Re-optimize the assignment

In this example, we first solve the assignment problem, then we change one of the cost parameter, and re-optimize. See how easily this is done in PyMathProg.

Below is the output, note how the optimal assignment changed after increasing the cost, and how the second time of solving the problem seems much easier.

GLPK Simplex Optimizer, v4.60
16 rows, 64 columns, 128 non-zeros
      0: obj =   0.000000000e+00 inf =   8.000e+00 (8)
      8: obj =   1.750000000e+02 inf =   0.000e+00 (0)
*    24: obj =   7.600000000e+01 inf =   0.000e+00 (0)
OPTIMAL LP SOLUTION FOUND
Total Cost = 76
Agent 0 gets Task 0
Agent 1 gets Task 7
Agent 2 gets Task 6
Agent 3 gets Task 4
Agent 4 gets Task 1
Agent 5 gets Task 5
Agent 6 gets Task 3
Agent 7 gets Task 2
Set cost c[0, 0] to higher value 23
GLPK Simplex Optimizer, v4.60
16 rows, 64 columns, 128 non-zeros
*    25: obj =   7.800000000e+01 inf =   0.000e+00 (0)
OPTIMAL LP SOLUTION FOUND
Total Cost = 78
Agent 0 gets Task 7
Agent 1 gets Task 0
Agent 2 gets Task 6
Agent 3 gets Task 4
Agent 4 gets Task 1
Agent 5 gets Task 5
Agent 6 gets Task 3
Agent 7 gets Task 2

Interaction with queens

This example will randomly pick a board size and then randomly place some queens on the board. The rest of the queens are then placed by the LP program to see if all queens can be placed without attaching each other. If that's impossible, then the random positions are said to be bad. Note that in that LP, no objective function is needed.

The traveling sales man(TSP)

Really, the too well known problem.

Zebra: using string as index

A logic puzzle, see how strings are used as indices here to enhance the readability of the model.

Magic numbers

Magic numbers, a very interesting problem.

Sudoku: want some AI?

Do you play Sudoku? This program might help you crack a Sudoku in seconds ;).

Super Sudoku

This example will provide a sample Super Sudoku: in addition to satisfying all the requirements of Sudoku, Super Sudoku also requires that the elements in each diagonal must be distinct, below is a sample Super Sudoku obtained by the code provided afterwards:

+-------+-------+-------+
| 2 1 3 | 4 6 5 | 8 7 9 |
| 4 5 6 | 7 8 9 | 2 1 3 |
| 7 8 9 | 2 1 3 | 4 5 6 |
+-------+-------+-------+
| 1 2 4 | 3 5 6 | 7 9 8 |
| 3 6 8 | 9 7 2 | 5 4 1 |
| 9 7 5 | 8 4 1 | 3 6 2 |
+-------+-------+-------+
| 8 4 2 | 1 9 7 | 6 3 5 |
| 6 3 1 | 5 2 4 | 9 8 7 |
| 5 9 7 | 6 3 8 | 1 2 4 |
+-------+-------+-------+

Machine Scheduling

Schedule jobs on a single machine, given the duration, earliest start time, and the latest finish time of each job. Jobs can not be interrupted, and the machine can only handle one job at one time.

The output schedule is as follows (your run may not be the same, as the data is randomly generated -- you might even get infeasible problem instances):

Duration [7, 4, 1, 7]
Earliest [13, 17, 19, 12]
Latest [22, 41, 28, 38]
status: opt
schedule:
job 0: 13.0 20.0
job 1: 20.0 24.0
job 2: 27.0 28.0
job 3: 31.0 38.0

Scheduling for Revenue

The same scheduling problem as above, but add a new spin to the problem: when it is not feasible to accept all jobs, try accept those that will maximize the total revenue.