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

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

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