# 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

• Supported versions: current(3.1) 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

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

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

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

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.

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¶

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

Warning

TODO