Random Placement Problem

Description
The random placement problem (RPP; for more details, see http://www.fi.muni.cz/~hanka/rpp) seeks to place a set of randomly generated rectangles (called objects) of different sizes into a larger rectangle (called placement area) in such a way that no objects overlap and all objects’ borders are parallel to the border of the placement area. In addition, a set of allowable placements can be randomly generated for each object. The ratio between the total area of all objects and the size of the placement area will be denoted as the filled area ratio.

RPP allows us to generate various instances of the problem similar to a trivial timetabling problem. The correspondence is as follows: the object corresponds to a course to be timetabled – the x-coordinate to its time, the y-coordinate to its classroom. For example, a course taking three hours corresponds to an object with dimensions 3x1 (the course should be taught in one classroom only). Each course can be placed only in a classroom of sufficient capacity – we can expect that the classrooms are ordered increasingly in their size so each object will have a lower bound on its y-coordinate.

MPP instances were generated as follows: First, the initial solution was computed. The changed problem differs from the initial problem by input perturbations. An input perturbation means that both x coordinate and y coordinate of a rectangle must differ from the initial values, i.e. x!=xinitial & y!=yinitial. For a single initial problem and for a given number of input perturbations, we can randomly generate various changed problems. In particular, for a given number of input perturbations, we randomly select a set of objects which should have input perturbations. The solution to MPP can be evaluated by the number of additional perturbations. They are given by subtraction of the final number of perturbations and the number of input perturbations.
Input Data Format
Each input problem (e.g., gen22.pl) has the following structure:
    objects([
      object(
        name( rect1 ),
        size( [ 2, 1 ] ),
        valid_positions( [ 0-38, 0-13 ] )
      ),
      object(
        name( rect2 ),
        size( [ 2, 1 ] ),
        valid_positions( [ 0-38, 0-13 ] )
      ),
    ...
      object(
        name( rect200 ),
        size( [ 2, 1 ] ),
        valid_positions( [ 0-38, 7-13 ] )
      )
    ] ).
Stating that the first rectangle (named rect1) has size 2x1 and its valid position are with x between 0 and 38, y between 0 and 13, and so on.
MPP instances contain an extra file with the solution (e.g., gen22.solution), with the following structure
    assigned([[rect1X,[17]], [rect1Y,[5]], [rect2X,[24]], [rect2Y,[4]], ... [rect200X,[37]], [rect200Y,[10]]]).
Which means that the first rectangle (named rect1) is to be placed at [17,5], second at [24,4] and so on.
There is also a file (e.g., gen22.mpp) describing which input placements are to be prohibited (not that if the input placement is prohibited, it means that also all values with the same X or Y coordinate are prohibited). It has the following structure:
    perturbation( 1, 0, [] ).
    perturbation( 2, 0, [] ).
    ...
    perturbation( 1, 2, [44,127] ).
    perturbation( 2, 2, [80,153] ).
    ...
    perturbation( 1, 4, [44,80,127,153] ).
    perturbation( 2, 4, [48,67,138,170] ).
    ...
Output Data Format
In the output folder, there are result.pl, stat.pl and stat.csv files. As for initial problem, result.pl has the following structure:
    result(1,1,0.156,200,248,
      unassigned(0/[]),
      assigned(400/[rect1X-36,rect1Y-8,rect2X-0,rect2Y-12,...])
    ).
For each solution, there is a term named result. It consists from:
  • problem number (e.g., 1 means gen1.pl)
  • test number (e.g., 1 means first run of that problem, up to RPP.NrTests
  • time needed to solve the problem
  • number of assigned variables (i.e., placed rectangles)
  • number of iterations
  • list of unassigned rectangles
  • list of assigned rectangles (i.e., rect2X-0 means that X-coordinate of 2nd recnangle (named rect2) is assigned to 0)
Next, the content of file stat.pl is:
    averages( problem( 1 ), time( 0.128 ), assigned( 200.000 ) ).
    deviations( problem( 1 ), time( 0.015 ), assigned( 0.000 ) ).
    averages( problem( 2 ), time( 0.162 ), assigned( 200.000 ) ).
    deviations( problem( 2 ), time( 0.021 ), assigned( 0.000 ) ).
For each problem there is an average number (term averages) and RMS (term deviations) of the time need to solve the problem (in seconds) and number of assigned variables.
Finally, the content of file stat.csv is:
    gen;time[s];timeRMS;assigned;assignedRMS;iters;itersRMS
    1;0.128;0.015;200.000;0.000;241.800;9.642
    2;0.162;0.021;200.000;0.000;363.800;59.057
    3;0.219;0.038;200.000;0.000;516.400;85.994
It is a table of problem number, average time (in seconds), RMS time, average number of assigned variables, RMS number of assigned variables, average number of iteratons and RMS number of iterations.

As for minimal perturbation problem (MPP), the content of above files is a bit different. File result.pl:
    result(1,0,0.078,200,260,
      unassigned(0/[]),
      perturbations(0/[])
    ).
    result(1,4,0.062,200,280,
      unassigned(0/[]),
      perturbations(10/[rect44X-3,rect44Y-11,rect80X-28,rect80Y-12,rect104X-5,rect104Y-11,rect127X-8,rect127Y-12,rect153X-33,rect153Y-11])
    ).
For each solution, there is a term named result. It consists from:
  • test number (e.g., 1 means first run of that problem, up to RPP.NrTests
  • number of input perturbations (it goes from 0 to 200, incremented by 4 in each step)
  • time needed to solve the problem
  • number of assigned variables (i.e., placed rectangles)
  • number of iterations
  • list of unassigned rectangles
  • list of perturbations rectangles (i.e., rectangles assigned to different than initial locations)
File stat.pl:
    averages( initperturbations( 0 ), time( 0.078 ), assigned( 200.000 ), perturbations( 0.000 ) ).
    deviations( initperturbations( 0 ), time( 0.000 ), assigned( 0.000 ), perturbations( 0.000 ) ).
    averages( initperturbations( 4 ), time( 0.062 ), assigned( 200.000 ), perturbations( 5.000 ) ).
    deviations( initperturbations( 4 ), time( 0.000 ), assigned( 0.000 ), perturbations( 0.000 ) ).
For each number of input perturbations there is an average number (term averages) and RMS (term deviations) of the time (in seconds), the number of assigned rectangles and the number of perturbations.
File stat.csv:
    pert;time[s];timeRMS;assigned;assignedRMS;perturbations;perturbationsRMS;iters;itersRMS
    0;0.078;0.000;200.000;0.000;0.000;0.000;260.000;0.000
    4;0.062;0.000;200.000;0.000;5.000;0.000;280.000;0.000
    8;0.078;0.000;200.000;0.000;9.000;0.000;408.000;0.000
For each number of input perturbations there is an average number (term averages) and RMS (term deviations) of the time (in seconds), the number of assigned rectangles, the number of perturbations and the number of iterations.
[back]