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AC dynamic in a 3-D spatial pattern

Spatial pattern is conceived as a cubic grid. Each cell of this grid takes 7 states: commerce (1), industry (2), housing (3), vacant (4), river (5), railway (6), and road (7).

A cell in a state that ranges 1-3 is considered as a built cell. In this case, the states are related to the activity that is carried out within each building. Three indexes identify a cell $c_{ijk}$. The first two indexes $i$ and $j$, relate to the position on the plane; the third one, $k$, represents height of the cell if the cell is built, or, otherwise stated, the floors number of the building as follows: $k=1$ represents the ground floor, $k=2$ the first floor of the building and so on. Only a cell with $k=1$ may take states ranging from 5-7.

Obviously a cell with $k>1$ cannot be built if the underlying cell is not already built. Building cost is related to each cell and changes with $k$. In addition for each activity, there is also a cost that is related to the floor level on which the activity occurs. In the latter case, if cost increases sensibly with the increasing of $k$, then built cells with $k>1$ will become rare.

The method for assigning states to cells is similar to that proposed by White, et al. (1997). Each cell is potentially capable of changing from state $p$ to a state $q$ with $1{\leq}p{\leq}4$ and $1{\leq}q{\leq}3$. This potential can be calculated using equation 1. States are assigned by beginning with the maximum potential and continuing until the global quantity of each state, exogenously established, has been reached. The potential ability of transformation from state $p$ to state $q$ in cell $c_{ijk}$ is calculated using the following equation:


\begin{displaymath}
P_{p,q}=
vs_{ijk}
(1+\sum_{r,d}m_{q,r,d}I_{d})
+H_{p}
-C_{k}
-F_{q,k},
\end{displaymath} (1)

where:

$P_{p,q}$: transition potential from state $p$ to state $q$ in cell $c_{ijk}$;

$m_{q,r,d}$: weight connected to the cells in state $r$ at distance $d$ from $c_{ijk}$ in relation to state $q$;

$I_{d}=1$ if he state of the cell distant $d$ from $c_{ijk}$ is equal to $r$, $I_{d}=0$ otherwise;

$H_q$: inertia parameter, $H_{q}>0$ if $q=p$, otherwise $H_{q}=0$;

$C_{k}$: building cost for a cell at floor $k$ in case $p=4$;

$F_{k,q}$: cost related to performing activity $q$ in floor $k$;

$s_{ijk}$: difficulty to build in relation to the slope of ground , $0<~s_{ijk}~<1$;

$v$: disturbance factor, $v=1+[-ln(rand)]^\alpha$.

           


next up previous contents
Next: Results of experiments Up: The dynamic of an Previous: Introduction   Contents
ferdinando semboloni
2000-11-06