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 combined with income class, and labour type l.
• r : housing zone (aka region), as defined by LMS, Municipality, or Corop.
• b : building type
• t : time-period. S_t = 0ⁱ indicates a state S at the start of the first time period.
• c : conversion option (aka construction option)

definitions

• Geography:
  • I_rⁱ: Incidence (membership) of building i to region r.

• Designer's input:
  • $\sideset{^c}{_b}{CS}$: Amount of construction of buildings (#residences for housing and #m² for labour related buildings) of type b for conversion option c per unit (ha) of land.
  • $\sideset{^c}{}{CE}_{b}$: Expenses (aka costs) of constructing one unit of b.
  • $\sideset{^c}{}{CE}$: Extra expenses (aka costs) for conversion c.
  • $\sideset{^c}{}{CB} := \sideset{^c}{}{CE} + \sum\limits_b \sideset{^c}{_b}{CE} \cdot \sideset{^c}{_b}{CS}$ : Cost of a full conversion of type c.

• household preferences input:
  • V^a: max price for actor type a.
  • $\sideset{^{b}}{^x_i}W$: WOZ value.
  • R_a**b: value-discount that actor a has for building type b, such as large household that don't fit into small houses.

• Input for time-period 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 of option c
  • $\sideset{^b_t}{i}N := \sum\limits{c} \sideset{^c_t}{_i}N \cdot {CS}^c_b$ : New Construction of building type b.
  • $\sideset{^{ab}_t}{_i}N$ : Allocation of new buildings
  • $\sideset{^c_t}{_i}C := \sideset{^c}{}{CB} + \sideset{^b_t}{^A_i}v \cdot \sideset{^b_t}{_i}S$ : Cost of a full conversion at i including cost of expropriation.
  • $\sideset{^c_t}{_i}Y := \sum\limits_b \sideset{^{b}_t}{^N_i}v \cdot \sideset{^c}{_b}{CS} - \sideset{^c_t}{_i}C$ : Yield of a full conversion at i.
  • $\sideset{^{ab}_t}{^A_i}v := \min \left( \sideset{^{b}}{^A_i}W + \sideset{^a_t}{r}\lambda + R{ab}, V^a \right)$ : value for a of a b at existing site i.
  • $\sideset{^{ab}_t}{^N_i}v :=$ : value of a b for a at new site i.
  • $\forall a: \sideset{^{b}_t}{^A_i}v \ge \sideset{^{ab}_t}{^A_i}v \space \bot \space \sideset{^{ab}_t}{_i}A \ge 0$; smoothed by $\sideset{^{b}_t}{^A_i}v := \log(\sum\limits_a \exp(\beta_3 \cdot \sideset{^{ab}_t}{^A_i}v )) \cdot \beta_3^{-1}$.
  • $\forall a: \sideset{^{b}_t}{^N_i}v \ge \sideset{^{ab}_t}{^N_i}v \space \bot \space \sideset{^{ab}_t}{_i}N \ge 0$; smoothed by $\sideset{^{b}_t}{^A_i}v:= \log(\sum\limits_a \exp(\beta_4 \cdot \sideset{^{ab}_t}{^N_i}v )) \cdot \beta_4^{-1}$.
  • building specific WOZ-value constraint: $\sideset{^{b}_t}{^x_i}v \le \sideset{^{b}}{^x_i}W$; this is sort of met when $\sideset{^a_t}{_r}\lambda \le 0$ and R_a**b ≤ 0.

scenario specification

• r: Corop region
• a: 13 Different Tigris Houshold types with varying count, job status, and age group(s) + 3 additional types, to be discussed.
• a: Clusters of observed recent (modus building year > 2000) representative building projects, including two extreme alternatives.

Conversion Options:

• C**S_b^c: Cluster Analyse Nederlandse Woningvoorraad, Jip Claassens (Juli 2018).
• C**B^c: obv Bouwkompas.

Input:

• $\sideset{^a_t}{_r}D$: Tigris
• $\sideset{^b_t}{_i}S$: Ruimtescanner
• V^a: Maximum price payable by actor a.

equilibrium equation

At each 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:

\( \sideset{^a_t}{_r}C := \sideset{^a_t}{_r}D - I^i_r \cdot \sum\limits_b \left( \sideset{^{ab}_t}{_i}A ( 1 - X_i ) + \sideset{^{ab}_t}{_i}N \right) \bot \sideset{^a_t}{_r}\lambda \le 0 \)

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

TODO: toevoegen ortho budget beschikbaarheid

• Allocation meets existing stock: $\forall t,b,i: \sum\limits_a \sideset{^{ab}_t}{_i}A \le \sideset{^b_t}{_i}S$: controlled by $\sideset{^b_t}{^A_i}\lambda$
• Allocation meets new stock: $\forall t,b,i: \sum\limits_{a} \sideset{^{ab}_t}{^N_i}N \le \sideset{^b_t}{_i}N$,
• Maximum conversion per site: $\forall t,i: \sum\limits_c \sideset{^{c}_t}{_i}N \le 1$: 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

Allocation of existing and new buildings:

$\sideset{^{ab}_t}{_i}A := { \exp ( \sideset{^{ab}_t}{^A_i}v - \sideset{^{b}_t}{^A_i}v ) }$

$\sideset{^{ab}_t}{_i}N := { \exp ( \sideset{^{ab}_t}{^N_i}v - \sideset{^{b}_t}{^N_i}v ) }$

Construction:

$\sideset{^c_t}{_i}N := \exp \left( \beta_2 \cdot \sideset{^c_t}{_i}Y \right) \cdot \sideset{_t}{^N_i}\lambda$ with $\sideset{_t}{^N_i}\lambda := \left( 1 + \sum\limits_c \exp \left( \beta_2 \cdot \sideset{^c_t}{_i}Y \right) \right)^{-1}$ $\sideset{_t}{_i}X := 1 - \left( \sideset{_t}{^N_i}\lambda \right)^{-1} = \sum\limits_c \sideset{^c_t}{_i}N$ The amount of conversion at site i $\sideset{^{ab}_t}{_i}A := \sideset{^b_t}{^A_i}S \cdot {{\sideset{^{ab}_t}{_i}v \cdot I^r_i \cdot \sideset{^a_t}{r}\lambda} \over {\sum\limits{a'}\sideset{^{a'b}_t}{_i}v \cdot I^r_i \cdot \sideset{^{a'}_t}{_r}\lambda}}$ with $\sideset{^b_t}{^A_i}\lambda := {\sideset{^b_t}{_i}S / {\sum\limits_a \sideset{^a_t}{_r}\lambda \cdot I^r_i \cdot \sideset{^{ab}_t}{_i}v}}$ $\sideset{^{ab}_t}{_i}N := {\sideset{^{ab}t}{^N_i}v \over {\sum\limits{a'} \sideset{^{a'b}_t}{^N_i}v}} \cdot {CS}^c_b \cdot \sideset{^c_t}{_i}N$ $\sideset{^{ab}_t}{^A_i}v := {woz}_i \cdot \sideset{^{ab}_t}{^A_i}\lambda I^r_i \cdot \sideset{^a_t}{_R}\lambda$