I'm stuck trying to solve this linear programming question.
You want to make a website with a list of features F, which are n elements long. Each feature has a corresponding value for how long it'll take to implement it. Since you dont know how to program, you enlist the help of your m programmer friends. Each of your friends have three attributes
- Preferences: the list of features they like to program, this is a subset of F
- Hours: The amount of time they are willing to work
- Cost: The amount of money they will charge per hour. Note that your friends will only charge you money if they work a feature thats not in their preferences.
To summarize For $i = 0$ to $n$,
- $F(i)$ = $i$-th feature
- $T_i$ = time it takes to program the $i$-th feature
For $i = 0$ to $m$,
- $M(i)$ = $i$-th programmer
- $P(i)$ = $i$-th programmer's preferred features
- $C(i)$ = \$/hr the $i$-th programmer will charge for working on features he doesn't like
- $h(i)$ = The number of hours the $i$-th program is willing to work
We need to come up with a linear programming formulation that'll finish all the features or find out if it's impossible.
Minimize the cost
The linear programming problems I have done so far are usually the farmer wants to plant 2 crops kind of things. So this is really new and hard for me. My approach was to have a set of inequalities for each programmer but the main problem here is that for some features they don't charge and some they do. I'm not sure how to incorporate that into the formulation correctly. Also, a point to keep in mind is that a programmer can work on a feature for an hour and leave it and let another programmer take over.