# pgRouting Manual (3.0-rc1)

## Vehicle Routing Functions - Category (Experimental)

Warning

Possible server crash

- These functions might create a server crash

Warning

Experimental functions

- They are not officially of the current release.
- They likely will not be officially be part of the next release:
  - The functions might not make use of ANY-INTEGER and ANY-NUMERICAL
  - Name might change.
  - Signature might change.
  - Functionality might change.
  - pgTap tests might be missing.
  - Might need c/c++ coding.
  - May lack documentation.
  - Documentation if any might need to be rewritten.
  - Documentation examples might need to be automatically generated.
  - Might need a lot of feedback from the comunity.
  - Might depend on a proposed function of pgRouting
  - Might depend on a deprecated function of pgRouting

- Pickup and delivery problem
  - pgr_pickDeliver - Experimental - Pickup & Delivery using a Cost Matrix
  - pgr_pickDeliverEuclidean - Experimental - Pickup & Delivery with Euclidean distances
- Distribution problem
  - pgr_vrpOneDepot - Experimental - From a single depot, distributes orders

Contents

- Vehicle Routing Functions - Category (Experimental)
  - Introduction
    - Characteristics
  - Pick & Delivery
  - Parameters
    - Pick & deliver
  - Inner Queries
    - Pick & Deliver Orders SQL
    - Pick & Deliver Vehicles SQL
    - Pick & Deliver Matrix SQL
  - Results
    - Description of the result (TODO Disussion: Euclidean & Matrix)
    - Description of the result (TODO Disussion: Euclidean & Matrix)
  - Handling Parameters
    - Capacity and Demand Units Handling
    - Locations
    - Time Handling
    - Factor Handling
  - See Also

Previous versions of this page

- Supported versions: current(3.0)

## Introduction

Vehicle Routing Problems VRP are NP-hard optimization problem, it generalises the travelling salesman problem (TSP).

- The objective of the VRP is to minimize the total route cost.
- There are several variants of the VRP problem,

pgRouting does not try to implement all variants.

### Characteristics

- Capacitated Vehicle Routing Problem CVRP where The vehicles have limited carrying capacity of the goods.
- Vehicle Routing Problem with Time Windows VRPTW where the locations have time windows within which the vehicle’s visits must be made.
- Vehicle Routing Problem with Pickup and Delivery VRPPD where a number of goods need to be moved from certain pickup locations to other delivery locations.

Limitations

- No multiple time windows for a location.
- Less vehicle used is considered better.
- Less total duration is better.
- Less wait time is better.

## Pick & Delivery

Problem: CVRPPDTW Capacitated Pick and Delivery Vehicle Routing problem with Time Windows

- Times are relative to 0
- The vehicles
  - have start and ending service duration times.
  - have opening and closing times for the start and ending locations.
  - have a capacity.
- The orders
  - Have pick up and delivery locations.
  - Have opening and closing times for the pickup and delivery locations.
  - Have pickup and delivery duration service times.
  - have a demand request for moving goods from the pickup location to the delivery location.
- Time based calculations:
  - Travel time between customers is \(distance / speed\)
  - Pickup and delivery order pair is done by the same vehicle.
  - A pickup is done before the delivery.

## Parameters

### Pick & deliver

Both implementations use the following same parameters:

Column | Type | Default | Description
--- | --- | --- | ---
orders_sql | `TEXT` |  | Pick & Deliver Orders SQL query containing the orders to be processed.
vehicles_sql | `TEXT` |  | Pick & Deliver Vehicles SQL query containing the vehicles to be used.
factor | `NUMERIC` | 1 | (Optional) Travel time multiplier. See Factor Handling
max_cycles | `INTEGER` | 10 | (Optional) Maximum number of cycles to perform on the optimization.

initial_sol

`INTEGER`

4

(Optional) Initial solution to be used.

- `1` One order per truck
- `2` Push front order.
- `3` Push back order.
- `4` Optimize insert.
- `5` Push back order that allows more orders to be inserted at the back
- `6` Push front order that allows more orders to be inserted at the front

The non euclidean implementation, additionally has:

Column | Type | Description
--- | --- | ---
matrix_sql | `TEXT` | Pick & Deliver Matrix SQL query containing the distance or travel times.

## Inner Queries

- Pick & Deliver Orders SQL
- Pick & Deliver Vehicles SQL
- Pick & Deliver Matrix SQL

return columns

- Description of return columns
- Description of the return columns for Euclidean version

### Pick & Deliver Orders SQL

In general, the columns for the orders SQL is the same in both implementation of pick and delivery:

Column | Type | Default | Description
--- | --- | --- | ---
id | ANY-INTEGER |  | Identifier of the pick-delivery order pair.
demand | ANY-NUMERICAL |  | Number of units in the order
p_open | ANY-NUMERICAL |  | The time, relative to 0, when the pickup location opens.
p_close | ANY-NUMERICAL |  | The time, relative to 0, when the pickup location closes.
d_service | ANY-NUMERICAL | 0 | The duration of the loading at the pickup location.
d_open | ANY-NUMERICAL |  | The time, relative to 0, when the delivery location opens.
d_close | ANY-NUMERICAL |  | The time, relative to 0, when the delivery location closes.
d_service | ANY-NUMERICAL | 0 | The duration of the loading at the delivery location.

For the non euclidean implementation, the starting and ending identifiers are needed:

Column | Type | Description
--- | --- | ---
p_node_id | ANY-INTEGER | The node identifier of the pickup, must match a node identifier in the matrix table.
d_node_id | ANY-INTEGER | The node identifier of the delivery, must match a node identifier in the matrix table.

For the euclidean implementation, pick up and delivery \((x,y)\) locations are needed:

Column | Type | Description
--- | --- | ---
p_x | ANY-NUMERICAL | \(x\) value of the pick up location
p_y | ANY-NUMERICAL | \(y\) value of the pick up location
d_x | ANY-NUMERICAL | \(x\) value of the delivery location
d_y | ANY-NUMERICAL | \(y\) value of the delivery location

### Pick & Deliver Vehicles SQL

In general, the columns for the vehicles_sql is the same in both implementation of pick and delivery:

Column | Type | Default | Description
--- | --- | --- | ---
id | ANY-INTEGER |  | Identifier of the pick-delivery order pair.
capacity | ANY-NUMERICAL |  | Number of units in the order
speed | ANY-NUMERICAL | 1 | Average speed of the vehicle.
start_open | ANY-NUMERICAL |  | The time, relative to 0, when the starting location opens.
start_close | ANY-NUMERICAL |  | The time, relative to 0, when the starting location closes.
start_service | ANY-NUMERICAL | 0 | The duration of the loading at the starting location.
end_open | ANY-NUMERICAL | start_open | The time, relative to 0, when the ending location opens.
end_close | ANY-NUMERICAL | start_close | The time, relative to 0, when the ending location closes.
end_service | ANY-NUMERICAL | start_service | The duration of the loading at the ending location.

For the non euclidean implementation, the starting and ending identifiers are needed:

Column | Type | Default | Description
--- | --- | --- | ---
start_node_id | ANY-INTEGER |  | The node identifier of the starting location, must match a node identifier in the matrix table.
end_node_id | ANY-INTEGER | start_node_id | The node identifier of the ending location, must match a node identifier in the matrix table.

For the euclidean implementation, starting and ending \((x,y)\) locations are needed:

Column | Type | Default | Description
--- | --- | --- | ---
start_x | ANY-NUMERICAL |  | \(x\) value of the coordinate of the starting location.
start_y | ANY-NUMERICAL |  | \(y\) value of the coordinate of the starting location.
end_x | ANY-NUMERICAL | start_x | \(x\) value of the coordinate of the ending location.
end_y | ANY-NUMERICAL | start_y | \(y\) value of the coordinate of the ending location.

### Pick & Deliver Matrix SQL

Warning

TODO

## Results

### Description of the result (TODO Disussion: Euclidean & Matrix)

    RETURNS SET OF
        (seq, vehicle_seq, vehicle_id, stop_seq, stop_type,
            travel_time, arrival_time, wait_time, service_time,  departure_time)
        UNION
        (summary row)

Column | Type | Description
--- | --- | ---
seq | INTEGER | Sequential value starting from 1.
vehicle_seq | INTEGER | Sequential value starting from 1 for current vehicles. The \(n_{th}\) vehicle in the solution.
vehicle_id | BIGINT | Current vehicle identifier.
stop_seq | INTEGER | Sequential value starting from 1 for the stops made by the current vehicle. The \(m_{th}\) stop of the current vehicle.

stop_type

INTEGER

Kind of stop location the vehicle is at:

- `1`: Starting location
- `2`: Pickup location
- `3`: Delivery location
- `6`: Ending location

order_id

BIGINT

Pickup-Delivery order pair identifier.

- `-1`: When no order is involved on the current stop location.

cargo | FLOAT | Cargo units of the vehicle when leaving the stop.

travel_time

FLOAT

Travel time from previous `stop_seq` to current `stop_seq`.

- `0` When `stop_type = 1`

arrival_time | FLOAT | Previous `departure_time` plus current `travel_time`.
wait_time | FLOAT | Time spent waiting for current location to open.
service_time | FLOAT | Service time at current location.

departure_time

FLOAT

\(arrival\_time + wait\_time + service\_time\).

- When `stop_type = 6` has the total_time used for the current vehicle.

Summary Row

Warning

TODO: Review the summary

Column | Type | Description
--- | --- | ---
seq | INTEGER | Continues the Sequential value
vehicle_seq | INTEGER | `-2` to indicate is a summary row
vehicle_id | BIGINT | Total Capacity Violations in the solution.
stop_seq | INTEGER | Total Time Window Violations in the solution.
stop_type | INTEGER | `-1`
order_id | BIGINT | `-1`
cargo | FLOAT | `-1`
travel_time | FLOAT | total_travel_time The sum of all the travel_time
arrival_time | FLOAT | `-1`
wait_time | FLOAT | total_waiting_time The sum of all the wait_time
service_time | FLOAT | total_service_time The sum of all the service_time
departure_time | FLOAT | total_solution_time = \(total\_travel\_time + total\_wait\_time + total\_service\_time\).

### Description of the result (TODO Disussion: Euclidean & Matrix)

    RETURNS SET OF
        (seq, vehicle_seq, vehicle_id, stop_seq, stop_type,
            travel_time, arrival_time, wait_time, service_time,  departure_time)
        UNION
        (summary row)

Column | Type | Description
--- | --- | ---
seq | INTEGER | Sequential value starting from 1.
vehicle_seq | INTEGER | Sequential value starting from 1 for current vehicles. The \(n_{th}\) vehicle in the solution.
vehicle_id | BIGINT | Current vehicle identifier.
stop_seq | INTEGER | Sequential value starting from 1 for the stops made by the current vehicle. The \(m_{th}\) stop of the current vehicle.

stop_type

INTEGER

Kind of stop location the vehicle is at:

- `1`: Starting location
- `2`: Pickup location
- `3`: Delivery location
- `6`: Ending location

order_id

BIGINT

Pickup-Delivery order pair identifier.

- `-1`: When no order is involved on the current stop location.

cargo | FLOAT | Cargo units of the vehicle when leaving the stop.

travel_time

FLOAT

Travel time from previous `stop_seq` to current `stop_seq`.

- `0` When `stop_type = 1`

arrival_time | FLOAT | Previous `departure_time` plus current `travel_time`.
wait_time | FLOAT | Time spent waiting for current location to open.
service_time | FLOAT | Service time at current location.

departure_time

FLOAT

\(arrival\_time + wait\_time + service\_time\).

- When `stop_type = 6` has the total_time used for the current vehicle.

Summary Row

Warning

TODO: Review the summary

Column | Type | Description
--- | --- | ---
seq | INTEGER | Continues the Sequential value
vehicle_seq | INTEGER | `-2` to indicate is a summary row
vehicle_id | BIGINT | Total Capacity Violations in the solution.
stop_seq | INTEGER | Total Time Window Violations in the solution.
stop_type | INTEGER | `-1`
order_id | BIGINT | `-1`
cargo | FLOAT | `-1`
travel_time | FLOAT | total_travel_time The sum of all the travel_time
arrival_time | FLOAT | `-1`
wait_time | FLOAT | total_waiting_time The sum of all the wait_time
service_time | FLOAT | total_service_time The sum of all the service_time
departure_time | FLOAT | total_solution_time = \(total\_travel\_time + total\_wait\_time + total\_service\_time\).

Where:

ANY-INTEGER: | SMALLINT, INTEGER, BIGINT
ANY-NUMERICAL: | SMALLINT, INTEGER, BIGINT, REAL, FLOAT

## Handling Parameters

To define a problem, several considerations have to be done, to get consistent results. This section gives an insight of how parameters are to be considered.

- Capacity and Demand Units Handling
- Locations
- Time Handling
- Factor Handling

### Capacity and Demand Units Handling

The capacity of a vehicle, can be measured in:

- Volume units like \(m^3\).
- Area units like \(m^2\) (when no stacking is allowed).
- Weight units like \(kg\).
- Number of boxes that fit in the vehicle.
- Number of seats in the vehicle

The demand request of the pickup-deliver orders must use the same units as the units used in the vehicle’s capacity.

To handle problems like: 10 (equal dimension) boxes of apples and 5 kg of feathers that are to be transported (not packed in boxes).

If the vehicle’s capacity is measured by boxes, a conversion of kg of feathers to equivalent number of boxes is needed. If the vehicle’s capacity is measured by kg, a conversion of box of apples to equivalent number of kg is needed.

Showing how the 2 possible conversions can be done

Let: - \(f_boxes\): number of boxes that would be used for 1 kg of feathers. - \(a_weight\): weight of 1 box of apples.

Capacity Units | apples | feathers
--- | --- | ---
boxes | 10 | \(5 * f\_boxes\)
kg | \(10 * a\_weight\) | 5

### Locations

- When using the Euclidean signatures:
  - The vehicles have \((x, y)\) pairs for start and ending locations.
  - The orders Have \((x, y)\) pairs for pickup and delivery locations.
- When using a matrix:
  - The vehicles have identifiers for the start and ending locations.
  - The orders have identifiers for the pickup and delivery locations.
  - All the identifiers are indices to the given matrix.

### Time Handling

The times are relative to 0

Suppose that a vehicle’s driver starts the shift at 9:00 am and ends the shift at 4:30 pm and the service time duration is 10 minutes with 30 seconds.

All time units have to be converted

Meaning of 0 | time units | 9:00 am | 4:30 pm | 10 min 30 secs
--- | --- | --- | --- | ---
0:00 am | hours | 9 | 16.5 | \(10.5 / 60 = 0.175\)
9:00 am | hours | 0 | 7.5 | \(10.5 / 60 = 0.175\)
0:00 am | minutes | \(9*60 = 54\) | \(16.5*60 = 990\) | 10.5
9:00 am | minutes | 0 | \(7.5*60 = 540\) | 10.5

### Factor Handling

Warning

TODO

## See Also

- https://en.wikipedia.org/wiki/Vehicle_routing_problem
- The queries use the Sample Data network.

Indices and tables

- Index
- Search Page