COMP572

Neural Networks

Ludovic Hilde

Summer 2008

06/25/08

Assignment 2

A. An introduction that describes the TSP problem and the general idea for the neural network approach. Include observations you made after seeing the simulations, including any ideas you have for what made, or could make, the network better (get shorter tours more consistently in fewer iterations, etc.).

The Traveling Salesman Problem (TSP) is defined with the following question: What is the shortest possible tour for a salesman to visit all cities once [Ci…Cj], starting and returning at city Ci? The neural network approach demonstrates how a highly interconnected network of simple processing units can solve a complex problem such as TSP. The local interactions of such neural networks lead to a particular global effect.

B. Post a copy of the code.

Click on the link below to access the source code of the TSP neural network simulation:

TSP neural network source code

C. Show the specific results for running the network twice for each case (case1, case2), with the following:

Case 1, initial activations in the 10^-10 range, and N= 20

1. City positions.

city0 at 0.7500, 0.5000

city1 at 0.7378, 0.5773

city2 at 0.7023, 0.6469

city3 at 0.6469, 0.7023

city4 at 0.5773, 0.7378

city5 at 0.5000, 0.7500

city6 at 0.4227, 0.7378

city7 at 0.3531, 0.7023

city8 at 0.2977, 0.6469

city9 at 0.2622, 0.5773

city10 at 0.2500, 0.5000

city11 at 0.2622, 0.4227

city12 at 0.2977, 0.3531

city13 at 0.3531, 0.2977

city14 at 0.4227, 0.2622

city15 at 0.5000, 0.2500

city16 at 0.5773, 0.2622

city17 at 0.6469, 0.2977

city18 at 0.7023, 0.3531

city19 at 0.7378, 0.4227

The position of each city is shown in the graph below:

2. Tour that resulted.

The shortest tour resulted at iteration 10.

Visited Cities: 13 19 3 17 11 18 10 12 14 15 16 20 9 5 6 1 7 4 2 8

The tour started at city 13, moved to 19, moved to 3, so on, and ended at 8. Clearly the fuzzy read out did not find an optimum solution, which would have been 13, 14, 15 … 11, and 12 for example.

3. Drawing of the tour.

4. Length of the tour.

The distances between each city of the tour depicted in 3 are as follows:

distance between cities 13 and 19: 0.4045084971874737

distance between cities 19 and 3: 0.2938926261462367

distance between cities 3 and 17: 0.4045084971874738

distance between cities 17 and 11: 0.4045084971874737

distance between cities 11 and 18: 0.4455032620941839

distance between cities 18 and 10: 0.47552825814757677

distance between cities 10 and 12: 0.15450849718747367

distance between cities 12 and 14: 0.15450849718747373

distance between cities 14 and 15: 0.07821723252011549

distance between cities 15 and 16: 0.07821723252011538

distance between cities 16 and 20: 0.29389262614623657

distance between cities 20 and 9: 0.49384417029756894

distance between cities 9 and 5: 0.29389262614623657

distance between cities 5 and 6: 0.07821723252011543

distance between cities 6 and 1: 0.3535533905932738

distance between cities 1 and 7: 0.4045084971874737

distance between cities 7 and 4: 0.22699524986977346

distance between cities 4 and 2: 0.1545084971874737

distance between cities 2 and 8: 0.40450849718747367

The total length of the tour is 5.597821884501222

5. Final activations of the network.

V0 contains the activation values of the first city (first row of the nxn matrix), V1 contains the activation values of the second city, and so on.

v0<

2.8122e-11, -6.8734e-11, -5.1469e-11, -5.7901e-11, -6.5325e-11,

4.3064e-12, 1.2549e-11, 1.8601e-11, 2.0914e-11, -4.1939e-11,

-7.5969e-12, 2.9033e-11, 1.6346e-11, -3.0572e-11, -1.1930e-11,

-3.9891e-11, -3.6278e-11, -4.0852e-11, -5.6820e-11, 3.0702e-11<

v1<

5.3852e-12, 1.4159e-12, -1.6419e-11, -5.5662e-11, -2.4749e-11,

1.9279e-13, -5.1314e-11, -4.1244e-11, 2.3695e-11, -3.5921e-11,

2.3235e-11, -1.6363e-11, -4.2910e-11, -2.3820e-11, 2.1512e-11,

-1.5666e-11, -3.5722e-11, -1.8121e-11, -4.8709e-11, -3.4464e-11>

v2<

4.2770e-12, -4.8791e-11, 4.1275e-12, -5.7651e-11, -3.3703e-11,

-9.2528e-12, -2.3930e-11, -4.4195e-11, -6.7118e-11, -2.9326e-11,

-5.9651e-11, 3.2411e-11, -2.2304e-11, 2.2159e-11, -4.1004e-11,

-1.8840e-11, 1.4341e-11, -3.2398e-11, 1.4870e-11, 2.6056e-11

v3<

1.1132e-11, 1.6541e-11, -3.3445e-11, 3.9676e-12, 6.7387e-12, >

4.4943e-12, 1.3097e-11, -5.3355e-11, 1.5665e-11, -5.3818e-11,

1.4788e-11, -4.9066e-11, 4.9757e-12, -3.6930e-11, -6.2769e-11,

-5.8185e-11, -3.2066e-11, -4.0395e-11, -3.4497e-11, -3.1343e-11>

v4<

-4.1902e-11, 5.8237e-12, 2.5541e-11, -6.8705e-12, -9.3335e-12,

-4.9393e-11, -5.2560e-11, -5.6821e-11, -3.8746e-11, 1.3761e-11,

2.2793e-11, -4.6476e-11, -3.7313e-11, -4.1347e-11, -5.7991e-12,

6.6801e-12, -2.7603e-11, 5.3378e-12, -3.3537e-11, -2.6311e-11

v5<

-2.1208e-12, -1.7832e-11, -4.5654e-11, 1.2777e-11, -3.9160e-11,

-1.0896e-11, -1.3874e-11, 7.0229e-13, -2.0454e-11, -7.9971e-12,

2.6747e-11, -1.8055e-11, -2.8860e-11, -5.5674e-11, 2.3767e-11,

-3.1571e-11, -5.9105e-11, -3.7098e-11, -3.3059e-12, -3.6068e-11>

v6<

8.8782e-12, -4.8899e-11, -4.3797e-11, -9.2222e-12, -2.2641e-11,

-1.7801e-11, -3.3843e-11, 2.4875e-11, 1.7594e-11, -5.6057e-11,

-4.0217e-11, -4.2212e-11, 1.5854e-11, -3.8928e-11, -6.8900e-11,

2.9235e-12, 3.6602e-11, -5.3127e-12, -1.7512e-11, -4.0929e-11>

v7<

-3.4915e-11, -1.5263e-11, -8.4984e-12, -4.5013e-11, 3.5883e-12,

-2.9947e-11, 2.9080e-12, -6.5938e-11, -4.6286e-11, 1.3157e-11,

-1.3382e-11, -5.5043e-11, 1.3251e-11, -2.3485e-11, 1.8476e-11,

-2.2296e-11, -3.4946e-11, -3.4955e-11, 1.9623e-11, 1.0073e-11

v8<

3.6072e-12, -2.4568e-11, -1.5751e-11, -3.0846e-11, 1.6619e-11,

3.8326e-12, -4.3756e-11, -1.5112e-11, -3.6366e-11, -4.6759e-11,

-3.9851e-12, -4.2025e-11, -4.6853e-11, 5.7986e-12, -1.3060e-11,

-1.8044e-11, -4.8727e-11, -1.3402e-11, -5.2458e-11, 3.5500e-11<

v9<

3.4152e-11, 1.9796e-13, -1.8623e-11, -6.0957e-12, -1.8231e-11,

8.0062e-12, -3.1473e-11, -3.3945e-11, -4.6245e-12, -6.5121e-11,

2.1267e-11, -6.1816e-11, -4.4908e-11, -3.6464e-11, -6.8556e-11,

-8.4006e-12, 3.5131e-11, 7.5097e-12, -1.9298e-12, -5.2567e-11

v10<

-7.3372e-12, -5.1978e-11, -5.9294e-11, -3.6417e-11, 7.0352e-12,

-5.4260e-11, 7.3736e-12, -1.9762e-11, -1.9702e-11, -3.6721e-11,

1.2579e-11, 2.0930e-11, -1.6339e-11, -4.5196e-11, 1.7302e-11,

-2.1683e-11, -4.6501e-11, -3.3896e-11, 1.3861e-11, 3.6826e-11

v11<

-2.1671e-11, 2.6281e-11, -4.4059e-11, -5.5314e-11, -6.0618e-11,

-2.6837e-11, 2.8266e-11, -1.0159e-13, -2.8502e-11, -1.4759e-11,

-4.7138e-11, -2.1686e-11, -5.6362e-11, 1.1950e-11, -5.4057e-11,

2.1691e-11, -3.2699e-11, -2.6841e-11, 1.7947e-11, 3.2094e-11

v12<

-3.8644e-11, -3.0147e-11, 1.8562e-11, 9.7555e-12, -6.7565e-11,

1.8022e-11, -5.6280e-11, -6.2838e-11, -1.1925e-11, -4.5918e-11,

-6.1549e-11, -3.4178e-11, -1.5473e-11, 1.4186e-11, 1.6251e-11, >

1.7996e-11, 6.4485e-12, 3.8989e-11, -1.5313e-11, -3.2938e-11

v13<

2.8998e-11, -6.3371e-11, -5.6138e-12, 1.9261e-11, 8.6597e-12, >

7.8634e-12, 1.9306e-11, 2.3791e-11, -5.5945e-11, -4.1649e-11,

-6.3419e-11, -5.6660e-11, -4.6916e-11, -8.8792e-12, -4.6316e-11,

1.3647e-11, -5.4048e-11, -4.3994e-12, -1.6580e-11, 2.3616e-11<

v14<

-3.8680e-11, 1.0649e-11, -3.7190e-11, 9.4983e-12, -1.3558e-11,

-4.0117e-11, -4.3240e-11, -1.5170e-11, 4.9719e-12, -5.4488e-11,

-4.8153e-11, 3.0341e-11, 2.1031e-11, -3.8548e-12, -4.9788e-11,

-2.9238e-11, 1.9030e-11, -4.0807e-11, 1.4871e-11, -2.5618e-11<

v15<

-2.6511e-11, 3.6062e-12, 2.7418e-12, -4.4909e-11, -9.4810e-12,

-4.1009e-11, -7.0201e-11, -3.8710e-11, -7.4558e-12, 1.5038e-11,

-7.5724e-12, 1.1361e-12, -5.7759e-12, 2.5418e-12, -7.5230e-12,

3.9666e-12, -2.8786e-11, 9.4653e-12, -4.7258e-11, -8.7460e-12

v16<

-2.9370e-11, 9.8231e-12, -3.1888e-11, 6.6332e-13, -3.8765e-12,

-5.7133e-11, -5.0269e-11, -5.2495e-11, -1.4857e-11, 1.2636e-11,

1.6733e-11, -5.6403e-11, -2.6690e-11, -1.4424e-11, 5.8470e-12,

-5.4664e-11, 3.5374e-11, 7.1321e-13, 1.8267e-11, -2.6575e-11

v17<

-3.1420e-11, -2.5432e-11, -4.2041e-12, -3.4631e-11, -1.6260e-11,

-3.4332e-11, 2.2769e-11, 5.1254e-12, 2.0256e-11, 2.5595e-11,

-2.2646e-11, -2.0170e-11, -1.3388e-11, -4.3867e-11, -4.9773e-11,

-4.8601e-11, -1.3496e-11, -3.8593e-11, -4.9818e-12, 9.7297e-12<

v18<

-5.6609e-12, -1.6920e-11, -1.4183e-12, -6.7014e-11, -4.8228e-11,

3.5297e-12, 1.6683e-11, 2.5034e-11, -2.1636e-11, 2.5913e-11,

9.1312e-12, -4.7396e-11, 2.2511e-11, 5.8173e-13, -7.8208e-12,

-3.1027e-11, -6.0199e-11, -1.7334e-11, -5.9261e-11, -2.0315e-11>

v19<

4.0531e-12, -5.0540e-11, 8.6964e-12, 1.7360e-11, -2.4594e-11,

-2.9914e-11, -4.9065e-11, -3.0844e-11, -2.9018e-12, -5.1454e-11,

-2.9870e-11, 2.8413e-11, -3.4810e-11, -3.8110e-11, 7.1807e-12,

5.8280e-12, -4.1314e-11, -1.3730e-12, -5.0551e-11, 1.8862e-11<

6. Number of iterations for convergence.

The convergence criteria defined in section D1 was reached after 10 iterations.

7. A plot of the energy as a function of iteration.

The energy function is given by: E = -1/2 { åi åx åj åy aix ajy wixjy }

I only showed the energy function for the first 200 iterations because my network converged very early (again based on the convergence criteria selected in 1D).

For Case 2, initial activations in the 10^-10 range, and N= 20

1. City positions.

city0 at 0.9785, 0.6582

city1 at 0.3009, 0.2213

city2 at 0.3835, 0.7523

city3 at 0.7291, 0.8414

city4 at 0.2581, 0.4346

city5 at 0.4865, 0.9485

city6 at 0.1658, 0.5352

city7 at 0.1804, 0.9600

city8 at 0.2171, 0.2859

city9 at 0.0660, 0.7274

city10 at 0.8053, 0.9515

city11 at 0.5761, 0.7838

city12 at 0.9963, 0.6768

city13 at 0.4041, 0.6005

city14 at 0.3086, 0.8276

city15 at 0.4623, 0.7483

city16 at 0.5192, 0.6259

city17 at 0.0117, 0.5886

city18 at 0.4287, 0.0708

city19 at 0.1398, 0.0112

The position of each city is shown in the graph below:

2. Tour that resulted.

The shortest tour resulted at iteration 12.

Visited Cities: 19 3 4 15 7 20 10 16 11 12 13 17 18 1 14 6 9 2 5 8

The tour started at city 19, moved to 3, moved to 4, so on, and ended at 8.

3. Drawing of the tour.

Looking on the graph, the fuzzy read out did not find an optimum solution based on the convergence criteria defined in section 1D.

4. Length of the tour.

The distances between each city of the tour depicted in 3 are as follows:

distance between cities 19 and 3: 0.6829095059821498

distance between cities 3 and 4: 0.3569464536555209

distance between cities 4 and 15: 0.42072371723885765

distance between cities 15 and 7: 0.3254523048456721

distance between cities 7 and 20: 0.524588489967892

distance between cities 20 and 10: 0.7199199024766738

distance between cities 10 and 16: 0.39684376112890585

distance between cities 16 and 11: 0.3986508010288919

distance between cities 11 and 12: 0.2839901453340677

distance between cities 12 and 13: 0.43353517684727

distance between cities 13 and 17: 0.4797354115107833

distance between cities 17 and 18: 0.5089401526517008

distance between cities 18 and 1: 0.9693596395085522

distance between cities 1 and 14: 0.57728434701347

distance between cities 14 and 6: 0.3575446722420205

distance between cities 6 and 9: 0.7152126666760608

distance between cities 9 and 2: 0.10576902706278865

distance between cities 2 and 5: 0.2174580167300065

distance between cities 5 and 8: 0.5311281259636907

The total length of the tour is 9.005992317864974

5. Final activations of the network.

I used the same notation as in CASE 1.

v0<

-3.4943e-11, -4.7956e-11, -4.0178e-11, -5.9744e-11, -6.7911e-11,

-6.2118e-11, 1.1693e-11, -3.9201e-11, -1.8415e-11, 1.8162e-11,

1.0666e-11, -9.8752e-12, -3.9794e-11, -2.8531e-12, -3.3535e-11,

-5.9131e-11, -1.1646e-11, -1.0498e-11, -4.3635e-12, -2.8287e-11>

v1<

8.0243e-12, 1.8407e-11, -4.1862e-11, -7.4636e-12, -3.4673e-12,

-4.2826e-12, -3.9620e-11, -4.8328e-11, 1.2075e-11, -4.9393e-11,

9.9087e-12, -6.3385e-11, -3.8725e-11, -3.9443e-11, 2.7122e-11,

3.1166e-11, -5.3593e-11, -3.1689e-11, -1.9138e-11, 1.9065e-11<

v2<

5.5526e-11, 1.7888e-11, -1.5763e-11, 4.5041e-12, -1.3747e-11,

-5.5365e-11, -8.5622e-12, -2.7769e-11, 3.6538e-11, -5.1956e-11,

-4.7301e-11, -3.0433e-11, 2.5776e-11, -4.7123e-11, 3.0754e-11,

-4.4256e-11, 1.2539e-11, 1.8429e-11, -4.3558e-12, 1.9869e-11

v3<

4.7156e-11, 2.6246e-11, -4.5213e-11, 6.7805e-13, -5.9383e-11,

-1.7010e-11, 2.6853e-11, -1.6572e-11, -6.5936e-12, -3.8174e-11,

-5.7926e-11, 1.0779e-11, 1.3586e-11, -6.0989e-11, -3.3974e-11,

-2.4184e-11, 1.4666e-11, -3.5613e-11, -3.6170e-11, 1.7153e-11<

v4<

-4.0870e-11, 2.3530e-11, -2.5465e-11, 8.4159e-12, -1.8713e-11,

2.7119e-11, -4.4589e-11, 8.7170e-13, -3.3363e-11, 1.0999e-11,

-4.3545e-12, 2.1541e-11, -1.8356e-11, -5.8968e-11, -1.6946e-11,

4.2236e-11, 3.8652e-11, -3.9804e-11, -1.7366e-11, -2.6694e-11>

v5<

1.1365e-11, -3.3741e-11, -6.3276e-11, 1.7001e-11, 1.7675e-11, >

-3.3421e-11, 1.8066e-11, -1.3410e-11, 9.6341e-12, 2.2095e-11, >

-2.8298e-11, 2.6425e-11, -1.6822e-12, 2.8604e-11, -4.9805e-12,

-2.2378e-11, -3.7776e-11, -1.5985e-11, -6.7816e-11, -4.5837e-11>

v6<

4.1640e-12, 2.6904e-11, 3.6128e-11, 2.3230e-11, -3.7014e-11,

-1.5823e-11, -2.9937e-11, -5.1548e-11, -5.5232e-11, 2.3459e-11,

-2.2011e-11, -3.5770e-11, -2.4316e-11, -1.5907e-11, 5.9955e-12,

2.7491e-12, -2.3610e-11, 1.8017e-12, 3.3989e-11, -7.0020e-12

v7<

-3.6808e-11, 2.2912e-11, -2.4169e-11, -1.5353e-11, -2.2734e-11,

6.3179e-12, -2.3851e-11, -4.2850e-11, -3.7713e-11, -4.9676e-11,

-5.3798e-11, -8.9774e-12, -5.2546e-12, -5.9909e-11, 1.3142e-12,

-5.3483e-11, -1.3283e-11, 1.4090e-11, -2.8667e-11, 5.5285e-11

v8<

3.0635e-12, 6.6163e-12, -5.7093e-11, -1.0789e-11, 1.0076e-11,

-3.0376e-11, -3.6236e-11, -6.0522e-11, -3.7995e-11, 1.4538e-11,

-6.2115e-11, 1.4203e-11, -1.1474e-12, 2.0701e-11, 9.4683e-13, >

-3.8470e-11, -2.7609e-12, 3.1167e-12, 1.9140e-11, 9.8117e-12

v9<

-3.0071e-12, -2.0356e-11, -4.6783e-11, 3.2327e-11, -3.9547e-11,

-7.0777e-11, -2.9128e-11, 7.2525e-12, 9.9622e-12, -2.7857e-11,

-2.2896e-11, -1.4496e-11, -1.9668e-12, -4.2116e-11, -2.7902e-11,

-5.7001e-13, -4.0556e-11, 3.1952e-11, 6.7848e-12, 4.4031e-12

v10<

-4.7505e-11, -3.3921e-11, -3.1465e-12, -7.3632e-11, 3.4553e-12,

1.0852e-12, 2.0264e-11, 1.2370e-11, -4.6261e-11, -4.9211e-11,

-4.5381e-11, -1.9016e-11, -1.0310e-11, -3.6264e-11, 1.5646e-11,

6.2487e-12, -5.2413e-11, 1.8592e-12, 2.0193e-11, -2.0588e-11

v11<

-1.5163e-11, -5.6563e-12, -5.2666e-11, -6.0306e-11, 2.6822e-11,

1.4640e-11, 2.8923e-11, -1.3040e-11, 3.4573e-11, 2.0366e-11, >

1.1698e-11, 2.0268e-12, -1.8469e-11, 1.8503e-11, -5.8161e-11,

-7.2413e-12, -2.1508e-11, -2.4879e-11, 2.1763e-11, -3.3071e-11<

v12<

-9.9662e-12, -5.4282e-11, -1.3621e-11, -2.1139e-11, -5.9411e-11,

1.6377e-11, -3.3798e-11, -4.4630e-11, 1.7938e-11, -5.0617e-11,

-2.9846e-11, -6.2852e-11, -2.2201e-11, -2.9910e-11, -3.7044e-11,

-8.9835e-12, -3.9033e-11, -5.5767e-11, -5.0111e-11, -2.9609e-11

v13<

-3.5575e-11, -1.8309e-11, -1.7904e-11, -5.3178e-12, -5.1810e-11,

2.3022e-11, 2.6172e-11, 1.2668e-11, -4.7810e-11, -2.8964e-11,

4.5323e-11, -1.6493e-11, -5.3118e-11, 4.0001e-11, -7.6457e-12,

-3.5119e-11, -3.8687e-11, 5.4255e-12, 2.7959e-11, 2.7476e-11

v14<

2.7061e-11, 4.1770e-11, -2.9786e-11, 7.9266e-12, 1.2658e-11, >

1.1320e-11, -4.1368e-11, -2.7473e-11, 1.6753e-11, -3.3941e-11,

-1.3846e-11, 3.8386e-12, -2.0796e-11, -1.9284e-11, -4.0477e-11,

-4.3302e-11, 3.4572e-11, -5.4660e-11, -2.7514e-11, -2.1662e-11>

v15<

-2.3487e-11, 6.0981e-12, -1.4972e-11, -5.1735e-11, -4.1098e-11,

3.9234e-11, -2.6812e-11, 7.4185e-12, -3.5453e-11, -4.3197e-11,

3.3984e-11, -2.6250e-12, 3.0173e-11, 2.4980e-11, 2.2356e-12,

-4.6318e-12, -8.7458e-12, -1.5617e-11, -3.4368e-11, -3.2458e-12>

v16<

4.0778e-12, -5.5503e-11, 2.2744e-11, 2.1621e-11, 7.1984e-12,

1.8609e-11, -5.0715e-11, 2.1894e-11, -1.4645e-11, 3.1672e-11,

-3.9991e-11, -3.0792e-11, -4.1031e-11, 2.4572e-11, 1.1432e-11, >

-1.3075e-11, -3.6558e-11, -3.2678e-11, -4.7320e-12, -2.4165e-12>

v17<

-2.8256e-11, -1.0553e-11, -4.9815e-11, -1.9501e-11, -1.6443e-11,

-6.4040e-11, -5.0433e-11, 1.7354e-11, -1.6345e-11, 3.4979e-11,

-2.7371e-11, -4.4215e-11, 5.1776e-12, -1.2216e-11, -7.3364e-12,

-1.7165e-11, -3.8375e-12, -3.9810e-11, -5.0602e-12, -1.1796e-11>

v18<

4.3354e-11, -1.8856e-11, -2.7439e-11, -5.2764e-12, -2.8890e-11,

7.8375e-13, -2.6994e-11, -2.5095e-11, -2.2664e-11, 3.6607e-12,

-1.7405e-11, 1.4596e-12, -6.4007e-11, -7.6054e-11, -9.7142e-12,

-5.3510e-11, -2.9582e-11, -2.1125e-11, -5.6257e-11, 1.2072e-12<

v19<

-4.0055e-11, -2.6177e-11, 2.3222e-11, 1.7804e-11, -4.5886e-11,

-6.2253e-11, -1.1593e-11, -4.9610e-11, -3.8283e-11, -5.5423e-11,

-4.5100e-11, -8.0029e-12, -5.3734e-11, -1.6457e-11, -6.7948e-11,

-3.3429e-11, -5.4524e-12, 7.8437e-12, -7.3913e-11, 4.6508e-12

6. Number of iterations for convergence.

The convergence criteria defined in section D1 was reached after 13 iterations.

7. A plot of the energy as a function of iteration.

CASE1 and CASE2 energy functions are very similar for the first 200 iterations of my network. Again, I only showed the energy function for the first 200 iterations because my network converged very early (based on the convergence criteria selected in 1D).

A. Convergence Issues:

1. What "convergence criteria" did you use? (How did your program know when to stop the simulation run?).

Initially, I adopted the convergence criteria described in the project, that is

If there is a unique neuron in each row and column with activation above the threshold, then halt the simulation and declare the corresponding permutation matrix to be the "output".

I set the threshold to axj = 0.7, and then checked for the existence of a permutation matrix.

The algorithm for the permutation matrix is as follows:

For each row

Check if one neuron has an activation value greater than 0.7

If there is one make sure that it is unique

For each column

Check if one neuron has an activation value greater than 0.7

If there is one make sure that it is unique

The algorithm of this convergence scheme is shown below:

For each iteration

If a permutation matrix exists

Display results of the shortest tour

End

Display the no convergence error message

The simulation would have stopped and outputted the results after detecting a permutation matrix of the activation values, or when it would have exceeded 10000 iterations, in which case the network would not have converged.

This convergence scheme did not work because the activation values were always close to 0.

As a result, I chose the following convergence criteria scheme instead:

For each iteration

Calculate the fuzzy tour as long as there are valid cms for each row

Record the shortest tour

End

Display results

2. It is assumed that you did more than a couple of simulation runs. Approximately what percent of the time did the network converge to an ambiguous state (no clear winner in each row and column)?

The network converged to an ambiguous state 100% of the time.

The center of mass calculations were defined for the first iterations because there were positive activation values in each row of the nxn matrix. However, these cms became undefined because the activation values became negative for all neurons after a few iterations. I was never able to reach the amount of iterations described in section 6 of “A Fuzzy Hopfield-Tank Traveling Salesman Problem Model”

The following table shows the number of iterations it took for the cms to become undefined based on the initial activations, number of cities, and the city positioning case.

Initial Activations	Cities	City Positioning	Iterations
10^-2	10	CASE 1	12
10^-5	10	CASE 1	9
10^-10	10	CASE 1	9
10^-20	10	CASE 1	9
10^-30	10	CASE 1	10
10^-2	20	CASE 1	22
10^-5	20	CASE 1	24
10^-10	20	CASE 1	22
10^-20	20	CASE 1	21
10^-30	20	CASE 1	22
10^-2	40	CASE 1	42
10^-5	40	CASE 1	46
10^-10	40	CASE 1	43
10^-20	40	CASE 1	60
10^-30	40	CASE 1	66
10^-2	10	CASE 2	9
10^-5	10	CASE 2	8
10^-10	10	CASE 2	10
10^-20	10	CASE 2	11
10^-30	10	CASE 2	10
10^-2	20	CASE 2	34
10^-5	20	CASE 2	22
10^-10	20	CASE 2	23
10^-20	20	CASE 2	28
10^-30	20	CASE 2	25
10^-2	40	CASE 2	42
10^-5	40	CASE 2	45
10^-10	40	CASE 2	45
10^-20	40	CASE 2	47
10^-30	40	CASE 2	45