christoph f. eick: using ec to solve transportation problems on initialization and mutation 1.values...

Christoph F. Eick: Using EC to Solve Transportation Problems

On Initialization and Mutation

1. Values tij have to be between 0 and min(source(i),distination(j))

2. I can compute delta_tij=min(Source(i)Sumjtij , destination(j)Sumitij), which is how much tij has to be increased/decreased to satisfy the constraints

3. Simple Initialization Procedure: Set tij to 0 initially, and visit tij in some random order and assign the maximum allowed value (delta_tij) to it---[[but tij is larger than min(source(i),distination(j)) (or less than 0) it is set to min(source(i),distination(j)) (set to 0)]]

4. Complex initialization procedure: Use multiple loops and assign a percentage of the maximum increase to tij, and set it to tij+delta_tij in the last iteration.

5. Reverse initialization procedure: assign min(source(i),destination(j) to tij and reduce it…

6. Mutation: Change tij by adding/subtracting a number from it, not violating condition 1, and reuse the simple (complex??) initialization procedure: negative delta_tij values result in a reduction of the amount transported.

5 510

11 9

1 9 14 1 4


Ideas for the Transportation Problem

If M1 and M2 are legal solutions, a*M1 + b*M2 (with a,b>0 a+b=1) are also legal solutions. This provides as with a quite natural crossover operator. This operator is called arithmetical crossover in the EC numerical optimization literature.

Boundary Mutation (that sets the value of one (possibly more) elements of the matrix to its minimum (0) or maximum possible value (min(source(i), dest(j))), might also have some merit.


Some Initial Thoughts on the Course Project

Have a general theme Compare at least 2 approaches (could be similar or the

second approach could be kind of trivial) Run algorithms at least 3 times (you might just be unlucky) Report the results of running the benchmark transparently

and completely Interpret your results (even if there is no clear evidence

pointing into one direct); explain your results (could be speculative)

Report the history of the project. What was expected, what was unexpected?


Conducting Experiments in General and in the Context of the Transportation Problem

Things to observe when running an EC-system Average fitness Best solution found so far Diversity in the current population (expensive) Degree of change from generation to generation Visualizing the current best solutions could be helpful Size of searched solutions; building blocks in the searched solutions Complexity: runtime, storage, number of genetic operators applied,… What parts of the search space are searched (hard to analyze)

Things to report when summarizing experiments Experimental Environment: Operators used and probabilities of their

application, selection method, population size, best found solution, best average fitness.

Observed Results: Best solution found, best fitness/average fitness over time, diversity over time.


Reverse Initialization Algorithm

Let row(I) the sum of elements in the I-th row

Let col(j) the sum of the elements in the j-th column

Let sour(I) the sum of the supplies of the I-th row

Let dest(j) the demand for the j-th colums

Let dest(j)=<col(j) and source(I)=<row(I) for I,j=…

Visit the matrix elements (possibly excluding some elements) in some randomly selected order and do the following with the visited element vij with its current value v: maxred= min(col(j)-dist(j), row(i)-sour(i)) r= min(v, maxred) vij=vij-r; row(i)=row(i)-r; col(j)=col(j);

2001 Material


Boundary Mutation

2 2 0 0 0 2 6

1 1 22 4 0 2 0 6

1 0 0 3 2 0 6

0 1 22 0 1 0 2 6

Boundary Mutation:

1. Selection an element of the matrix

2. Set it to its maximum possible value (4 in the example)

3. Rerun a reverse initialization algorithm (the normal initialization algorithm) that reduces the elements of a matrix until the source and destination amounts are correct.

2001 Material

christoph f. eick: using ec to solve transportation problems on initialization and mutation 1.values...

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