Defining the Linear Programming Model for Traveling Salesman in Python
<p>Using Python and <code>PuLP</code> library, how can we create the linear programming model to solve the Traveling Salesman Problem (TSP)?</p>

<p>From Wikipedia, the objective function and constraints are </p>

<p><a href="http://i.stack.imgur.com/Z9wuZ.png" rel="nofollow"><img src="http://i.stack.imgur.com/Z9wuZ.png" alt="enter image description here"></a></p>

<p><strong>Problem:</strong> Here is my partial attempt where I am stuck.  </p>

<ol>
<li><p>I did not include the final constraint in the code because I dont know how to define it. I believe this constraint with u variables are for preventing sub-cycles in the solution</p></li>
<li><p>Also, solving for the current model gives decision variables such as <code>x0_0</code> and <code>x1_1</code> being equal to <code>1.0</code> which is definitely wrong... I can't figure out why this is so even though I had</p>

<pre><code>    if i == j:
        upperBound = 0
</code></pre></li>
</ol>

<p><strong>Python Code</strong></p>

<pre><code>import pulp

def get_dist(tsp):
    with open(tsp, 'rb') as tspfile:
        r = csv.reader(tspfile, delimiter='\t')
        d = [row for row in r]

    d = d[1:] # skip header row
    locs = set([r[0] for r in d]) | set([r[1] for r in d])
    loc_map = {l:i for i, l in enumerate(locs)}
    idx_map = {i:l for i, l in enumerate(locs)}
    dist = [(loc_map[r[0]], loc_map[r[1]], r[2]) for r in d]
    return dist, idx_map


def dist_from_coords(dist, n):
    points = []
    for i in range(n):
        points.append([0] * n)
    for i, j, v in dist:
        points[i][j] = points[j][i] = float(v)
    return points


def find_tour():
    tsp_file = '/Users/test/' + 'my-waypoints-dist-dur.tsv'
    coords, idx_map = get_dist(tsp_file)
    n = len(idx_map)
    dist = dist_from_coords(coords, n)


    # Define the problem
    m = pulp.LpProblem('TSP', pulp.LpMinimize)


    # Create variables
    # x[i,j] is 1 if edge i-&gt;j is on the optimal tour, and 0 otherwise
    # Also forbid loops
    x = {}
    for i in range(n):
        for j in range(n):
            lowerBound = 0
            upperBound = 1

            # Forbid loops
            if i == j:
                upperBound = 0
                # print i,i

            x[i,j] = pulp.LpVariable('x' + str(i) + '_' + str(j), lowerBound, upperBound, pulp.LpBinary)
            # x[j,i] = x[i,j]


    # Define the objective function to minimize
    m += pulp.lpSum([dist[i][j] * x[i,j] for i in range(n) for j in range(n)])


    # Add degree-2 constraint
    for i in range(n):
        m += pulp.lpSum([x[i,j] for j in range(n)]) == 2


    # Solve and display results
    status = m.solve()
    print pulp.LpStatus[status]
    for i in range(n):
        for j in range(n):
            if pulp.value(x[i,j]) &gt;0:
                print str(i) + '_' + str(j) + ': ' + str( pulp.value(x[i,j]) )

find_tour()
</code></pre>

<p><strong>my-waypoints-dist-dur.tsv</strong></p>

<p>The data file <a href="https://gist.github.com/anonymous/43ba3f6337556eae4ce8104b8ed5992b" rel="nofollow">can be found here</a>.</p>

<p><strong>Result</strong></p>

<pre><code>0_0: 1.0
0_5: 1.0
1_1: 1.0
1_15: 1.0
2_2: 1.0
2_39: 1.0
3_3: 1.0
3_26: 1.0
4_4: 1.0
4_42: 1.0
5_5: 1.0
5_33: 1.0
6_6: 1.0
6_31: 1.0
7_7: 1.0
7_38: 1.0
8_8: 1.0
8_24: 1.0
9_9: 1.0
9_26: 1.0
10_4: 1.0
10_10: 1.0
11_11: 1.0
11_12: 1.0
12_11: 1.0
12_12: 1.0
13_13: 1.0
13_17: 1.0
14_14: 1.0
14_18: 1.0
15_1: 1.0
15_15: 1.0
16_3: 1.0
16_16: 1.0
17_13: 1.0
17_17: 1.0
18_14: 1.0
18_18: 1.0
19_19: 1.0
19_20: 1.0
20_4: 1.0
20_20: 1.0
21_21: 1.0
21_25: 1.0
22_22: 1.0
22_27: 1.0
23_21: 1.0
23_23: 1.0
24_8: 1.0
24_24: 1.0
25_21: 1.0
25_25: 1.0
26_26: 1.0
26_43: 1.0
27_27: 1.0
27_38: 1.0
28_28: 1.0
28_47: 1.0
29_29: 1.0
29_31: 1.0
30_30: 1.0
30_34: 1.0
31_29: 1.0
31_31: 1.0
32_25: 1.0
32_32: 1.0
33_28: 1.0
33_33: 1.0
34_30: 1.0
34_34: 1.0
35_35: 1.0
35_42: 1.0
36_36: 1.0
36_47: 1.0
37_36: 1.0
37_37: 1.0
38_27: 1.0
38_38: 1.0
39_39: 1.0
39_44: 1.0
40_40: 1.0
40_43: 1.0
41_41: 1.0
41_45: 1.0
42_4: 1.0
42_42: 1.0
43_26: 1.0
43_43: 1.0
44_39: 1.0
44_44: 1.0
45_15: 1.0
45_45: 1.0
46_40: 1.0
46_46: 1.0
47_28: 1.0
47_47: 1.0



...
</code></pre>

<hr>

<p><strong>Updated Code</strong></p>

<pre><code>import csv
import pulp


def get_dist(tsp):
    with open(tsp, 'rb') as tspfile:
        r = csv.reader(tspfile, delimiter='\t')
        d = [row for row in r]

    d = d[1:] # skip header row
    locs = set([r[0] for r in d]) | set([r[1] for r in d])
    loc_map = {l:i for i, l in enumerate(locs)}
    idx_map = {i:l for i, l in enumerate(locs)}
    dist = [(loc_map[r[0]], loc_map[r[1]], r[2]) for r in d]
    return dist, idx_map


def dist_from_coords(dist, n):
    points = []
    for i in range(n):
        points.append([0] * n)
    for i, j, v in dist:
        points[i][j] = points[j][i] = float(v)
    return points


def find_tour():
    tsp_file = '/Users/test/' + 'my-waypoints-dist-dur.tsv'
    coords, idx_map = get_dist(tsp_file)
    n = len(idx_map)
    dist = dist_from_coords(coords, n)


    # Define the problem
    m = pulp.LpProblem('TSP', pulp.LpMinimize)


    # Create variables
    # x[i,j] is 1 if edge i-&gt;j is on the optimal tour, and 0 otherwise
    # Also forbid loops
    x = {}
    for i in range(n+1):
        for j in range(n+1):
            lowerBound = 0
            upperBound = 1

            # Forbid loops
            if i == j:
                upperBound = 0
                # print i,i

            # Create the decision variable and First constraint
            x[i,j] = pulp.LpVariable('x' + str(i) + '_' + str(j), lowerBound, upperBound, pulp.LpBinary)
            # x[j,i] = x[i,j]


    # Define the objective function to minimize
    m += pulp.lpSum([dist[i][j] * x[i,j] for i in range(1,n+1) for j in range(1,n+1)])


    # Add degree-2 constraint (3rd and 4th)
    for i in range(1,n+1):
        m += pulp.lpSum([x[i,j] for j in range(1,n+1)]) == 2


    # Add the last (5th) constraint (prevents subtours)
    u = []
    for i in range(1, n+1):
        u.append(pulp.LpVariable('u_' + str(i), cat='Integer'))
    for i in range(1, n-1):
        for j in range(i+1, n+1):
            m += pulp.lpSum([ u[i] - u[j] + n*x[i,j]]) &lt;= n-1


    # status = m.solve()
    # print pulp.LpStatus[status]
    # for i in range(n):
    #   for j in range(n):
    #       if pulp.value(x[i,j]) &gt;0:
    #           print str(i) + '_' + str(j) + ': ' + str( pulp.value(x[i,j]) )

find_tour()
</code></pre>

<p>The last constraint is <strong>not</strong> a single constraint. You should add one constraint for each pair of indices <code>i, j</code> that satisfy that condition:</p>

<pre><code>for i in range(n-1):
    for j in range(i+1, n):
        m += pulp.lpSum([ u[i] - u[j] + n*x[i,j]]) &lt;= n-1
</code></pre>

<p>However you first have to declare the <code>u_i</code> variables as integers, passing the <code>cat='Integer'</code> argument to <a href="https://pythonhosted.org/PuLP/pulp.html" rel="nofollow"><code>LpVariable</code></a>:</p>

<pre><code>u = []
for i in range(n):
    u.append(pulp.LpVariable('u_' + str(i), cat='Integer'))
</code></pre>

python algorithm python-2.7 linear-programming pulp