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

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

Used in pgr_pickDeliverEuclidean - Experimental

Column Type Description
Orders SQL TEXT Orders SQL as described below.
Vehicles SQL TEXT Vehicles SQL as described below.

Used in pgr_pickDeliver - Experimental

Column Type Description
Orders SQL TEXT Orders SQL as described below.
Vehicles SQL TEXT Vehicles SQL as described below.
Matrix SQL TEXT Matrix SQL as described below.

Pick-Deliver optional parameters

Column Type Default Description
factor NUMERIC 1 Travel time multiplier. See Factor handling
max_cycles INTEGER 10 Maximum number of cycles to perform on the optimization.
initial_sol INTEGER 4

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

Inner Queries

Orders SQL

Common columns for the orders SQL in both implementations:

Column Type 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.
[p_service] ANY-NUMERICAL

The duration of the loading at the pickup location.

  • When missing: 0 time units are used
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

The duration of the unloading at the delivery location.

  • When missing: 0 time units are used

Where:

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

For pgr_pickDeliver - Experimental the pickup and delivery identifiers of the locations are needed:

Column Type Description
p_node_id ANY-INTEGER The node identifier of the pickup, must match a vertex identifier in the Matrix SQL.
d_node_id ANY-INTEGER The node identifier of the delivery, must match a vertex identifier in the Matrix SQL.

Where:

ANY-INTEGER:SMALLINT, INTEGER, BIGINT

For pgr_pickDeliverEuclidean - Experimental the \((x, y)\) values of the 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

Where:

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

Vehicles SQL

Common columns for the vehicles SQL in both implementations:

Column Type Description
id ANY-NUMERICAL Identifier of the vehicle.
capacity ANY-NUMERICAL Maiximum capacity units
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

The duration of the loading at the starting location.

  • When missing: A duration of \(0\) time units is used.
[end_open] ANY-NUMERICAL

The time, relative to 0, when the ending location opens.

  • When missing: The value of start_open is used
[end_close] ANY-NUMERICAL

The time, relative to 0, when the ending location closes.

  • When missing: The value of start_close is used
[end_service] ANY-NUMERICAL

The duration of the loading at the ending location.

  • When missing: A duration in start_service is used.

For pgr_pickDeliver - Experimental the starting and ending identifiers of the locations are needed:

Column Type Description
start_node_id ANY-INTEGER The node identifier of the start location, must match a vertex identifier in the Matrix SQL.
[end_node_id] ANY-INTEGER

The node identifier of the end location, must match a vertex identifier in the Matrix SQL.

  • When missing: end_node_id is used.

Where:

ANY-INTEGER:SMALLINT, INTEGER, BIGINT

For pgr_pickDeliverEuclidean - Experimental the \((x, y)\) values of the locations are needed:

Column Type Description
start_x ANY-NUMERICAL \(x\) value of the starting location
start_y ANY-NUMERICAL \(y\) value of the starting location
[end_x] ANY-NUMERICAL

\(x\) value of the ending location

  • When missing: start_x is used.
[end_y] ANY-NUMERICAL

\(y\) value of the ending location

  • When missing: start_y is used.

Where:

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

Matrix SQL

Set of (start_vid, end_vid, agg_cost)

Column Type Description
start_vid BIGINT Identifier of the starting vertex.
end_vid BIGINT Identifier of the ending vertex.
agg_cost FLOAT Aggregate cost from start_vid to end_vid.

Return columns

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.

  • Value \(-2\) indicates it is the summary row.
vehicle_id BIGINT

Current vehicle identifier.

  • Sumary row has the total capacity violations.
    • A capacity violation happens when overloading or underloading a vehicle.
stop_seq INTEGER

Sequential value starting from 1 for the stops made by the current vehicle. The \(m_{th}\) stop of the current vehicle.

  • Sumary row has the total time windows violations.
    • A time window violation happens when arriving after the location has closed.
stop_type INTEGER
  • Kind of stop location the vehicle is at
    • \(-1\): at the solution summary row
    • \(1\): Starting location
    • \(2\): Pickup location
    • \(3\): Delivery location
    • \(6\): Ending location and indicates the vehicle’s summary row
order_id BIGINT

Pickup-Delivery order pair identifier.

  • Value \(-1\): When no order is involved on the current stop location.
cargo FLOAT

Cargo units of the vehicle when leaving the stop.

  • Value \(-1\) on solution summary row.
travel_time FLOAT

Travel time from previous stop_seq to current stop_seq.

  • Summary has the total traveling time:
    • The sum of all the travel_time.
arrival_time FLOAT

Time spent waiting for current location to open.

  • \(-1\): at the solution summary row.
  • \(0\): at the starting location.
wait_time FLOAT

Time spent waiting for current location to open.

  • Summary row has the total waiting time:
    • The sum of all the wait_time.
service_time FLOAT

Service duration at current location.

  • Summary row has the total service time:
    • The sum of all the service_time.
departure_time FLOAT
  • The time at which the vehicle departs from the stop.
    • \(arrival\_time + wait\_time + service\_time\).
  • The ending location has the total time used by the current vehicle.
  • Summary row has the total solution time:
    • \(total\ traveling\ time + total\ waiting\ time + total\ service\ time\).

Summary Row

Column Type Description
seq INTEGER Continues the sequence
vehicle_seq INTEGER Value \(-2\) indicates it is the summary row.
vehicle_id BIGINT

total capacity violations:

  • A capacity violation happens when overloading or underloading a vehicle.
stop_seq INTEGER

total time windows violations:

  • A time window violation happens when arriving after the location has closed.
stop_type INTEGER \(-1\)
order_id BIGINT \(-1\)
cargo FLOAT \(-1\)
travel_time FLOAT

total traveling 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

Summary row has the total solution time:

  • \(total\ traveling\ time + total\ waiting\ time + total\ service\ time\).

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

Showing how the 2 possible conversions can be done

Let: - \(f\_boxes\): number of boxes needed 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 pgr_pickDeliverEuclidean - Experimental:
    • The vehicles have \((x, y)\) pairs for start and ending locations.
    • The orders Have \((x, y)\) pairs for pickup and delivery locations.
  • When using pgr_pickDeliver - Experimental:
    • 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. All time units have to be converted to a 0 reference and the same time units.

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.

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

Factor handling

factor acts as a multiplier to convert from distance values to time units the matrix values or the euclidean values.

  • When the values are already in the desired time units
    • factor should be 1
    • When factor > 1 the travel times are faster
    • When factor < 1 the travel times are slower

For the pgr_pickDeliverEuclidean - Experimental:

Working with time units in seconds, and x/y in lat/lon: Factor: would depend on the location of the points and on the average velocity say 25m/s is the velocity.

Latitude Conversion Factor
45 1 longitude degree is (78846.81m)/(25m/s) 3153 s
0 1 longitude degree is (111319.46 m)/(25m/s) 4452 s

For the pgr_pickDeliver - Experimental:

Given \(v = d / t\) therefore \(t = d / v\) And the factor becomes \(1 / v\)

Where:

v:Velocity
d:Distance
t:Time

For the following equivalences \(10m/s \approx 600m/min \approx 36 km/hr\)

Working with time units in seconds and the matrix been in meters: For a 1000m lenght value on the matrix:

Units velocity Conversion Factor Result
seconds \(10 m/s\) \(\frac{1}{10m/s}\) \(0.1s/m\) \(1000m * 0.1s/m = 100s\)
minutes \(600 m/min\) \(\frac{1}{600m/min}\) \(0.0016min/m\) \(1000m * 0.0016min/m = 1.6min\)
Hours \(36 km/hr\) \(\frac{1}{36 km/hr}\) \(0.0277hr/km\) \(1km * 0.0277hr/km = 0.0277hr\)