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

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

return columns

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.

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

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