Ruimtescanner L in formules

Mathematical description of the Ruimtescanner L, which is a simplicifation of Ruimtescanner XL

See also: Transformation Potential II.

= Indices =
 * \( i \) : building and site index
 * \( a \) : actor type, a union of household type \( h \), a classification of socio-economic characteristics, and labour type \( l \).
 * \( r \) : housing zone (aka region), as defined by LMS or Municipality.
 * \( b \) : building type
 * \( t \) : time-period. \(S^i_{t=0}\) indicates a state \(S\) at the start of the first time period.
 * \( c \) : conversion option (aka construction option)

= Definitions =
 * Geography:
 * \( I^i_r \): Incidence (membership) of building \( i \) to region \( r \).


 * Designer's input:
 * \( {CS}^{c}_{b} \): Amount of construction of buildings (#residences for housing and #m\(^2\) for labour related buildings) of type \( b \) for conversion option \( c \) per unit (ha) of land.
 * \( {CE}^{c}_{b} \): Expenses (aka costs) of constructing one unit of \( b \).
 * \( {CE}^{c} \): Extra expenses (aka costs) for conversion \(c\).
 * \( \sideset{^c}{_i}{CB} := CE^c + \sum_b CE^{c}_{b} \cdot {CS}^{c}_{b}\) : Cost of a full conversion at site \(i\).


 * Input for stage t:
 * \( \sideset{^a_t}{_r}D \) : Demand of actor \(a\) in region \(r\).
 * \( \sideset{^b_t}{_i}S \) : Stock of accommodation of type \(b\) at site \(i\).


 * Dynamic state variables
 * \( \sideset{^{ab}_t}{_i}A \) : Allocation of existing buildings
 * \( \sideset{^c_t}{_i}N \) : New Construction
 * \( \sideset{^c_t}{_i}C := \sideset{^c}{_i}{CB} + \sideset{^b_t}{^A_i}v \cdot \sideset{^b_t}{_i}S \) : Cost of a full conversion at \(i\) including cost of exproriation
 * \( \sideset{^{ab}_t}{^A_i}v \) : value of a \(b\) for \(a\) at existing site \(i\).
 * \( \sideset{^{ab}_t}{^N_i}v \) : value of a \(b\) for \(a\) at new site \(i\).
 * \( \sideset{^{b}_t}{^A_i}v := \log(\sum_a \exp(\beta_3 \cdot \sideset{^{ab}_t}{^A_i}v ))\)
 * \( \sideset{^{b}_t}{^N_i}v := \log(\sum_a \exp(\beta_4 \cdot \sideset{^{ab}_t}{^N_i}v ))\)

= Equilibrium Equation = At each final state of a time period, we try to find those prices \( \sideset{^a_t}{_r}\lambda \) that reflect market equilibrium, i.e. that the (conversions of the) supply meet the demand.

Excess households are assumed to be located at regional campings, for which a Camping residue \(C\) is defined as: \( \sideset{^a_t}{_r}C := \sideset{^a_t}{_r}D - I^i_r \cdot \left( \sideset{^{ab}_t}{_i}A ( 1 - X_i ) + \sideset{^{ab}_t}{_i}N \right) \)

Vacant building stock will be represented as a negative Camping residue.

The control variable for making \(\sideset{^a_t}{_r}C = 0\), is therefore \( \sideset{^a_t}{_r}\lambda \)

= constraints =
 * Allocation meets existing stock: \( \forall t,b,i: \sum_a \sideset{^{ab}_t}{_i}A \le \sideset{^b_t}{_i}S \): controlled by \( \sideset{^b_t}{^A_i}\lambda \)
 * Maximum conversion per site: \( \forall t,i: \sum_c \sideset{^{c}_t}{_i}N \le 1\): controlled by \( \sideset{_t}{^N_i}\lambda\)
 * Conversion's financial feasibility: \( \forall t,c,i: \sideset{^c_t}{_i}N \ge 0 \perp \sideset{^c_t}{_i}C \ge \sideset{^{b}_t}{^N_i}v \times \sideset{^b_t}{_i}S \),

= smoothed-out version = \( \sideset{^c_t}{_i}N := \exp \left( \beta_2 \left( \sideset{^c_t}{_i}C - \sum_b \sideset{^{b}_t}{^N_i}v \cdot \sideset{^b_t}{_i}S \right) \right) \cdot \sideset{^b_t}{^N_i}\lambda \)

\( \sideset{_t}{_i}X := 1 - \sideset{_t}{^N_i}\lambda = \sum_c \sideset{^c_t}{_i}N \) The amount of conversion at site \(i\)

with \( \sideset{_t}{^N_i}\lambda := 1 / \left( 1 + \sum_c \exp \left( \beta_2 \left( \sideset{^c_t}{_i}C - \sum_b \sideset{^{b}_t}{^N_i}v \cdot \sideset{^b_t}{_i}S \right) \right) \right) \)

\( \sideset{^{ab}_t}{_i}A := \sideset{^b_t}{^A_i}S \cdot {{\sideset{^{ab}_t}{_i}v \cdot I^r_i \cdot \sideset{^a_t}{_r}\lambda} \over {\sum_{a'}\sideset{^{a'b}_t}{_i}v \cdot I^r_i \cdot \sideset{^{a'}_t}{_r}\lambda}} \)

with \( \sideset{^b_t}{^A_i}\lambda := {\sideset{^b_t}{_i}S / {\sum_a \sideset{^a_t}{_r}\lambda \cdot I^r_i \cdot \sideset{^{ab}_t}{_i}v}} \)

\( \sideset{^{ab}_t}{_i}N := {\sideset{^{ab}_t}{^N_i}v \over {\sum_{a'} \sideset{^{a'b}_t}{^N_i}v}} \cdot {CS}^c_b \cdot \sideset{^c_t}{_i}N\)

\( \sideset{^{ab}_t}{^A_i}v := {woz}_i \cdot \sideset{^{ab}_t}{^A_i}\lambda I^r_i \cdot \sideset{^a_t}{_R}\lambda \)