Mathematical Model
Overview
This document presents the complete mathematical formulation of the TransComp optimization model. The model is designed as a linear programming problem that minimizes the total cost of providing transportation services while satisfying various technical, operational, and policy constraints.
The optimization framework considers multiple dimensions:
- Temporal: Multi-year planning horizon with dynamic fleet evolution
- Spatial: Origin-destination pairs connected by multiple routes
- Modal: Different transportation modes (road, rail, etc.)
- Technological: Various vehicle technologies and fuel types
- Economic: Investment costs, operational costs, and budget constraints
Mathematical Formulation
Nomenclature
Sets, Decision Variables and Parameters
The following tables define the mathematical notation used throughout the model formulation.
Sets
| Notation | Description | Unit |
|---|---|---|
| $y \in \mathcal{Y}$ | Planning years | - |
| $p \in \mathcal{P}$ | Product types (including passengers) | - |
| $m \in \mathcal{M}$ | Transportation modes | - |
| $r \in \mathcal{R}$ | Origin-destination pairs | - |
| $k \in \mathcal{K}$ | Routes between origin-destination pairs | - |
| $v \in \mathcal{V}$ | Vehicle types | - |
| $t \in \mathcal{T}$ | Drive-train technology and fuel combinations | - |
| $l \in \mathcal{L}_t$ | Fuel supply options for technology $t$ | - |
| $e \in \mathcal{E}$ | Geographic locations/edges | - |
| $ic \in \mathcal{IC}$ | Income classes | - |
| $g \in \mathcal{G}$ | Vehicle fleet generations | - |
| $\mathcal{V}_k$ | Sequence of edges for route $k$: $(e_1, e_2, e_3, \ldots, e_I)$ | - |
| $\mathcal{U}_k$ | Set of indexed edges: $\{(i, e_i) \mid e_i \in \mathcal{V}_k, 1 \leq i \leq I \}$ | - |
| $\mathcal{Y}_y$ | Years up to year $y$: $\{0, \ldots, y\}$ | - |
| $\mathcal{E}_{kmtg}$ | Edges within driving range of technology $t$ for route $k$ | - |
Decision Variables
| Notation | Description | Unit |
|---|---|---|
| $f_{yprkmvtg}$ | Transport volumes using technology $t$ on mode $m$, route $k$ in year $y$ | T |
| $h_{yprmtg}$ | Vehicle fleet size for mode $m$ with technology $t$ and generation $g$ | vehicles |
| $h^{+}_{yprmtg}$ | Vehicle fleet additions (investments) for mode $m$ with technology $t$ | vehicles |
| $h^{-}_{yprmtg}$ | Vehicle fleet reductions (retirements) for mode $m$ with technology $t$ | vehicles |
| $h^{exist}_{yprmtg}$ | Existing vehicle fleet at start of year $y$ | vehicles |
| $s_{ypkmvtle}$ | Fueling demand during peak hour at edge $e$ for technology $t$ | kWh |
| $s_{ypkmtln}$ | Fueling demand during peak hour at node $n$ for technology $t$ | kWh |
| $q^{+, mode\_infr}_{yme}$ | Installed mode infrastructure capacity for mode $m$ on edge $e$ in year $y$ | kW |
| $q^{+, fuel\_infr}_{yte}$ | Installed fueling infrastructure capacity for technology $t$ on edge $e$ in year $y$ | kW |
| $q^{+, supply\_infr}_{yle}$ | Installed supply infrastructure capacity for supply option $l$ on edge $e$ in year $y$ | kW |
Parameters
| Notation | Description | Unit |
|---|---|---|
| $LoS_{yktv}$ | Level of service (travel time) for route $k$ and technology $t$ | h |
| $F_{yrp}$ | Transport demand between O-D pair $r$ for product $p$ in year $y$ | T |
| $D^{spec}_{yvtg}$ | Specific energy consumption of technology $t$ and generation $g$ | kWh/km |
| $W_{yvtg}$ | Average payload capacity of vehicle with technology $t$ and generation $g$ | T |
| $L^a_{gmvt}$ | Maximum annual mileage of vehicle | km |
| $L_k$ | Length of route $k$ | km |
| $C^{CAPEX}_{yvtg}$ | Capital expenditure for vehicle technology $t$ and generation $g$ | € |
| $C^{fuel\_infr}_{yte}$ | Investment cost for fueling infrastructure | € |
| $C^{mode\_infr}_{yme}$ | Investment cost for mode infrastructure | € |
| $VoT_{ykvt,ic}$ | Value of time for income class $ic$ | €/h |
Objective Function
The model minimizes the total system cost, which includes infrastructure investments, vehicle costs, operational expenses, and penalty costs:
\[\min_{x} Z = C^{infrastructure} + C^{vehicle} + C^{transport} + C^{intangible} + C^{penalty}\]
Where each cost component is defined as follows:
Infrastructure Costs
Total costs for fueling, mode, and supply infrastructure investments and operations:
\[C^{infrastructure} = \sum_{t} \sum_{y} \left( \sum_{e} C^{fuel\_infr}_{yte} q^{+, fuel\_infr}_{yte} + \sum_{y' \in \mathcal{Y}^y} C_{yte}^{fuel\_infr, OM, fix}\left( Q^{fuel\_infr}_{te} + q^{+, fuel\_infr}_{y'te}\right) \right) \\ + \sum_{m} \sum_{y} \left( \sum_{e} C^{mode\_infr}_{yme} q^{+, mode\_infr}_{yme} + \sum_{y' \in \mathcal{Y}^y} C_{yme}^{mode\_infr, OM, fix}\left( Q^{mode\_infr}_{me} + q^{+, mode\_infr}_{y'me}\right) \right) \\ + \sum_{l} \sum_{y} \left( \sum_{e} C^{supply\_infr}_{yle} q^{+, supply\_infr}_{yle} + \sum_{y' \in \mathcal{Y}^y} C_{yle}^{supply\_infr, OM, fix}\left( Q^{supply\_infr}_{le} + q^{+, supply\_infr}_{y'le}\right) \right)\]
Vehicle Costs
Capital and operational costs for vehicle fleets, including fuel costs:
\[C^{vehicle} = \sum_y \sum_m \sum_v \sum_t \sum_g \left( C^{CAPEX}_{yvtg} h^{+}_{yprvtg} + C^{OM, fix}_{yvtg} h_{yprvtg} + \sum_l \sum_{e \in E^k} C^{fuel}_{yle} s_{ypkmvtle} \right)\]
Transport Activity Costs
Distance-based operational costs:
\[C^{transport} = \sum_y \left( C^{OM, fix, dist}_{mvt} f_{yprkmvtg} + \sum_k C^{OM, var, dist}_{mvt} L_k f_{yprkmvtg} \right)\]
Intangible Costs
User costs related to travel time and service quality:
\[C^{intangible} = \sum_y \sum_m \sum_r \sum_{kvt} VoT_{ykvt, ic} \cdot LoS^{f}_{ykvt} \cdot f_{yprkmvtg}\]
Where the level of service includes:
\[LoS^f_{yk} = \frac{L_k}{Speed_{yvmt}} + FuelingTime_{ykvmt} + WaitingTime_{ykm}\]
Penalty Costs
Costs for violating budget or other soft constraints:
\[C^{penalty} = \sum_y \sum_p \sum_r penalty^{budget}_{pry}\]
Constraints
The optimization model includes several categories of constraints that ensure feasible and realistic solutions:
Demand Coverage Constraint
Ensures that all transport demand is satisfied:
\[\sum_{kmvtg} f_{yprkmvtg} = F_{yrp} \quad \forall y \in \mathcal{Y}, r \in \mathcal{R}, p \in \mathcal{P}\]
This fundamental constraint guarantees that the total transport flow across all modes, routes, vehicles, technologies, and generations equals the exogenous demand for each origin-destination pair and product type.
Vehicle Stock Evolution Constraints
These constraints model the dynamic evolution of vehicle fleets over time, accounting for existing stock, new investments, and retirements.
Fleet Balance Equation
\[h_{yprmvtg} = h^{exist}_{yprmvt(g-1)} + h^{+}_{yprmvtg} - h^{-}_{yprmvtg} \quad \forall y \in \mathcal{Y}\setminus \{y_0\}, r, p, m, v, t, g\]
Initial Fleet Definition
\[h^{exist}_{yprmvt(g-1)} = h_{(y-1)prmvtg} \quad \forall y = y_0, r, p, m, v, t, g\]
These constraints ensure continuity in fleet evolution, where the fleet size in each year equals the previous year's fleet plus new additions minus retirements.
Fueling Demand Constraints
These constraints link transport activity to energy demand and ensure adequate fueling infrastructure.
Energy Demand Calculation
\[\sum_{l \in \mathcal{L}_t} \sum_{e \in \mathcal{E}_{k}} s_{ypkmvtle} = \sum_{g \in \mathcal{G}} \sum_{a \in \mathcal{A}^p} \sum_{e \in \mathcal{E}_{k}} \sum_{n \in \mathcal{N}_{k}} \gamma \frac{D^{spec}_{yt} L_{ke}}{W_{ymvt}} f_{ypakmvtg} \quad \forall y, p, k, m, t\]
This constraint calculates the fueling demand based on transport volumes, specific energy consumption, and route characteristics.
Vehicle Range Constraint
\[\sum_{l \in \mathcal{L}_t} \sum_{e \in \mathcal{U}_{ke}} s_{ypkmvtle} \leq \gamma \sum_{g \in \mathcal{G}} \frac{1}{W_{gmvt}} f_{ypkmvtg} \cdot Q^{tank}_{gmvt} \quad \forall y, p, k, m, t\]
This ensures that vehicles do not exceed their fuel tank capacity and range limitations.
Technology Shift Constraints
These constraints limit the speed of technological transitions to reflect realistic market dynamics and policy constraints.
Vehicle Stock Technology Shift
\[\pm \left( \sum_g h_{yprmvt} - \sum_g h_{(y-1)prmvt} \right) \leq \alpha \sum_{gvt} h_{yprmvt} + \beta \sum_g h_{(y-1)prmvtg} \quad \forall y \in \mathcal{Y}\setminus \{y_0\}, r, m, v, t\]
For the initial year:
\[\pm \left( \sum_g h_{yprmvtg} - \sum_g h^{exist}_{yprmvtg} \right) \leq \alpha \sum_{gvt} h_{yprmvt} + \beta \sum_g h^{exist}_{yprmvtg} \quad \forall y \in \{y_0\}, r, m, v, t\]
These constraints prevent unrealistic rapid shifts in vehicle technology adoption by limiting year-over-year changes.
Mode Shift Constraints
Similar to technology shift, these constraints limit the speed of modal transitions.
Transport Volume Mode Shift
\[\left( \sum_{kg} f_{yprkmvtg} - \sum_{kg} f_{(y-1)prkmvtg} \right) \leq \alpha F_{yrp} + \sum_{kg} f_{(y-1)prkmvtg} \quad \forall y \in \mathcal{Y}\setminus \{y_0\}, r, m\]
This constraint ensures that mode share changes occur gradually, reflecting user behavior and infrastructure limitations.
Infrastructure Expansion Constraints
These constraints ensure adequate infrastructure capacity to support transport activity.
Mode Infrastructure Capacity
\[Q^{mode\_infr}_{me} + \sum_{y \in \mathcal{Y}_y} q^{+, mode\_infr}_{yme} \geq \gamma \sum_{e \in \mathcal{K}_e} \sum_{p \in \mathcal{P}} \sum_{t \in T_{m}} f_{ypkmtg} \quad \forall y, m, e\]
Fueling Infrastructure Capacity
\[Q^{fuel\_infr}_{te} + \sum_{y \in \mathcal{Y}_y} q^{+, fuel\_infr}_{yte} \geq \sum_{k \in \mathcal{K}_e} \sum_{p \in \mathcal{P}} \sum_{m \in \mathcal{M}} \sum_{l \in \mathcal{L}_t} s_{ypkmtle} \quad \forall y, t, e\]
Supply Infrastructure Capacity
\[Q^{supply\_infr}_{le} + \sum_{y \in \mathcal{Y}_y} q^{+, supply\_infr}_{yle} \geq \sum_{e \in \mathcal{K}_e} \sum_{p \in \mathcal{P}} \sum_{m \in M} \sum_{t \in \mathcal{T}_l} s_{ypkmtle} \quad \forall y, l, e\]
These constraints ensure that infrastructure capacity (existing plus new investments) meets the demand generated by transport activities.
Monetary Budget Constraints
These constraints model financial limitations on investments, either as hard constraints or with penalty functions.
Total Budget Constraint (with penalties)
\[\sum_y C^{CAPEX}_{yvtg} \cdot h^{+}_{yr} \leq Budget_{ic} \cdot f \cdot |Y| + penalty^{+, budget}\]
\[\sum_y C^{CAPEX}_{yvtg} \cdot h^{+}_{yr} \geq Budget_{ic} \cdot f \cdot |Y| - penalty^{-, budget}\]
Periodic Budget Constraints
\[\sum_{y'\in Y_i} C^{CAPEX}_{y'vtg} \cdot h^{+}_{y'r} \leq Budget_{ic} \cdot f \cdot \tau^i + penalty^{+, budget}\]
\[\sum_{y'\in Y_i} C^{CAPEX}_{y'vtg} \cdot h^{+}_{y'r} \geq Budget_{ic} \cdot f \cdot \tau^i - penalty^{-, budget}\]
These constraints can enforce budget limits either as hard constraints (when penalties are prohibitively expensive) or as soft constraints (allowing violations with associated costs), enabling flexible policy modeling.