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

\(36 km/hr\)

\(\frac{1}{36 km/hr}\)

\(0.0277hr/km\)

\(1km * 0.0277hr/km = 0.0277hr\)

See Also

Indices and tables