 1/2 on the RHS wouldn't the LHS read (x_ig + x_wg) / x_ii instead, since you're given the information in the form of (x_ig + x_wg) ≤ (1/5)x_ii?
And what do you mean about some costs missing?
Also how is the x_ig constraint wrong? I took that part of the problem to mean that he doesn't want to spend more than 20% of his total time (50 hours) improving the house, on gardening, 20% of 50 hours is 10 hours so doesn't that just mean he doesn't want to spend more than 10 hours gardening, hence x_ig ≤ 10 hours ?

 

[mfb]

if the last constraint had 1/2 on the RHS wouldn't the LHS read (x_ig + x_wg) / x_ii instead

No, that would be wrong again. Currently you require insulation to be at least twice the time spent on gardening, but the requirement is "at least half".

And what do you mean about some costs missing?

You only accounted for two tasks of the investor, but he can do all of them.

Also how is the x_ig constraint wrong? I took that part of the problem to mean that he doesn't want to spend more than 20% of his total time (50 hours) improving the house, on gardening, 20% of 50 hours is 10 hours so doesn't that just mean he doesn't want to spend more than 10 hours gardening, hence x_ig ≤ 10 hours ?

He doesn't want to spend more than 20% of the time he spends on gardening. It literally has the bold part in the problem statement. He doesn't have to spend 50 hours.