Sorting People into Groups based on Votes

Question

I have a problem with finding a algorithm for sorting a dataset of people. I try to explain as detailed as possible:

The story starts with a survey. A bunch of people, l

Answer 1

You can solve this optimally by formulating it as a min cost network flow problem.

Add a node for each person, and one for each project.

Set cost for a flow between a person and a project according to their preferences.

(As Networkx provides a min cost flow, but not max cost flow I have set the costs to be negative.)

For example, using Networkx and Python:

import networkx as nx

G=nx.DiGraph()

prefs={'Tom':['Project1','Project2','Project3'],
       'Dick':['Project2','Project1','Project3'],
       'Harry':['Project1','Project3','Project1']}

capacities={'Project1':2,'Project2':10,'Project3':4}

num_persons=len(prefs)
G.add_node('dest',demand=num_persons)
A=[]
for person,projectlist in prefs.items():
    G.add_node(person,demand=-1)
    for i,project in enumerate(projectlist):
        if i==0:
            cost=-100 # happy to assign first choices
        elif i==1:
            cost=-60 # slightly unhappy to assign second choices
        else:
            cost=-30 # very unhappy to assign third choices
        G.add_edge(person,project,capacity=1,weight=cost) # Edge taken if person does this project

for project,c in capacities.items():
        G.add_edge(project,'dest',capacity=c,weight=0)

flowdict = nx.min_cost_flow(G)
for person in prefs:
    for project,flow in flowdict[person].items():
        if flow:
            print person,'joins',project

In this code Tom's number 1 choice is Project1, followed by Project2, then Project3.

The capacities dictionary specifies the upper limit on how many people can join each project.